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1 | %$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cfr0/c_cfr0.tex,v 1.18 2004/03/12 11:12:35 dtashley Exp $ | %$Header$ |
2 | ||
3 | \chapter{\ccfrzerolongtitle{}} | \chapter{\ccfrzerolongtitle{}} |
4 | ||
5 | \label{ccfr0} | \label{ccfr0} |
6 | ||
7 | \beginchapterquote{``I began by saying that there is probably less difference | \beginchapterquote{``I began by saying that there is probably less difference |
8 | between the positions of a mathematician and a physicist | between the positions of a mathematician and a physicist |
9 | than is generally supposed, and that the most important | than is generally supposed, and that the most important |
10 | seems to me to be this, that the mathematician is in much | seems to me to be this, that the mathematician is in much |
11 | more direct contact with reality \ldots{} mathematical | more direct contact with reality \ldots{} mathematical |
12 | objects are so much more what they seem. A chair or a | objects are so much more what they seem. A chair or a |
13 | star is not in the least what it seems to be; the more we think | star is not in the least what it seems to be; the more we think |
14 | of it, the fuzzier its outlines become in the haze of sensation | of it, the fuzzier its outlines become in the haze of sensation |
15 | which surround it; but `2' or `317' has nothing to do with | which surround it; but `2' or `317' has nothing to do with |
16 | sensation, and its properties stand out the more clearly the more | sensation, and its properties stand out the more clearly the more |
17 | closely we scrutinize it.''} | closely we scrutinize it.''} |
18 | {G. H. Hardy \cite{bibref:b:mathematiciansapology:1940}, pp. 128-130} | {G. H. Hardy \cite{bibref:b:mathematiciansapology:1940}, pp. 128-130} |
19 | \index{Hardy, G. H.} | \index{Hardy, G. H.} |
20 | ||
21 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
22 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
23 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
24 | \section{Introduction} | \section{Introduction} |
25 | %Section tag: INT0 | %Section tag: INT0 |
26 | \label{ccfr0:sint0} | \label{ccfr0:sint0} |
27 | \index{continued fraction} | \index{continued fraction} |
28 | \index{continued fraction!definition} | \index{continued fraction!definition} |
29 | ||
30 | A \emph{finite simple continued fraction} is a fraction of the form | A \emph{finite simple continued fraction} is a fraction of the form |
31 | ||
32 | \begin{equation} | \begin{equation} |
33 | \label{eq:ccfr0:int0:00} | \label{eq:ccfr0:int0:00} |
34 | a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 | a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 |
35 | + \cfrac{1}{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots + \cfrac{1}{a_n}}}} | + \cfrac{1}{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots + \cfrac{1}{a_n}}}} |
36 | = | = |
37 | [a_0; a_1, a_2, \ldots , a_n] , | [a_0; a_1, a_2, \ldots , a_n] , |
38 | \end{equation} | \end{equation} |
39 | ||
40 | \noindent{}where $a_0 \in \vworkintsetnonneg$ and | \noindent{}where $a_0 \in \vworkintsetnonneg$ and |
41 | $a_i \in \vworkintsetpos$, $i > 0$. Each integer | $a_i \in \vworkintsetpos$, $i > 0$. Each integer |
42 | $a_i$ is called an \index{continued fraction!element}\emph{element} or | $a_i$ is called an \index{continued fraction!element}\emph{element} or |
43 | \index{continued fraction!partial quotient}\emph{partial quotient} | \index{continued fraction!partial quotient}\emph{partial quotient} |
44 | of the continued fraction. | of the continued fraction. |
45 | We require, except in the case of | We require, except in the case of |
46 | the continued fraction representation of an integer, | the continued fraction representation of an integer, |
47 | that the final element $a_n$ not be equal | that the final element $a_n$ not be equal |
48 | to 1.\footnote{\label{footnote:ccfr0:sint0:00}The reason for | to 1.\footnote{\label{footnote:ccfr0:sint0:00}The reason for |
49 | this restriction is discussed later.} | this restriction is discussed later.} |
50 | ||
51 | Continued fractions are quite unwieldly to write and typeset, | Continued fractions are quite unwieldly to write and typeset, |
52 | and so a continued fraction in the form of (\ref{eq:ccfr0:int0:00}) | and so a continued fraction in the form of (\ref{eq:ccfr0:int0:00}) |
53 | is written as $[a_0; a_1, a_2, \ldots , a_n]$. Note that the | is written as $[a_0; a_1, a_2, \ldots , a_n]$. Note that the |
54 | separator between $a_0$ and $a_1$ is a semicolon (`;'), and that all other | separator between $a_0$ and $a_1$ is a semicolon (`;'), and that all other |
55 | separators are commas (`,'). In some works, commas are used exclusively; and in | separators are commas (`,'). In some works, commas are used exclusively; and in |
56 | other works, the first element is $a_1$ rather than $a_0$. Throughout this | other works, the first element is $a_1$ rather than $a_0$. Throughout this |
57 | work, the notational conventions illustrated in (\ref{eq:ccfr0:int0:00}) are | work, the notational conventions illustrated in (\ref{eq:ccfr0:int0:00}) are |
58 | followed. | followed. |
59 | ||
60 | In this chapter, the framework of continued fractions is presented in the | In this chapter, the framework of continued fractions is presented in the |
61 | context of finding rational numbers in $F_N$, the Farey series of order $N$, | context of finding rational numbers in $F_N$, the Farey series of order $N$, |
62 | enclosing an arbitrary $r_I \in \vworkrealsetnonneg$. The continued fraction | enclosing an arbitrary $r_I \in \vworkrealsetnonneg$. The continued fraction |
63 | algorithm presented (Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn}) | algorithm presented (Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn}) |
64 | is $O(log N)$, and so is suitable for finding the best rational | is $O(log N)$, and so is suitable for finding the best rational |
65 | approximations in $F_N$ even when $N$ is very large. Because our emphasis | approximations in $F_N$ even when $N$ is very large. Because our emphasis |
66 | is on practical applications rather than number theory, we don't include more | is on practical applications rather than number theory, we don't include more |
67 | information than is necessary to understand the applications we have in | information than is necessary to understand the applications we have in |
68 | mind. | mind. |
69 | ||
70 | The study of continued | The study of continued |
71 | fractions is a topic from number theory (a branch of mathematics). It may be | fractions is a topic from number theory (a branch of mathematics). It may be |
72 | counterintuitive to anyone but a number theorist that continued fractions | counterintuitive to anyone but a number theorist that continued fractions |
73 | can be used to economically find best rational approximations, | can be used to economically find best rational approximations, |
74 | or that continued fractions are anything but | or that continued fractions are anything but |
75 | a parlor curiosity. C.D. Olds (\cite{bibref:b:OldsClassic}, p. 3) comments: | a parlor curiosity. C.D. Olds (\cite{bibref:b:OldsClassic}, p. 3) comments: |
76 | ||
77 | \index{Olds, C. D.} | \index{Olds, C. D.} |
78 | ||
79 | \begin{quote} | \begin{quote} |
80 | At first glance, nothing seems simpler or less significant than writing a number, | At first glance, nothing seems simpler or less significant than writing a number, |
81 | for example $\frac{9}{7}$, in the form | for example $\frac{9}{7}$, in the form |
82 | ||
83 | \begin{equation} | \begin{equation} |
84 | \frac{9}{7} = 1 + \frac{2}{7} = 1 + \frac{1}{\cfrac{7}{2}} | \frac{9}{7} = 1 + \frac{2}{7} = 1 + \frac{1}{\cfrac{7}{2}} |
85 | = 1 + \cfrac{1}{3 + \cfrac{1}{2}} | = 1 + \cfrac{1}{3 + \cfrac{1}{2}} |
86 | = 1 + \cfrac{1}{3 + \cfrac{1}{1 + \cfrac{1}{1}}}. | = 1 + \cfrac{1}{3 + \cfrac{1}{1 + \cfrac{1}{1}}}. |
87 | \end{equation} | \end{equation} |
88 | ||
89 | It turns out, however, that fractions of this form, called ``continued | It turns out, however, that fractions of this form, called ``continued |
90 | fractions'', provide much insight into mathematical problems, particularly into | fractions'', provide much insight into mathematical problems, particularly into |
91 | the nature of numbers. | the nature of numbers. |
92 | ||
93 | Continued fractions were studied by the great mathematicians of the seventeenth | Continued fractions were studied by the great mathematicians of the seventeenth |
94 | and eighteenth centuries and are a subject of active investigation today. | and eighteenth centuries and are a subject of active investigation today. |
95 | \end{quote} | \end{quote} |
96 | ||
97 | ||
98 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
99 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
100 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
101 | \section{History Of Continued Fractions} | \section{History Of Continued Fractions} |
102 | %Section tag: HST0 | %Section tag: HST0 |
103 | \label{cfr0:hst0} | \label{cfr0:hst0} |
104 | \index{continued fraction!history of} | \index{continued fraction!history of} |
105 | ||
106 | The only work we are aware of that explicitly treats the history | The only work we are aware of that explicitly treats the history |
107 | of continued fractions is \cite{bibref:b:HistoryCfPadeApproxBrezinski}. | of continued fractions is \cite{bibref:b:HistoryCfPadeApproxBrezinski}. |
108 | Although the history of continued fractions is complex, | Although the history of continued fractions is complex, |
109 | two points are clear. First, it is clear that Euclid's \index{Euclid} | two points are clear. First, it is clear that Euclid's \index{Euclid} |
110 | GCD algorithm \index{Euclid!GCD algorithm} | GCD algorithm \index{Euclid!GCD algorithm} |
111 | (Algorithm \ref{alg:ccfr0:sega0:euclidsgcdalgorithm}), | (Algorithm \ref{alg:ccfr0:sega0:euclidsgcdalgorithm}), |
112 | which was known no later than around 300 B.C., | which was known no later than around 300 B.C., |
113 | represents the historical origin of the continued fraction. Second, | represents the historical origin of the continued fraction. Second, |
114 | it is clear that the utility of the apparatus of continued fractions | it is clear that the utility of the apparatus of continued fractions |
115 | in finding best rational approximations---specifically the properties | in finding best rational approximations---specifically the properties |
116 | of convergents---was understood by the 17th century. | of convergents---was understood by the 17th century. |
117 | ||
118 | In this section, we present some excerpts from | In this section, we present some excerpts from |
119 | \cite{bibref:b:HistoryCfPadeApproxBrezinski} | \cite{bibref:b:HistoryCfPadeApproxBrezinski} |
120 | which show the very early use of continued fractions to obtain best rational | which show the very early use of continued fractions to obtain best rational |
121 | approximations with a numerator and denominator less than certain | approximations with a numerator and denominator less than certain |
122 | prescribed limits. | prescribed limits. |
123 | We simply demonstrate that the technique we present was known by | We simply demonstrate that the technique we present was known by |
124 | the 17th century (with the possible exception of the | the 17th century (with the possible exception of the |
125 | second component of Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}), | second component of Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}), |
126 | and we don't attempt to describe the other uses | and we don't attempt to describe the other uses |
127 | of continued fractions or the significance of continued fractions | of continued fractions or the significance of continued fractions |
128 | in mathematics or number theory. | in mathematics or number theory. |
129 | ||
130 | Although we present best rational | Although we present best rational |
131 | approximations in the context of being able to effectively use | approximations in the context of being able to effectively use |
132 | processor integer multiplication and division instructions, | processor integer multiplication and division instructions, |
133 | earlier historical work was aimed at either | earlier historical work was aimed at either |
134 | providing rational approximations to irrational numbers ($\sqrt{2}$ or $\pi$, | providing rational approximations to irrational numbers ($\sqrt{2}$ or $\pi$, |
135 | for example), or at determining optimal numbers of gear teeth | for example), or at determining optimal numbers of gear teeth |
136 | (in mechanical systems). Naturally, the need for best rational approximations | (in mechanical systems). Naturally, the need for best rational approximations |
137 | in the context of computer arithmetic is a relatively recent | in the context of computer arithmetic is a relatively recent |
138 | development. | development. |
139 | ||
140 | In the introduction of \cite{bibref:b:HistoryCfPadeApproxBrezinski}, | In the introduction of \cite{bibref:b:HistoryCfPadeApproxBrezinski}, |
141 | Brezinski \index{Brezenski, Claude} hints at the broad application and importance | Brezinski \index{Brezenski, Claude} hints at the broad application and importance |
142 | of continued fractions: | of continued fractions: |
143 | ||
144 | \begin{quote} | \begin{quote} |
145 | The history of continued fractions is certainly one of the longest | The history of continued fractions is certainly one of the longest |
146 | among those of mathematical concepts, since it begins with | among those of mathematical concepts, since it begins with |
147 | Euclid's algorithm \index{Euclid!GCD algorithm} for the greatest common divisor at least | Euclid's algorithm \index{Euclid!GCD algorithm} for the greatest common divisor at least |
148 | three centuries B.C. As it is often the case and like | three centuries B.C. As it is often the case and like |
149 | Monsieur Jourdain in Moli\`ere's ``le bourgeois gentilhomme'' | Monsieur Jourdain in Moli\`ere's ``le bourgeois gentilhomme'' |
150 | (who was speaking in prose though he did not know he was doing so), | (who was speaking in prose though he did not know he was doing so), |
151 | continued fractions were used for many centuries before their real | continued fractions were used for many centuries before their real |
152 | discovery. | discovery. |
153 | ||
154 | The history of continued fractions and Pad\'e approximants is also | The history of continued fractions and Pad\'e approximants is also |
155 | quite important, since they played a leading role in the development | quite important, since they played a leading role in the development |
156 | of some branches of mathematics. For example, they were the basis | of some branches of mathematics. For example, they were the basis |
157 | for the proof of the transcendence of $\pi$ in 1882, an open | for the proof of the transcendence of $\pi$ in 1882, an open |
158 | problem for more than two thousand years, and also for our modern | problem for more than two thousand years, and also for our modern |
159 | spectral theory of operators. Actually they still are of great | spectral theory of operators. Actually they still are of great |
160 | interest in many fields of pure and applied mathematics and in | interest in many fields of pure and applied mathematics and in |
161 | numerical analysis, where they provide computer approximations to | numerical analysis, where they provide computer approximations to |
162 | special functions and are connected to some convergence acceleration | special functions and are connected to some convergence acceleration |
163 | methods. Continued fractions are also used in number theory, | methods. Continued fractions are also used in number theory, |
164 | computer science, automata, electronics, etc. \ldots{} | computer science, automata, electronics, etc. \ldots{} |
165 | \end{quote} | \end{quote} |
166 | ||
167 | Notice that Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} | Notice that Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} |
168 | has two components. First, it is shown that the highest-order | has two components. First, it is shown that the highest-order |
169 | convergent with an acceptable denominator is closer to $a/b$ than | convergent with an acceptable denominator is closer to $a/b$ than |
170 | any Farey neighbor to this convergent (thus, this convergent must be | any Farey neighbor to this convergent (thus, this convergent must be |
171 | either a left or right Farey neighbor of $a/b$). Second, it is shown | either a left or right Farey neighbor of $a/b$). Second, it is shown |
172 | what the other Farey neighbor must be. It is historically clear | what the other Farey neighbor must be. It is historically clear |
173 | that the properties of convergents as best rational approximations were | that the properties of convergents as best rational approximations were |
174 | understood by the 17th century (this is the \emph{first} part of | understood by the 17th century (this is the \emph{first} part of |
175 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}). However, it | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}). However, it |
176 | is not historically clear when the \emph{second} part of | is not historically clear when the \emph{second} part of |
177 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} was discovered. | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} was discovered. |
178 | ||
179 | Even in Khinchin's \index{Khinchin, A. Ya.} classic work, | Even in Khinchin's \index{Khinchin, A. Ya.} classic work, |
180 | \cite{bibref:b:KhinchinClassic}, Theorem 15, p. 22, Khinchin stops | \cite{bibref:b:KhinchinClassic}, Theorem 15, p. 22, Khinchin stops |
181 | just short of the result presented as the second part of | just short of the result presented as the second part of |
182 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}. Khinchin writes: | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}. Khinchin writes: |
183 | ||
184 | \begin{quote} | \begin{quote} |
185 | THEOREM 15. \em Every best approximation of a number is a convergent | THEOREM 15. \em Every best approximation of a number is a convergent |
186 | or an intermediate fraction of the continued fraction representing | or an intermediate fraction of the continued fraction representing |
187 | that number. | that number. |
188 | \end{quote} | \end{quote} |
189 | ||
190 | \noindent{}Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} goes | \noindent{}Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} goes |
191 | slightly farther than Khinchin's \emph{THEOREM 15}, above. | slightly farther than Khinchin's \emph{THEOREM 15}, above. |
192 | Khinchin \index{Khinchin, A. Ya.} states | Khinchin \index{Khinchin, A. Ya.} states |
193 | that a best approximation will be a convergent or an intermediate | that a best approximation will be a convergent or an intermediate |
194 | fraction---but Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} | fraction---but Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} |
195 | goes slightly farther to indicate \emph{exactly which} intermediate fraction | goes slightly farther to indicate \emph{exactly which} intermediate fraction |
196 | is potentially the best approximation. Khinchin's \emph{THEOREM 15} | is potentially the best approximation. Khinchin's \emph{THEOREM 15} |
197 | is correct, but could be strengthened. Khinchin's work | is correct, but could be strengthened. Khinchin's work |
198 | was first published in 1935. This raises the [unlikely] possibility | was first published in 1935. This raises the [unlikely] possibility |
199 | that the second part of Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} | that the second part of Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} |
200 | had not been published even as recently as 1935, although | had not been published even as recently as 1935, although |
201 | we (the authors) don't have the ability to confirm or | we (the authors) don't have the ability to confirm or |
202 | refute this. | refute this. |
203 | ||
204 | In \cite{bibref:b:HistoryCfPadeApproxBrezinski}, p. 70, Brezinski | In \cite{bibref:b:HistoryCfPadeApproxBrezinski}, p. 70, Brezinski |
205 | \index{Brezenski, Claude} | \index{Brezenski, Claude} |
206 | writes: | writes: |
207 | ||
208 | \begin{quote} | \begin{quote} |
209 | In the same period, algorithms equivalent to continued fractions were | In the same period, algorithms equivalent to continued fractions were |
210 | still used to find approximate values for ratios and to simplify | still used to find approximate values for ratios and to simplify |
211 | fractions. We have already mentioned Albert Girard. | fractions. We have already mentioned Albert Girard. |
212 | ||
213 | Among the other authors who treated the subject, the most prominent | Among the other authors who treated the subject, the most prominent |
214 | is Daniel SCHWENTER \index{Schwenter, Daniel} | is Daniel SCHWENTER \index{Schwenter, Daniel} |
215 | (N\"urnberg, 31.1.1585 - Altdorf, 19.1.1636), | (N\"urnberg, 31.1.1585 - Altdorf, 19.1.1636), |
216 | who wrote two books ``\emph{Geometriae practicae novae et auctae | who wrote two books ``\emph{Geometriae practicae novae et auctae |
217 | tractatus}'' published in 1627 and ``\emph{Delicae | tractatus}'' published in 1627 and ``\emph{Delicae |
218 | Physico-mathematicae}'' which appeared in 1636 followed by a | Physico-mathematicae}'' which appeared in 1636 followed by a |
219 | second edition in 1651. | second edition in 1651. |
220 | ||
221 | In his first book, Schwenter found approximations of 177/233 by | In his first book, Schwenter found approximations of 177/233 by |
222 | finding their g.c.d. and gave the successive convergents | finding their g.c.d. and gave the successive convergents |
223 | 79/104, 19/25, 3/4, 1/1, and 0/1. His calculations were arranged | 79/104, 19/25, 3/4, 1/1, and 0/1. His calculations were arranged |
224 | in a table\footnote{The table is reproduced in | in a table\footnote{The table is reproduced in |
225 | \cite{bibref:b:HistoryCfPadeApproxBrezinski}, | \cite{bibref:b:HistoryCfPadeApproxBrezinski}, |
226 | but is omitted here.} \ldots{} although he gave no explanation of the method. | but is omitted here.} \ldots{} although he gave no explanation of the method. |
227 | \end{quote} | \end{quote} |
228 | ||
229 | On p. 84, Brezenski \index{Brezenski, Claude} writes: | On p. 84, Brezenski \index{Brezenski, Claude} writes: |
230 | ||
231 | \begin{quote} | \begin{quote} |
232 | Wallis \index{Wallis, John} also made use of continued fractions in his book | Wallis \index{Wallis, John} also made use of continued fractions in his book |
233 | ``\emph{A treatise of algebra both historical and practical}'' | ``\emph{A treatise of algebra both historical and practical}'' |
234 | (published in 1685), to approximate ratios with large | (published in 1685), to approximate ratios with large |
235 | numerators and denominators: | numerators and denominators: |
236 | ||
237 | \emph{``Before I leave the business of Decimal Parts, and the | \emph{``Before I leave the business of Decimal Parts, and the |
238 | advantages which in practice may there cause; I have thought fit | advantages which in practice may there cause; I have thought fit |
239 | here to insert a Process Of Reducing Fractions or Proportions to | here to insert a Process Of Reducing Fractions or Proportions to |
240 | smaller termes, retaining as near as may be, the just value.} | smaller termes, retaining as near as may be, the just value.} |
241 | ||
242 | \emph{It was occasion'd by a Problem sent me (as I remember) about the | \emph{It was occasion'd by a Problem sent me (as I remember) about the |
243 | Year 1663 or 1664, by Dr. Lamplugh the present Bishop of Exeter from | Year 1663 or 1664, by Dr. Lamplugh the present Bishop of Exeter from |
244 | (his Wives Father) Dr. Davenant then one of the Prebends | (his Wives Father) Dr. Davenant then one of the Prebends |
245 | Residentaries of the Church of Salisbury, a very worthy Person, of | Residentaries of the Church of Salisbury, a very worthy Person, of |
246 | great Learning and Modesty; as I mire inderstand from persons | great Learning and Modesty; as I mire inderstand from persons |
247 | well acquainted with him, and by divers Writings of his which I have seen, | well acquainted with him, and by divers Writings of his which I have seen, |
248 | though I never had the opportunity of being personally acquainted with him, | though I never had the opportunity of being personally acquainted with him, |
249 | otherwise than by Letter. And amongst | otherwise than by Letter. And amongst |
250 | his other Learning, he was very well skilled the Mathematicks, | his other Learning, he was very well skilled the Mathematicks, |
251 | and a diligent Proficient therein.} | and a diligent Proficient therein.} |
252 | ||
253 | \emph{He sent me (as it abovesaid) a Fraction (which what it was I | \emph{He sent me (as it abovesaid) a Fraction (which what it was I |
254 | do not now particulary remember) who's Numerator and Denominator | do not now particulary remember) who's Numerator and Denominator |
255 | were, each of them of about six or seven places; and Proposed to | were, each of them of about six or seven places; and Proposed to |
256 | find the nearest Fraction in value to it, whose Denominator should | find the nearest Fraction in value to it, whose Denominator should |
257 | not be greater than 999.''} | not be greater than 999.''} |
258 | ||
259 | \begin{center} | \begin{center} |
260 | \rule{3in}{0.3mm} \\\ | \rule{3in}{0.3mm} \\\ |
261 | \end{center} | \end{center} |
262 | ||
263 | \begin{center} | \begin{center} |
264 | \emph{The Problem} \\ | \emph{The Problem} \\ |
265 | \end{center} | \end{center} |
266 | ||
267 | \emph{A Fraction (or Proportion) being assigned, to sind one as near as | \emph{A Fraction (or Proportion) being assigned, to sind one as near as |
268 | may be equal to it, in Numbers non exceeding a Number given, and in | may be equal to it, in Numbers non exceeding a Number given, and in |
269 | the smallest Terms.} | the smallest Terms.} |
270 | ||
271 | \emph{As (for instance), the Fraction $\frac{2684769}{8376571}$ (or the | \emph{As (for instance), the Fraction $\frac{2684769}{8376571}$ (or the |
272 | Proportion of 2684769 to 8376571) being assigned, to sind one equal to it | Proportion of 2684769 to 8376571) being assigned, to sind one equal to it |
273 | (if it may be) or at least the next Greater, or the next Lesser, | (if it may be) or at least the next Greater, or the next Lesser, |
274 | which may be expressed in Numbers not greater than 999; that is, in numbers | which may be expressed in Numbers not greater than 999; that is, in numbers |
275 | not exceeding three places.} | not exceeding three places.} |
276 | ||
277 | \begin{center} | \begin{center} |
278 | \rule{3in}{0.3mm} \\ | \rule{3in}{0.3mm} \\ |
279 | \end{center} | \end{center} |
280 | ||
281 | \emph{If the Fraction sought (whose terms are not to be greater than | \emph{If the Fraction sought (whose terms are not to be greater than |
282 | a Number given) be the Next Greater than a Fraction Proposed; divide the | a Number given) be the Next Greater than a Fraction Proposed; divide the |
283 | proposed Fractions Denominator by its Numerator: If the Next-Lesser, then | proposed Fractions Denominator by its Numerator: If the Next-Lesser, then |
284 | the Numerator by the Denominator, continuing the Quotient in Decimal | the Numerator by the Denominator, continuing the Quotient in Decimal |
285 | Parts, to such an Accuracy as shall be sufficient; which Quotient | Parts, to such an Accuracy as shall be sufficient; which Quotient |
286 | for the Next-Greater, is to be the Denominator answering to the | for the Next-Greater, is to be the Denominator answering to the |
287 | Numerator 1: But for the next Lesser, it is to be | Numerator 1: But for the next Lesser, it is to be |
288 | the Numerator answering to the Denominator 1: Completing a Fraction | the Numerator answering to the Denominator 1: Completing a Fraction |
289 | as near as shall be necessary to that Proposed, which Fraction I | as near as shall be necessary to that Proposed, which Fraction I |
290 | call to First Fraction Compleat: And the same wanting the Appendage of | call to First Fraction Compleat: And the same wanting the Appendage of |
291 | Decimal parts, I call, the First Fraction Cartail'd.} | Decimal parts, I call, the First Fraction Cartail'd.} |
292 | ||
293 | \emph{Khen by this Appendage of the First Fraction, | \emph{Khen by this Appendage of the First Fraction, |
294 | divide 1 Integer, and by the Integer Number which is Next-Less then | divide 1 Integer, and by the Integer Number which is Next-Less then |
295 | the sull Quotient, (that is, in case such Quotient be just an | the sull Quotient, (that is, in case such Quotient be just an |
296 | Interger Number, by the Integer Next-Less than it; but is it be an Interger | Interger Number, by the Integer Next-Less than it; but is it be an Interger |
297 | with Decimal parts annexed, than by that Integer | with Decimal parts annexed, than by that Integer |
298 | without those} | without those} |
299 | ||
300 | \emph{Decimal parts;) multiply both Terms of the first Fraction Compleat, | \emph{Decimal parts;) multiply both Terms of the first Fraction Compleat, |
301 | (the Numerator and the Denominator;) And the Products of such | (the Numerator and the Denominator;) And the Products of such |
302 | Multiplication, I call the Continual Increments of those Terms respectively. | Multiplication, I call the Continual Increments of those Terms respectively. |
303 | And so much as the Appendage of Decimal parts in such Continual Increment | And so much as the Appendage of Decimal parts in such Continual Increment |
304 | wants of 1 Integer, I call the Complements of the Appendage of the | wants of 1 Integer, I call the Complements of the Appendage of the |
305 | continual Increment.} | continual Increment.} |
306 | ||
307 | \emph{Then both to the Numerator and the Denominator of the First | \emph{Then both to the Numerator and the Denominator of the First |
308 | Fraction, add (respectively) its continual Increment, which make the Terms | Fraction, add (respectively) its continual Increment, which make the Terms |
309 | of the Second Fraction; and these again (respectively) | of the Second Fraction; and these again (respectively) |
310 | increased by the same Continual | increased by the same Continual |
311 | Increment, make the Terms of the Third Fraction: And so onward, | Increment, make the Terms of the Third Fraction: And so onward, |
312 | as long as the Fraction so arising hath an Appendage, which is not less | as long as the Fraction so arising hath an Appendage, which is not less |
313 | than the Complement of the Appendage of the Continual Increment.} | than the Complement of the Appendage of the Continual Increment.} |
314 | ||
315 | \emph{But where such Appendage becomes less than that Complement, | \emph{But where such Appendage becomes less than that Complement, |
316 | that Fraction I call the Last of the First Order; which also is to | that Fraction I call the Last of the First Order; which also is to |
317 | be the First of the Second Order.''} | be the First of the Second Order.''} |
318 | \end{quote} | \end{quote} |
319 | ||
320 | Although Wallis' archaic English above is difficult to decipher, | Although Wallis' archaic English above is difficult to decipher, |
321 | it appears that Wallis is describing the process of | it appears that Wallis is describing the process of |
322 | obtaining the convergents and intermediate fractions of | obtaining the convergents and intermediate fractions of |
323 | the continued fraction representation of a rational number. | the continued fraction representation of a rational number. |
324 | ||
325 | On p. 86, Brezenski writes: | On p. 86, Brezenski writes: |
326 | ||
327 | \begin{quote} | \begin{quote} |
328 | We have already mentioned the Dutch mathematician and astronomer | We have already mentioned the Dutch mathematician and astronomer |
329 | Christiaan HUYGENS \index{Huygens, Christiaan} (The Hague, 14.4.1629 - The Hague, 8.6.1695). | Christiaan HUYGENS \index{Huygens, Christiaan} (The Hague, 14.4.1629 - The Hague, 8.6.1695). |
330 | He made several contributions to continued fractions and related | He made several contributions to continued fractions and related |
331 | matters. | matters. |
332 | ||
333 | In 1682, Huygens built an automatic planetarium. To this end, | In 1682, Huygens built an automatic planetarium. To this end, |
334 | he used continued fractions, as described in his book | he used continued fractions, as described in his book |
335 | ``\emph{Descriptio automati planetarii}'', which was published | ``\emph{Descriptio automati planetarii}'', which was published |
336 | after his death (The Hague, 1698). In one year the earth | after his death (The Hague, 1698). In one year the earth |
337 | covers 359$^{\circ}$ $45'$ $40''$ $30'''$ and Saturn 12$^{\circ}$ | covers 359$^{\circ}$ $45'$ $40''$ $30'''$ and Saturn 12$^{\circ}$ |
338 | $13'$ $34''$ $18'''$, which gives the ratio 77708431/2640858. | $13'$ $34''$ $18'''$, which gives the ratio 77708431/2640858. |
339 | ||
340 | For finding the smallest integers whose ratio is close to the preceding | For finding the smallest integers whose ratio is close to the preceding |
341 | one, he divided the greatest number by the smallest, then the smallest | one, he divided the greatest number by the smallest, then the smallest |
342 | by the first remainder, and so on, which is Euclid's algorithm. | by the first remainder, and so on, which is Euclid's algorithm. |
343 | He thus got | He thus got |
344 | ||
345 | \begin{equation} | \begin{equation} |
346 | 29 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{1 + | 29 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{1 + |
347 | \cfrac{1}{5 + \cfrac{1}{1 + \cfrac{1}{4 + \ldots{}}}}}}} | \cfrac{1}{5 + \cfrac{1}{1 + \cfrac{1}{4 + \ldots{}}}}}}} |
348 | \nonumber | \nonumber |
349 | \end{equation} | \end{equation} |
350 | ||
351 | for the ratio. | for the ratio. |
352 | ||
353 | The fourth convergent of this continued fraction is 206/7, which | The fourth convergent of this continued fraction is 206/7, which |
354 | gave him the number of teeth for the gears of his planetarium, only | gave him the number of teeth for the gears of his planetarium, only |
355 | producing an error of $40'$ in a century! [H. 177], [H. 272]. | producing an error of $40'$ in a century! [H. 177], [H. 272]. |
356 | ||
357 | In a work, undated but not after 1687, he treats the general problem: | In a work, undated but not after 1687, he treats the general problem: |
358 | ||
359 | \emph{``Etant donn\'es deux grands nombres ayant entr'eux un | \emph{``Etant donn\'es deux grands nombres ayant entr'eux un |
360 | certain rapport, en trouver d'autres plus petits pour les dents | certain rapport, en trouver d'autres plus petits pour les dents |
361 | des roues qui ne soient pas incommodes par leurs grandeurs et qui | des roues qui ne soient pas incommodes par leurs grandeurs et qui |
362 | aient entr'eux \`a peu pr\`es le m\^eme rapport, | aient entr'eux \`a peu pr\`es le m\^eme rapport, |
363 | de telle facon qu'aucun couple de nombres plus petits ne | de telle facon qu'aucun couple de nombres plus petits ne |
364 | fournisse un rapport plus approchant de la vraie | fournisse un rapport plus approchant de la vraie |
365 | valeur.''}\footnote{English translation \index{Raspide, Sandrine@de Raspide, Sandrine} | valeur.''}\footnote{English translation \index{Raspide, Sandrine@de Raspide, Sandrine} |
366 | \cite{bibref:i:sandrinederaspide}: | \cite{bibref:i:sandrinederaspide}: |
367 | \emph{If we consider two large numbers forming a given ratio, | \emph{If we consider two large numbers forming a given ratio, |
368 | we need to find another set of smaller numbers for the teeth of the gearwheels, | we need to find another set of smaller numbers for the teeth of the gearwheels, |
369 | which are not inconvenient in their size and which bear the very same ratio | which are not inconvenient in their size and which bear the very same ratio |
370 | between them, in such a way that no other pair of smaller numbers | between them, in such a way that no other pair of smaller numbers |
371 | brings a ratio closer to the actual value.}} | brings a ratio closer to the actual value.}} |
372 | ||
373 | Thus Huygens was conscious of the property of best approximation | Thus Huygens was conscious of the property of best approximation |
374 | exhibited by continued fractions. He explained his method | exhibited by continued fractions. He explained his method |
375 | as follows: | as follows: |
376 | ||
377 | \emph{``Pour trouver donc des nombres plus petits qui expriment | \emph{``Pour trouver donc des nombres plus petits qui expriment |
378 | approximativement ce rapport, je divise le plus grand des nombres | approximativement ce rapport, je divise le plus grand des nombres |
379 | par le plus petit, puis le plus petit par le reste de la premi\`ere | par le plus petit, puis le plus petit par le reste de la premi\`ere |
380 | division et ensuite ce reste par le noveau reste \ldots{} | division et ensuite ce reste par le noveau reste \ldots{} |
381 | Poursuivant ce calcul aussi longtemps que possible, on parvient | Poursuivant ce calcul aussi longtemps que possible, on parvient |
382 | enfin par la division \`a un reste 1.''}\footnote{English | enfin par la division \`a un reste 1.''}\footnote{English |
383 | translation \index{Raspide, Sandrine@de Raspide, Sandrine} | translation \index{Raspide, Sandrine@de Raspide, Sandrine} |
384 | \cite{bibref:i:sandrinederaspide}: \emph{Thus to find some smaller numbers that | \cite{bibref:i:sandrinederaspide}: \emph{Thus to find some smaller numbers that |
385 | approximately express this ratio, I divide the largest of | approximately express this ratio, I divide the largest of |
386 | the numbers by the smallest, then the smallest by the | the numbers by the smallest, then the smallest by the |
387 | remainder of the first division and then this remainder by | remainder of the first division and then this remainder by |
388 | the new remainder, continuing this calculation as long as possible, | the new remainder, continuing this calculation as long as possible, |
389 | we finally end up with a division into a remainder of 1.}} | we finally end up with a division into a remainder of 1.}} |
390 | ||
391 | Then he makes the following comments: | Then he makes the following comments: |
392 | ||
393 | \emph{``Or, lorsqu'on n\'eglige \`a partir d'une fraction | \emph{``Or, lorsqu'on n\'eglige \`a partir d'une fraction |
394 | quelconque les derniers termes de la s\'erie et celles qui | quelconque les derniers termes de la s\'erie et celles qui |
395 | la suivent, et qu'on r\'eduit les autres plus le | la suivent, et qu'on r\'eduit les autres plus le |
396 | nombre entier \`a un commun d\'enominateur, le rapport de ce | nombre entier \`a un commun d\'enominateur, le rapport de ce |
397 | dernier au num\'erateur, sera voisin de celui du plus | dernier au num\'erateur, sera voisin de celui du plus |
398 | petit nombre donn\'e au plus grand; et la diff\'erence | petit nombre donn\'e au plus grand; et la diff\'erence |
399 | sera si faible qu'il serait impossible d'obtenir un meilleur accord | sera si faible qu'il serait impossible d'obtenir un meilleur accord |
400 | avec des nombres plus petits.''}\footnote{English | avec des nombres plus petits.''}\footnote{English |
401 | translation \index{Raspide, Sandrine@de Raspide, Sandrine} | translation \index{Raspide, Sandrine@de Raspide, Sandrine} |
402 | \cite{bibref:i:sandrinederaspide}: | \cite{bibref:i:sandrinederaspide}: |
403 | \emph{However, when, from an ordinary fraction, we neglect | \emph{However, when, from an ordinary fraction, we neglect |
404 | the last terms of the run and the ones that follow, and when | the last terms of the run and the ones that follow, and when |
405 | we reduce the others plus the integer to a common denominator, | we reduce the others plus the integer to a common denominator, |
406 | the ratio of the latter to the numerator will be in the neighborhood | the ratio of the latter to the numerator will be in the neighborhood |
407 | of the smallest given number to the largest; and the difference will | of the smallest given number to the largest; and the difference will |
408 | be so small that it would be impossible to obtain a better | be so small that it would be impossible to obtain a better |
409 | approximation with smaller numbers.}} | approximation with smaller numbers.}} |
410 | ||
411 | He proves this result and applies it to the continued fraction | He proves this result and applies it to the continued fraction |
412 | for $\pi$. | for $\pi$. |
413 | ||
414 | Let us give the opinion of the French astronomer | Let us give the opinion of the French astronomer |
415 | \index{Delambre, Jean Baptiste Joseph}Jean Baptiste | \index{Delambre, Jean Baptiste Joseph}Jean Baptiste |
416 | Joseph DELAMBRE (Amiens, 19.9.1749 - Paris, 19.8.1822), about | Joseph DELAMBRE (Amiens, 19.9.1749 - Paris, 19.8.1822), about |
417 | this part of Huygens' work. It is quite interesting [H. 108]: | this part of Huygens' work. It is quite interesting [H. 108]: |
418 | ||
419 | \emph{`` \ldots{}; enfin il d\'ecrit son plan\'etaire.}\footnote{English | \emph{`` \ldots{}; enfin il d\'ecrit son plan\'etaire.}\footnote{English |
420 | translation \index{Raspide, Sandrine@de Raspide, Sandrine} | translation \index{Raspide, Sandrine@de Raspide, Sandrine} |
421 | \cite{bibref:i:sandrinederaspide}: | \cite{bibref:i:sandrinederaspide}: |
422 | \emph{\ldots{}; finally, he describes his planetarium.}} | \emph{\ldots{}; finally, he describes his planetarium.}} |
423 | ||
424 | \emph{Ces sortes de machines ne sont que des objets de | \emph{Ces sortes de machines ne sont que des objets de |
425 | curiosit\'e pour les amateurs, ils sont absolument inutiles | curiosit\'e pour les amateurs, ils sont absolument inutiles |
426 | \`a l'Astronomie; celle \index{Huygens, Christiaan}d'Huygens | \`a l'Astronomie; celle \index{Huygens, Christiaan}d'Huygens |
427 | \'etait destin\'ee \`a montrer | \'etait destin\'ee \`a montrer |
428 | les mouvements elliptiques des plan\`etes, suivant les id\'ees | les mouvements elliptiques des plan\`etes, suivant les id\'ees |
429 | de \index{Kepler, Johannes}K\'epler. Le probl\`eme \`a r\'esoudre \'etait celui-ci: | de \index{Kepler, Johannes}K\'epler. Le probl\`eme \`a r\'esoudre \'etait celui-ci: |
430 | Etant donn\'e deux grands nombres, trouver deux autres nombres plus | Etant donn\'e deux grands nombres, trouver deux autres nombres plus |
431 | pitits et plus commodes, qui soient \`a peu pr\`es dans la m\^eme raison. | pitits et plus commodes, qui soient \`a peu pr\`es dans la m\^eme raison. |
432 | Il y emploie les fractions continues, et sans donner la | Il y emploie les fractions continues, et sans donner la |
433 | th\'eorie analytique de ces fractions, il les applique \`a des | th\'eorie analytique de ces fractions, il les applique \`a des |
434 | exemples. Il trouve ainsi le nombre des dents qui'il convient de donner | exemples. Il trouve ainsi le nombre des dents qui'il convient de donner |
435 | aux roues.}\footnote{English translation \index{Raspide, Sandrine@de Raspide, Sandrine} | aux roues.}\footnote{English translation \index{Raspide, Sandrine@de Raspide, Sandrine} |
436 | \cite{bibref:i:sandrinederaspide}: \emph{These kinds of machines are | \cite{bibref:i:sandrinederaspide}: \emph{These kinds of machines are |
437 | mere objects of curiosity for the amateurs, completely useless to astronomy; | mere objects of curiosity for the amateurs, completely useless to astronomy; |
438 | Huygens' machine was meant to demonstrate the elliptic movements of the | Huygens' machine was meant to demonstrate the elliptic movements of the |
439 | planets, following Kepler's ideas. The problem to solve was the following: | planets, following Kepler's ideas. The problem to solve was the following: |
440 | given two large numbers, we need to find two other numbers, smaller and | given two large numbers, we need to find two other numbers, smaller and |
441 | more convenient, which are more or less in the same ratio. To achieve | more convenient, which are more or less in the same ratio. To achieve |
442 | this, Huygens uses continued ratios, and, without giving the analytic | this, Huygens uses continued ratios, and, without giving the analytic |
443 | theory of these ratios, he applies it to some examples. Thus, he is able | theory of these ratios, he applies it to some examples. Thus, he is able |
444 | to determine the number of teeth needed for the gearwheels.}} | to determine the number of teeth needed for the gearwheels.}} |
445 | ||
446 | \emph{Cette propri\'et\'e des fractions continues, para\^{\i}t \`a | \emph{Cette propri\'et\'e des fractions continues, para\^{\i}t \`a |
447 | \index{Lagrange, Joseph-Louis}Lagrange, une des principales d\'ecouvertes | \index{Lagrange, Joseph-Louis}Lagrange, une des principales d\'ecouvertes |
448 | d'Huygens. Cet \'eloge | d'Huygens. Cet \'eloge |
449 | un peu exag\'er\'e fut sans doute dict\'e \`a | un peu exag\'er\'e fut sans doute dict\'e \`a |
450 | Lagrange par l'usage qu'il a su faire de ces fractions dans l'Analyse. | Lagrange par l'usage qu'il a su faire de ces fractions dans l'Analyse. |
451 | Quelques g\'eom\`etres ont paru douter des avantages de ces fractions et | Quelques g\'eom\`etres ont paru douter des avantages de ces fractions et |
452 | de l'utilit\'e | de l'utilit\'e |
453 | qu'elles peuvent avoir dans les recherches analytiques. Quant au | qu'elles peuvent avoir dans les recherches analytiques. Quant au |
454 | probl\`eme des rouages, il nous semble qu'on peut le r\'esoudre d'une | probl\`eme des rouages, il nous semble qu'on peut le r\'esoudre d'une |
455 | mani\`ere plus simple et plus commode par l'Arithm\'etique ordinaire. | mani\`ere plus simple et plus commode par l'Arithm\'etique ordinaire. |
456 | Nous avons d\'ej\`a appliqu\'e notre m\'ethode | Nous avons d\'ej\`a appliqu\'e notre m\'ethode |
457 | aux intercalations du calendrier. Nous allons l'appliquer aux deux | aux intercalations du calendrier. Nous allons l'appliquer aux deux |
458 | exemples choisis par Huyhens.''}\footnote{English | exemples choisis par Huyhens.''}\footnote{English |
459 | translation \index{Raspide, Sandrine@de Raspide, Sandrine} | translation \index{Raspide, Sandrine@de Raspide, Sandrine} |
460 | \cite{bibref:i:sandrinederaspide}: | \cite{bibref:i:sandrinederaspide}: |
461 | \emph{The property of continued fractions seems, to Lagrange, | \emph{The property of continued fractions seems, to Lagrange, |
462 | one of the main discoveries of Huygens. This slightly overdone | one of the main discoveries of Huygens. This slightly overdone |
463 | praise was probably induced in Lagrange for the use that he made of | praise was probably induced in Lagrange for the use that he made of |
464 | the fractions in his Analysis. Some surveyors seemed to have questioned | the fractions in his Analysis. Some surveyors seemed to have questioned |
465 | the advantages of these fractions and their use in analytical research. | the advantages of these fractions and their use in analytical research. |
466 | As far as the gearing problem is concerned, it seems to us that we can | As far as the gearing problem is concerned, it seems to us that we can |
467 | solve it in a simpler and easier way with ordinary arithmetic. | solve it in a simpler and easier way with ordinary arithmetic. |
468 | We have already applied our methodology to the intercalation of the calendar. | We have already applied our methodology to the intercalation of the calendar. |
469 | We are going to apply it to the two examples chosen by Huygens.}} | We are going to apply it to the two examples chosen by Huygens.}} |
470 | ||
471 | Delambre concludes: | Delambre concludes: |
472 | ||
473 | \emph{``Les fractions continues ne m'ont jamais paru qu'une chose | \emph{``Les fractions continues ne m'ont jamais paru qu'une chose |
474 | curieuse qui, au reste, ne servait qu'\`a obscurcir et compliquer et je | curieuse qui, au reste, ne servait qu'\`a obscurcir et compliquer et je |
475 | n'en ai jamais fait d'usage que pour m'en d\'emontrer | n'en ai jamais fait d'usage que pour m'en d\'emontrer |
476 | l'inutilit\'e.''}\footnote{English translation | l'inutilit\'e.''}\footnote{English translation |
477 | \index{Raspide, Sandrine@de Raspide, Sandrine} | \index{Raspide, Sandrine@de Raspide, Sandrine} |
478 | \cite{bibref:i:sandrinederaspide}: | \cite{bibref:i:sandrinederaspide}: |
479 | \emph{Continued fractions never appeared to me as something more | \emph{Continued fractions never appeared to me as something more |
480 | than a mere curiosity that, at the end of the day, only serves | than a mere curiosity that, at the end of the day, only serves |
481 | to darken and complicate matters, and I only used them to | to darken and complicate matters, and I only used them to |
482 | demonstrate their uselessness.}} | demonstrate their uselessness.}} |
483 | ||
484 | This was not a prophetic view! | This was not a prophetic view! |
485 | \end{quote} | \end{quote} |
486 | ||
487 | Thus, it is clear that the use of continued fraction convergents | Thus, it is clear that the use of continued fraction convergents |
488 | as best rational approximations dates back to at least the | as best rational approximations dates back to at least the |
489 | 17th century (this is the first part of | 17th century (this is the first part of |
490 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}). However, | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}). However, |
491 | the details of the historical appearance of the second part of | the details of the historical appearance of the second part of |
492 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} (the formula | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} (the formula |
493 | for the other Farey neighbor, Eq. \ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | for the other Farey neighbor, Eq. \ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
494 | are not known to the authors. | are not known to the authors. |
495 | ||
496 | ||
497 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
498 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
499 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
500 | \section{Overview Of The Apparatus} | \section{Overview Of The Apparatus} |
501 | %Section tag: PAR0 | %Section tag: PAR0 |
502 | ||
503 | The apparatus of continued fractions is best viewed as | The apparatus of continued fractions is best viewed as |
504 | an alternate apparatus for representing real numbers. | an alternate apparatus for representing real numbers. |
505 | Knowledge of the first $n$ partial quotients of | Knowledge of the first $n$ partial quotients of |
506 | the continued fraction representation of a real number | the continued fraction representation of a real number |
507 | $x$ is equivalent to the knowledge that the number lies | $x$ is equivalent to the knowledge that the number lies |
508 | in a certain partition (Eqns. | in a certain partition (Eqns. |
509 | \ref{eq:ccfr0:spar0:01}, | \ref{eq:ccfr0:spar0:01}, |
510 | \ref{eq:ccfr0:spar0:02}, and \ref{eq:ccfr0:spar0:03}). With additional | \ref{eq:ccfr0:spar0:02}, and \ref{eq:ccfr0:spar0:03}). With additional |
511 | partial quotients, the partitions become more restrictive. | partial quotients, the partitions become more restrictive. |
512 | ||
513 | \begin{equation} | \begin{equation} |
514 | \label{eq:ccfr0:spar0:01} | \label{eq:ccfr0:spar0:01} |
515 | (x=[a_0] \vee x=[a_0; \ldots ] ) \leftrightarrow (a_0 \leq x < a_0 + 1) | (x=[a_0] \vee x=[a_0; \ldots ] ) \leftrightarrow (a_0 \leq x < a_0 + 1) |
516 | \end{equation} | \end{equation} |
517 | ||
518 | \begin{equation} | \begin{equation} |
519 | \label{eq:ccfr0:spar0:02} | \label{eq:ccfr0:spar0:02} |
520 | (x=[a_0; a_1] \vee x=[a_0; a_1, \ldots ] ) \leftrightarrow | (x=[a_0; a_1] \vee x=[a_0; a_1, \ldots ] ) \leftrightarrow |
521 | \left( | \left( |
522 | { | { |
523 | a_0 + \frac{1}{a_1 + 1} < x \leq a_0 + \frac{1}{a_1} | a_0 + \frac{1}{a_1 + 1} < x \leq a_0 + \frac{1}{a_1} |
524 | } | } |
525 | \right) | \right) |
526 | \end{equation} | \end{equation} |
527 | ||
528 | \begin{equation} | \begin{equation} |
529 | \label{eq:ccfr0:spar0:03} | \label{eq:ccfr0:spar0:03} |
530 | \begin{array}{c} | \begin{array}{c} |
531 | (x=[a_0; a_1, a_2] \vee x=[a_0; a_1, a_2, \ldots ] ) \vspace{0.05in}\\ | (x=[a_0; a_1, a_2] \vee x=[a_0; a_1, a_2, \ldots ] ) \vspace{0.05in}\\ |
532 | \updownarrow \vspace{0.0in}\\ | \updownarrow \vspace{0.0in}\\ |
533 | \left( | \left( |
534 | { | { |
535 | a_0+\cfrac{1}{a_1 + \cfrac{1}{a_2}} \leq x < a_0+\cfrac{1}{a_1 + \cfrac{1}{a_2+1}} | a_0+\cfrac{1}{a_1 + \cfrac{1}{a_2}} \leq x < a_0+\cfrac{1}{a_1 + \cfrac{1}{a_2+1}} |
536 | } | } |
537 | \right) | \right) |
538 | \end{array} | \end{array} |
539 | \end{equation} | \end{equation} |
540 | ||
541 | Algorithms for finding the continued fraction representation | Algorithms for finding the continued fraction representation |
542 | of a real number are best viewed as algorithms for | of a real number are best viewed as algorithms for |
543 | determining in which partition a real number lies. However, what is | determining in which partition a real number lies. However, what is |
544 | special (for our purposes) is that the partitions imposed by the | special (for our purposes) is that the partitions imposed by the |
545 | apparatus of continued fractions have a special relationship | apparatus of continued fractions have a special relationship |
546 | with best rational approximations---namely, that all numbers (both | with best rational approximations---namely, that all numbers (both |
547 | rational and irrational) with the same partial quotients up to a | rational and irrational) with the same partial quotients up to a |
548 | point also have the same Farey neighbors up to a certain order. | point also have the same Farey neighbors up to a certain order. |
549 | Stated more colloquially, the apparatus of continued fractions | Stated more colloquially, the apparatus of continued fractions |
550 | hacks up the real number line in a way that is especially meaningful | hacks up the real number line in a way that is especially meaningful |
551 | for finding best rational approximations. | for finding best rational approximations. |
552 | ||
553 | ||
554 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
555 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
556 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
557 | \section[CF Representation Of Rationals] | \section[CF Representation Of Rationals] |
558 | {Continued Fraction Representation Of Rational Numbers} | {Continued Fraction Representation Of Rational Numbers} |
559 | %Section tag: CRN0 | %Section tag: CRN0 |
560 | ||
561 | Without proof, we present the following algorithm, Algorithm | Without proof, we present the following algorithm, Algorithm |
562 | \ref{alg:ccfr0:scrn0:akgenalg}, for | \ref{alg:ccfr0:scrn0:akgenalg}, for |
563 | determining the continued fraction representation (i.e. the partial | determining the continued fraction representation (i.e. the partial |
564 | quotients) of a non-negative | quotients) of a non-negative |
565 | rational number $a/b$. | rational number $a/b$. |
566 | ||
567 | \begin{vworkalgorithmstatementpar}{Continued Fraction Representation Of | \begin{vworkalgorithmstatementpar}{Continued Fraction Representation Of |
568 | A Rational Number \mbox{\boldmath $a/b$}} | A Rational Number \mbox{\boldmath $a/b$}} |
569 | \label{alg:ccfr0:scrn0:akgenalg} | \label{alg:ccfr0:scrn0:akgenalg} |
570 | \begin{alglvl0} | \begin{alglvl0} |
571 | \item $k:=-1$. | \item $k:=-1$. |
572 | \item $divisor_{-1} := a$. | \item $divisor_{-1} := a$. |
573 | \item $remainder_{-1} := b$. | \item $remainder_{-1} := b$. |
574 | ||
575 | \item Repeat | \item Repeat |
576 | ||
577 | \begin{alglvl1} | \begin{alglvl1} |
578 | \item $k := k + 1$. | \item $k := k + 1$. |
579 | \item $dividend_k := divisor_{k-1}$. | \item $dividend_k := divisor_{k-1}$. |
580 | \item $divisor_k := remainder_{k-1}$. | \item $divisor_k := remainder_{k-1}$. |
581 | \item $a_k := dividend_k \; div \; divisor_k$. | \item $a_k := dividend_k \; div \; divisor_k$. |
582 | \item $remainder_k := dividend_k \; mod \; divisor_k$. | \item $remainder_k := dividend_k \; mod \; divisor_k$. |
583 | \end{alglvl1} | \end{alglvl1} |
584 | ||
585 | \item Until ($remainder_k = 0$). | \item Until ($remainder_k = 0$). |
586 | \end{alglvl0} | \end{alglvl0} |
587 | \end{vworkalgorithmstatementpar} | \end{vworkalgorithmstatementpar} |
588 | %\vworkalgorithmfooter{} | %\vworkalgorithmfooter{} |
589 | ||
590 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
591 | \label{ex:ccfr0:scrn0:01} | \label{ex:ccfr0:scrn0:01} |
592 | Find the continued fraction partial quotients of | Find the continued fraction partial quotients of |
593 | $67/29$.\footnote{This example is reproduced from | $67/29$.\footnote{This example is reproduced from |
594 | Olds \cite{bibref:b:OldsClassic}, p. 8.} | Olds \cite{bibref:b:OldsClassic}, p. 8.} |
595 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
596 | \begin{vworkexampleparsection}{Solution} | \begin{vworkexampleparsection}{Solution} |
597 | \begin{table} | \begin{table} |
598 | \caption{Continued Fraction Partial Quotients Of $67/29$ (Example \ref{ex:ccfr0:scrn0:01})} | \caption{Continued Fraction Partial Quotients Of $67/29$ (Example \ref{ex:ccfr0:scrn0:01})} |
599 | \label{tbl:ccfr0:scrn0:01} | \label{tbl:ccfr0:scrn0:01} |
600 | \begin{center} | \begin{center} |
601 | \begin{tabular}{|c|c|c|c|c|} | \begin{tabular}{|c|c|c|c|c|} |
602 | \hline | \hline |
603 | \small{Index} & \small{$dividend_k$} & \small{$divisor_k$} & \small{$a_k$} & \small{$remainder_k$} \\ | \small{Index} & \small{$dividend_k$} & \small{$divisor_k$} & \small{$a_k$} & \small{$remainder_k$} \\ |
604 | \small{($k$)} & & & & \\ | \small{($k$)} & & & & \\ |
605 | \hline | \hline |
606 | \hline | \hline |
607 | \small{-1} & \small{N/A} & \small{67} & \small{N/A} & \small{29} \\ | \small{-1} & \small{N/A} & \small{67} & \small{N/A} & \small{29} \\ |
608 | \hline | \hline |
609 | \small{0} & \small{67} & \small{29} & \small{2} & \small{9} \\ | \small{0} & \small{67} & \small{29} & \small{2} & \small{9} \\ |
610 | \hline | \hline |
611 | \small{1} & \small{29} & \small{9} & \small{3} & \small{2} \\ | \small{1} & \small{29} & \small{9} & \small{3} & \small{2} \\ |
612 | \hline | \hline |
613 | \small{2} & \small{9} & \small{2} & \small{4} & \small{1} \\ | \small{2} & \small{9} & \small{2} & \small{4} & \small{1} \\ |
614 | \hline | \hline |
615 | \small{3} & \small{2} & \small{1} & \small{2} & \small{0} \\ | \small{3} & \small{2} & \small{1} & \small{2} & \small{0} \\ |
616 | \hline | \hline |
617 | \end{tabular} | \end{tabular} |
618 | \end{center} | \end{center} |
619 | \end{table} | \end{table} |
620 | Table \ref{tbl:ccfr0:scrn0:01} shows the application of | Table \ref{tbl:ccfr0:scrn0:01} shows the application of |
621 | Algorithm \ref{alg:ccfr0:scrn0:akgenalg} to find the | Algorithm \ref{alg:ccfr0:scrn0:akgenalg} to find the |
622 | continued fraction partial quotients of $67/29$. From | continued fraction partial quotients of $67/29$. From |
623 | Table \ref{tbl:ccfr0:scrn0:01}, the continued fraction | Table \ref{tbl:ccfr0:scrn0:01}, the continued fraction |
624 | representation of $67/29$ is $[2;3,4,2]$. | representation of $67/29$ is $[2;3,4,2]$. |
625 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
626 | \vworkexamplefooter{} | \vworkexamplefooter{} |
627 | ||
628 | The process of obtaining the continued fraction representation of | The process of obtaining the continued fraction representation of |
629 | a rational number is a | a rational number is a |
630 | process of obtaining each partial quotient $a_i$, and then processing the | process of obtaining each partial quotient $a_i$, and then processing the |
631 | remainder at each step to obtain further partial quotients. Noting that | remainder at each step to obtain further partial quotients. Noting that |
632 | the dividend and divisor at each step come from previous remainders | the dividend and divisor at each step come from previous remainders |
633 | (except for $k=0$ and $k=1$) allows us to simplify notation from | (except for $k=0$ and $k=1$) allows us to simplify notation from |
634 | Algorithm \ref{alg:ccfr0:scrn0:akgenalg}. If $r_i$ is used to | Algorithm \ref{alg:ccfr0:scrn0:akgenalg}. If $r_i$ is used to |
635 | denote the remainder from the division that produced $a_i$, the following | denote the remainder from the division that produced $a_i$, the following |
636 | recursive equations come immediately. | recursive equations come immediately. |
637 | ||
638 | \begin{equation} | \begin{equation} |
639 | \label{eq:ccfr0:scrn0:00a} | \label{eq:ccfr0:scrn0:00a} |
640 | \frac{a}{b} | \frac{a}{b} |
641 | = | = |
642 | a_0 + \frac{r_0}{b} | a_0 + \frac{r_0}{b} |
643 | = | = |
644 | a_0 + \frac{1}{\frac{b}{r_0}} | a_0 + \frac{1}{\frac{b}{r_0}} |
645 | , \; 0 < r_0 < b | , \; 0 < r_0 < b |
646 | \end{equation} | \end{equation} |
647 | ||
648 | \begin{equation} | \begin{equation} |
649 | \label{eq:ccfr0:scrn0:00b} | \label{eq:ccfr0:scrn0:00b} |
650 | \frac{b}{r_0} | \frac{b}{r_0} |
651 | = | = |
652 | a_1 + \frac{r_1}{r_0} | a_1 + \frac{r_1}{r_0} |
653 | , \; 0 < r_1 < r_0 | , \; 0 < r_1 < r_0 |
654 | \end{equation} | \end{equation} |
655 | ||
656 | \begin{equation} | \begin{equation} |
657 | \label{eq:ccfr0:scrn0:00c} | \label{eq:ccfr0:scrn0:00c} |
658 | \frac{r_0}{r_1} | \frac{r_0}{r_1} |
659 | = | = |
660 | a_2 + \frac{r_2}{r_1} | a_2 + \frac{r_2}{r_1} |
661 | , \; 0 < r_2 < r_1 | , \; 0 < r_2 < r_1 |
662 | \end{equation} | \end{equation} |
663 | ||
664 | \noindent{}Finally, nearing the termination of | \noindent{}Finally, nearing the termination of |
665 | Algorithm \ref{alg:ccfr0:scrn0:akgenalg}: | Algorithm \ref{alg:ccfr0:scrn0:akgenalg}: |
666 | ||
667 | \begin{equation} | \begin{equation} |
668 | \label{eq:ccfr0:scrn0:00d} | \label{eq:ccfr0:scrn0:00d} |
669 | \frac{r_{n-3}}{r_{n-2}} | \frac{r_{n-3}}{r_{n-2}} |
670 | = | = |
671 | a_{n-1} + \frac{r_{n-1}}{r_{n-2}} | a_{n-1} + \frac{r_{n-1}}{r_{n-2}} |
672 | , \; 0 < r_{n-1} < r_{n-2} | , \; 0 < r_{n-1} < r_{n-2} |
673 | \end{equation} | \end{equation} |
674 | ||
675 | \begin{equation} | \begin{equation} |
676 | \label{eq:ccfr0:scrn0:00e} | \label{eq:ccfr0:scrn0:00e} |
677 | \frac{r_{n-2}}{r_{n-1}} | \frac{r_{n-2}}{r_{n-1}} |
678 | = | = |
679 | a_n | a_n |
680 | \end{equation} | \end{equation} |
681 | ||
682 | A natural question to ask is whether Algorithm \ref{alg:ccfr0:scrn0:akgenalg} | A natural question to ask is whether Algorithm \ref{alg:ccfr0:scrn0:akgenalg} |
683 | will always terminate---that is, whether we can always find a continued | will always terminate---that is, whether we can always find a continued |
684 | fraction representation of a rational number. We present this result | fraction representation of a rational number. We present this result |
685 | as Lemma \ref{lem:ccfr0:scrn0:alwaysterminates}. | as Lemma \ref{lem:ccfr0:scrn0:alwaysterminates}. |
686 | ||
687 | \begin{vworklemmastatement} | \begin{vworklemmastatement} |
688 | \label{lem:ccfr0:scrn0:alwaysterminates} | \label{lem:ccfr0:scrn0:alwaysterminates} |
689 | Algorithm \ref{alg:ccfr0:scrn0:akgenalg} will always terminate: that is, | Algorithm \ref{alg:ccfr0:scrn0:akgenalg} will always terminate: that is, |
690 | every rational number has a finite continued fraction representation | every rational number has a finite continued fraction representation |
691 | $[a_0; a_1, \ldots{} , a_n]$. | $[a_0; a_1, \ldots{} , a_n]$. |
692 | \end{vworklemmastatement} | \end{vworklemmastatement} |
693 | \begin{vworklemmaproof} | \begin{vworklemmaproof} |
694 | Note in Algorithm \ref{alg:ccfr0:scrn0:akgenalg} and in | Note in Algorithm \ref{alg:ccfr0:scrn0:akgenalg} and in |
695 | (\ref{eq:ccfr0:scrn0:00a}) through (\ref{eq:ccfr0:scrn0:00e}) | (\ref{eq:ccfr0:scrn0:00a}) through (\ref{eq:ccfr0:scrn0:00e}) |
696 | that the remainder of one round becomes the divisor of the | that the remainder of one round becomes the divisor of the |
697 | next round, hence the remainders must form a decreasing sequence | next round, hence the remainders must form a decreasing sequence |
698 | ||
699 | \begin{equation} | \begin{equation} |
700 | \label{eq:lem:ccfr0:scrn0:alwaysterminates} | \label{eq:lem:ccfr0:scrn0:alwaysterminates} |
701 | r_0 > r_1 > r_2 > \ldots{} > r_{n-2} > r_{n-1} , | r_0 > r_1 > r_2 > \ldots{} > r_{n-2} > r_{n-1} , |
702 | \end{equation} | \end{equation} |
703 | ||
704 | because in general a remainder must be less than the divisor in | because in general a remainder must be less than the divisor in |
705 | the division that created it. | the division that created it. |
706 | \end{vworklemmaproof} | \end{vworklemmaproof} |
707 | \vworklemmafooter{} | \vworklemmafooter{} |
708 | ||
709 | ||
710 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
711 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
712 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
713 | \section{Convergents And Intermediate Fractions} | \section{Convergents And Intermediate Fractions} |
714 | %Section tag: CNV0 | %Section tag: CNV0 |
715 | \label{ccfr0:scnf0} | \label{ccfr0:scnf0} |
716 | ||
717 | Lemma \ref{lem:ccfr0:scrn0:alwaysterminates} shows that every | Lemma \ref{lem:ccfr0:scrn0:alwaysterminates} shows that every |
718 | rational number has a finite continued fraction representation. A second | rational number has a finite continued fraction representation. A second |
719 | reasonable question to ask is whether every finite simple continued | reasonable question to ask is whether every finite simple continued |
720 | fraction corresponds to a rational number. The most convincing way | fraction corresponds to a rational number. The most convincing way |
721 | to answer that question would be to devise a concrete procedure for | to answer that question would be to devise a concrete procedure for |
722 | [re-]constructing a rational number from its continued fraction | [re-]constructing a rational number from its continued fraction |
723 | representation. | representation. |
724 | ||
725 | Given a finite continued fraction $[a_0; a_1, \ldots{}, a_n]$, it is | Given a finite continued fraction $[a_0; a_1, \ldots{}, a_n]$, it is |
726 | obvious that a rational number can be constructed using the same | obvious that a rational number can be constructed using the same |
727 | algebraic technique that would be applied by hand. Such a technique | algebraic technique that would be applied by hand. Such a technique |
728 | involves ``reconstruction from the right'' because we would begin | involves ``reconstruction from the right'' because we would begin |
729 | by using $a_n$ and then work backwards to $a_0$. We illustrate | by using $a_n$ and then work backwards to $a_0$. We illustrate |
730 | the most obvious technique with an example. | the most obvious technique with an example. |
731 | ||
732 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
733 | \label{ex:ccfr0:scnv0:abreconstructionfromright:01} | \label{ex:ccfr0:scnv0:abreconstructionfromright:01} |
734 | Find a rational number $a/b$ corresponding to the | Find a rational number $a/b$ corresponding to the |
735 | continued fraction $[2;3,4,2]$. | continued fraction $[2;3,4,2]$. |
736 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
737 | \begin{vworkexampleparsection}{Solution} | \begin{vworkexampleparsection}{Solution} |
738 | The most obvious technique is to write out the continued fraction and then to | The most obvious technique is to write out the continued fraction and then to |
739 | algebraically simplify the continued fraction from the bottom up (this | algebraically simplify the continued fraction from the bottom up (this |
740 | is what we call ``working from the right'', as we begin with $a_n$). | is what we call ``working from the right'', as we begin with $a_n$). |
741 | (\ref{eq:ccfr0:scnv0:ex:abreconstructionfromright:00}) through | (\ref{eq:ccfr0:scnv0:ex:abreconstructionfromright:00}) through |
742 | (\ref{eq:ccfr0:scnv0:ex:abreconstructionfromright:02}) | (\ref{eq:ccfr0:scnv0:ex:abreconstructionfromright:02}) |
743 | illustrate this technique. | illustrate this technique. |
744 | ||
745 | \begin{equation} | \begin{equation} |
746 | \label{eq:ccfr0:scnv0:ex:abreconstructionfromright:00} | \label{eq:ccfr0:scnv0:ex:abreconstructionfromright:00} |
747 | [2;3,4,2] = | [2;3,4,2] = |
748 | 2 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{2}}} | 2 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{2}}} |
749 | \end{equation} | \end{equation} |
750 | ||
751 | \begin{equation} | \begin{equation} |
752 | \label{eq:ccfr0:scnv0:ex:abreconstructionfromright:01} | \label{eq:ccfr0:scnv0:ex:abreconstructionfromright:01} |
753 | [2;3,4,2] = | [2;3,4,2] = |
754 | 2 + \cfrac{1}{3 + \cfrac{2}{9}} | 2 + \cfrac{1}{3 + \cfrac{2}{9}} |
755 | \end{equation} | \end{equation} |
756 | ||
757 | \begin{equation} | \begin{equation} |
758 | \label{eq:ccfr0:scnv0:ex:abreconstructionfromright:02} | \label{eq:ccfr0:scnv0:ex:abreconstructionfromright:02} |
759 | [2;3,4,2] = | [2;3,4,2] = |
760 | 2 + \frac{9}{29} = \frac{67}{29} | 2 + \frac{9}{29} = \frac{67}{29} |
761 | \end{equation} | \end{equation} |
762 | ||
763 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
764 | \vworkexamplefooter{} | \vworkexamplefooter{} |
765 | ||
766 | Although converting a continued fraction $[a_0; a_1, \ldots{}, a_n]$ | Although converting a continued fraction $[a_0; a_1, \ldots{}, a_n]$ |
767 | to a rational number working ``from the right'' is the most | to a rational number working ``from the right'' is the most |
768 | intuitively obvious technique because it mirrors how a | intuitively obvious technique because it mirrors how a |
769 | continued fraction would most naturally be simplified by | continued fraction would most naturally be simplified by |
770 | hand, it is also possible to convert a continued fraction to | hand, it is also possible to convert a continued fraction to |
771 | a rational number ``from the left''. In all subsequent | a rational number ``from the left''. In all subsequent |
772 | discussions we embrace the ``from the left'' technique because | discussions we embrace the ``from the left'' technique because |
773 | it allows us to more economically calculate \emph{convergents}, which | it allows us to more economically calculate \emph{convergents}, which |
774 | have special properties, and which we now describe. | have special properties, and which we now describe. |
775 | ||
776 | \index{continued fraction!convergent} | \index{continued fraction!convergent} |
777 | The \emph{kth order convergent} of a continued fraction | The \emph{kth order convergent} of a continued fraction |
778 | $[a_0; a_1, \ldots{}, a_n]$ is the irreducible rational number | $[a_0; a_1, \ldots{}, a_n]$ is the irreducible rational number |
779 | corresponding to $[a_0; a_1, \ldots{}, a_k]$, $k \leq n$. | corresponding to $[a_0; a_1, \ldots{}, a_k]$, $k \leq n$. |
780 | In other words, the $k$th order convergent is the irreducible rational number | In other words, the $k$th order convergent is the irreducible rational number |
781 | corresponding to the first $k+1$ partial quotients of a | corresponding to the first $k+1$ partial quotients of a |
782 | continued fraction.\footnote{``$k+1$'' because the notational | continued fraction.\footnote{``$k+1$'' because the notational |
783 | numbering | numbering |
784 | for partial quotients starts at 0 rather than 1.} | for partial quotients starts at 0 rather than 1.} |
785 | ||
786 | An $n$th order continued fraction $[a_0; a_1, \ldots{}, a_n]$ | An $n$th order continued fraction $[a_0; a_1, \ldots{}, a_n]$ |
787 | has $n+1$ convergents, $[a_0]$, | has $n+1$ convergents, $[a_0]$, |
788 | $[a_0; a_1]$, \ldots{}, and $[a_0; a_1, \ldots{}, a_n]$. | $[a_0; a_1]$, \ldots{}, and $[a_0; a_1, \ldots{}, a_n]$. |
789 | We denote the $k$th order convergent as $s_k$, with numerator | We denote the $k$th order convergent as $s_k$, with numerator |
790 | $p_k$ and denominator $q_k$. | $p_k$ and denominator $q_k$. |
791 | ||
792 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
793 | \label{ex:ccfr0:scnv0:convergentexample:01} | \label{ex:ccfr0:scnv0:convergentexample:01} |
794 | Find all convergents of $[2;3,4,2]$. | Find all convergents of $[2;3,4,2]$. |
795 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
796 | \begin{vworkexampleparsection}{Solution}\hspace{-0.4em}\footnote{Canonically, it is | \begin{vworkexampleparsection}{Solution}\hspace{-0.4em}\footnote{Canonically, it is |
797 | required that all convergents be irreducible. Any valid method can be used to | required that all convergents be irreducible. Any valid method can be used to |
798 | calculate convergents---including algebraic simplification---so long as the | calculate convergents---including algebraic simplification---so long as the |
799 | rational numbers obtained are irreducible.} | rational numbers obtained are irreducible.} |
800 | \begin{equation} | \begin{equation} |
801 | s_0 = [a_0] = [2] = 2 = \frac{2}{1} = \frac{p_0}{q_0} | s_0 = [a_0] = [2] = 2 = \frac{2}{1} = \frac{p_0}{q_0} |
802 | \end{equation} | \end{equation} |
803 | ||
804 | \begin{equation} | \begin{equation} |
805 | s_1 = [a_0;a_1] = [2;3] = 2 + \frac{1}{3} = \frac{7}{3} = \frac{p_1}{q_1} | s_1 = [a_0;a_1] = [2;3] = 2 + \frac{1}{3} = \frac{7}{3} = \frac{p_1}{q_1} |
806 | \end{equation} | \end{equation} |
807 | ||
808 | \begin{equation} | \begin{equation} |
809 | s_2 = [a_0;a_1,a_2] = | s_2 = [a_0;a_1,a_2] = |
810 | [2;3,4] = | [2;3,4] = |
811 | 2 + \cfrac{1}{3 + \cfrac{1}{4}} = \frac{30}{13} = \frac{p_2}{q_2} | 2 + \cfrac{1}{3 + \cfrac{1}{4}} = \frac{30}{13} = \frac{p_2}{q_2} |
812 | \end{equation} | \end{equation} |
813 | ||
814 | \begin{equation} | \begin{equation} |
815 | s_3 = [a_0;a_1,a_2,a_3] = [2;3,4,2] = | s_3 = [a_0;a_1,a_2,a_3] = [2;3,4,2] = |
816 | 2 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{2}}} = \frac{67}{29} = \frac{p_3}{q_3} | 2 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{2}}} = \frac{67}{29} = \frac{p_3}{q_3} |
817 | \end{equation} | \end{equation} |
818 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
819 | \vworkexamplefooter{} | \vworkexamplefooter{} |
820 | ||
821 | We now move on to the question of how to convert a continued fraction | We now move on to the question of how to convert a continued fraction |
822 | to a rational number ``from the left''. We present the | to a rational number ``from the left''. We present the |
823 | canonical algorithm for construction of convergents ``from the left''. | canonical algorithm for construction of convergents ``from the left''. |
824 | In addition | In addition |
825 | to producing irreducible rational numbers (we prove this property | to producing irreducible rational numbers (we prove this property |
826 | later), the algorithm is | later), the algorithm is |
827 | convenient because it is economical---lower-order convergents | convenient because it is economical---lower-order convergents |
828 | are used in the calculation of higher-order convergents and there | are used in the calculation of higher-order convergents and there |
829 | are no wasted calculations. | are no wasted calculations. |
830 | ||
831 | \begin{vworktheoremstatementpar}{Rule For Canonical Construction Of Continued Fraction | \begin{vworktheoremstatementpar}{Rule For Canonical Construction Of Continued Fraction |
832 | Convergents} | Convergents} |
833 | \label{thm:ccfr0:scnv0:canonicalconvergentconstruction} | \label{thm:ccfr0:scnv0:canonicalconvergentconstruction} |
834 | The numerators $p_i$ and the denominators $q_i$ of the $i$th | The numerators $p_i$ and the denominators $q_i$ of the $i$th |
835 | convergent $s_i$ of the continued fraction | convergent $s_i$ of the continued fraction |
836 | $[a_0;a_1, \ldots{} , a_n]$ satisfy the equations | $[a_0;a_1, \ldots{} , a_n]$ satisfy the equations |
837 | ||
838 | \begin{eqnarray} | \begin{eqnarray} |
839 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01} |
840 | p_i & = & a_i p_{i-1} + p_{i-2} \\ | p_i & = & a_i p_{i-1} + p_{i-2} \\ |
841 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02} |
842 | q_i & = & a_i q_{i-1} + q_{i-2} | q_i & = & a_i q_{i-1} + q_{i-2} |
843 | \end{eqnarray} | \end{eqnarray} |
844 | ||
845 | \noindent{}with the initial values | \noindent{}with the initial values |
846 | ||
847 | \begin{eqnarray} | \begin{eqnarray} |
848 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:03} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:03} |
849 | p_0 = a_0, & & p_1 = a_0 a_1 + 1, \\ | p_0 = a_0, & & p_1 = a_0 a_1 + 1, \\ |
850 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:04} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:04} |
851 | q_0 = 1, & & q_1 = a_1 . | q_0 = 1, & & q_1 = a_1 . |
852 | \end{eqnarray} | \end{eqnarray} |
853 | \end{vworktheoremstatementpar} | \end{vworktheoremstatementpar} |
854 | \begin{vworktheoremproof}\hspace{-0.4em}\footnote{Reproduced nearly | \begin{vworktheoremproof}\hspace{-0.4em}\footnote{Reproduced nearly |
855 | verbatim from \cite{bibref:b:OldsClassic}, Theorem 1.3, pp. 21-23.} | verbatim from \cite{bibref:b:OldsClassic}, Theorem 1.3, pp. 21-23.} |
856 | The proof is inductive. First, the case of $i=2$ is verified, | The proof is inductive. First, the case of $i=2$ is verified, |
857 | then an inductive step is used to show that the theorem applies | then an inductive step is used to show that the theorem applies |
858 | for $i \geq 3$. | for $i \geq 3$. |
859 | ||
860 | To create a canonical form, we assign | To create a canonical form, we assign |
861 | $s_0 = [a_0] = p_0/q_0 = a_0/1$. Thus, in all cases, $p_0 = a_0$ | $s_0 = [a_0] = p_0/q_0 = a_0/1$. Thus, in all cases, $p_0 = a_0$ |
862 | and $q_0 = 1$. Similarly, to create a unique canonical form, | and $q_0 = 1$. Similarly, to create a unique canonical form, |
863 | ||
864 | \begin{equation} | \begin{equation} |
865 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:05} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:05} |
866 | s_1 = [a_0;a_1] = a_0 + \frac{1}{a_1} = \frac{a_0 a_1 + 1}{a_1} = \frac{p_1}{q_1} , | s_1 = [a_0;a_1] = a_0 + \frac{1}{a_1} = \frac{a_0 a_1 + 1}{a_1} = \frac{p_1}{q_1} , |
867 | \end{equation} | \end{equation} |
868 | ||
869 | \noindent{}and canonically, $p_1 = a_0 a_1 + 1$ and $q_1 = a_1$. | \noindent{}and canonically, $p_1 = a_0 a_1 + 1$ and $q_1 = a_1$. |
870 | ||
871 | For $i=2$, we need to verify that the algebraic results coincide with the | For $i=2$, we need to verify that the algebraic results coincide with the |
872 | claims of the theorem. Simplifying $s_2$ algebraically leads to | claims of the theorem. Simplifying $s_2$ algebraically leads to |
873 | ||
874 | \begin{equation} | \begin{equation} |
875 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:06} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:06} |
876 | \begin{array}{c} | \begin{array}{c} |
877 | s_2 = [a_0;a_1,a_2] = | s_2 = [a_0;a_1,a_2] = |
878 | a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2}} = | a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2}} = |
879 | a_0 + \cfrac{1}{\cfrac{a_1 a_2 + 1}{a_2}} = a_0 + \cfrac{a_2}{a_1 a_2 + 1} \\ | a_0 + \cfrac{1}{\cfrac{a_1 a_2 + 1}{a_2}} = a_0 + \cfrac{a_2}{a_1 a_2 + 1} \\ |
880 | \\ | \\ |
881 | =\cfrac{a_0 (a_1 a_2 + 1) + a_2}{a_1 a_2 + 1} | =\cfrac{a_0 (a_1 a_2 + 1) + a_2}{a_1 a_2 + 1} |
882 | =\cfrac{a_2(a_0 a_1 + 1) + a_0}{a_2 a_1 + 1} . | =\cfrac{a_2(a_0 a_1 + 1) + a_0}{a_2 a_1 + 1} . |
883 | \end{array} | \end{array} |
884 | \end{equation} | \end{equation} |
885 | ||
886 | \noindent{}On the other hand, applying the recursive formula | \noindent{}On the other hand, applying the recursive formula |
887 | claimed by the theorem (Eqns. | claimed by the theorem (Eqns. |
888 | \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}, | \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}, |
889 | \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}) yields | \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}) yields |
890 | ||
891 | \begin{equation} | \begin{equation} |
892 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:07} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:07} |
893 | s_2 = \frac{a_2 p_1 + p_0}{a_2 q_1 + q_0} | s_2 = \frac{a_2 p_1 + p_0}{a_2 q_1 + q_0} |
894 | = \frac{a_2(a_0 a_1 + 1) + a_0}{a_2 (a_1) + 1} , | = \frac{a_2(a_0 a_1 + 1) + a_0}{a_2 (a_1) + 1} , |
895 | \end{equation} | \end{equation} |
896 | ||
897 | which, on inspection, is consistent with the results of | which, on inspection, is consistent with the results of |
898 | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:06}). | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:06}). |
899 | ||
900 | We now prove the inductive step. Assume that | We now prove the inductive step. Assume that |
901 | the recursive relationships supplied as | the recursive relationships supplied as |
902 | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}) | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}) |
903 | and (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}) | and (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}) |
904 | hold up through some $s_k = p_k/q_k$. We would like to show that | hold up through some $s_k = p_k/q_k$. We would like to show that |
905 | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}) | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}) |
906 | and (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}) | and (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}) |
907 | then hold for $s_{k+1}$. | then hold for $s_{k+1}$. |
908 | ||
909 | $s_k$ is a fraction of the form | $s_k$ is a fraction of the form |
910 | ||
911 | \begin{equation} | \begin{equation} |
912 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:07b} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:07b} |
913 | s_k | s_k |
914 | = | = |
915 | [a_0; a_1, a_2, \ldots , a_k] | [a_0; a_1, a_2, \ldots , a_k] |
916 | = | = |
917 | a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 | a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 |
918 | + \cfrac{1}{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots + | + \cfrac{1}{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots + |
919 | \cfrac{1}{a_{k-1} + \cfrac{1}{a_k}}}}} . | \cfrac{1}{a_{k-1} + \cfrac{1}{a_k}}}}} . |
920 | \end{equation} | \end{equation} |
921 | ||
922 | In order to form $s_{k+1}$, note that we can replace $a_k$ by | In order to form $s_{k+1}$, note that we can replace $a_k$ by |
923 | $a_k + 1/a_{k + 1}$. (Note that there is no requirement | $a_k + 1/a_{k + 1}$. (Note that there is no requirement |
924 | in Eqns. \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}, | in Eqns. \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}, |
925 | \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}, | \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}, |
926 | \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:06}, | \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:06}, |
927 | \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:07}, | \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:07}, |
928 | or \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:07b} | or \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:07b} |
929 | that the partial quotients $a_i$ be integers.) | that the partial quotients $a_i$ be integers.) |
930 | In other words, we can form a | In other words, we can form a |
931 | $k$th order continued fraction having the same value as a | $k$th order continued fraction having the same value as a |
932 | $k+1$th order continued fraction by substituting | $k+1$th order continued fraction by substituting |
933 | $a_k := a_k + \frac{1}{a_{k + 1}}$. Using this substitution | $a_k := a_k + \frac{1}{a_{k + 1}}$. Using this substitution |
934 | we can calculate $s_{k+1}$ using the same recursive | we can calculate $s_{k+1}$ using the same recursive |
935 | relationship shown to be valid in calculating $s_k$: | relationship shown to be valid in calculating $s_k$: |
936 | ||
937 | \begin{equation} | \begin{equation} |
938 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:08} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:08} |
939 | \begin{array}{c} | \begin{array}{c} |
940 | s_{k+1} = | s_{k+1} = |
941 | \cfrac | \cfrac |
942 | {\left( {a_k+\cfrac{1}{a_{k+1}} } \right) p_{k-1} + p_{k-2}} | {\left( {a_k+\cfrac{1}{a_{k+1}} } \right) p_{k-1} + p_{k-2}} |
943 | {\left( {a_k+\cfrac{1}{a_{k+1}} } \right) q_{k-1} + q_{k-2}} | {\left( {a_k+\cfrac{1}{a_{k+1}} } \right) q_{k-1} + q_{k-2}} |
944 | = | = |
945 | \cfrac | \cfrac |
946 | {(a_k a_{k+1} + 1) p_{k-1} + a_{k+1} p_{k-2}} | {(a_k a_{k+1} + 1) p_{k-1} + a_{k+1} p_{k-2}} |
947 | {(a_k a_{k+1} + 1) q_{k-1} + a_{k+1} q_{k-2}} \\ | {(a_k a_{k+1} + 1) q_{k-1} + a_{k+1} q_{k-2}} \\ |
948 | \\ | \\ |
949 | = | = |
950 | \cfrac | \cfrac |
951 | {a_{k+1} (a_k p_{k-1} + p_{k-2}) + p_{k-1}} | {a_{k+1} (a_k p_{k-1} + p_{k-2}) + p_{k-1}} |
952 | {a_{k+1} (a_k q_{k-1} + q_{k-2}) + q_{k-1}} | {a_{k+1} (a_k q_{k-1} + q_{k-2}) + q_{k-1}} |
953 | \end{array} | \end{array} |
954 | \end{equation} | \end{equation} |
955 | ||
956 | Now, we can use the assumption that the recursive relationships | Now, we can use the assumption that the recursive relationships |
957 | hold for $s_k$, i.e. | hold for $s_k$, i.e. |
958 | ||
959 | \begin{eqnarray} | \begin{eqnarray} |
960 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:09} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:09} |
961 | p_k = a_k p_{k-1} + p_{k-2} \\ | p_k = a_k p_{k-1} + p_{k-2} \\ |
962 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:10} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:10} |
963 | q_k = a_k q_{k-1} + q_{k-2} | q_k = a_k q_{k-1} + q_{k-2} |
964 | \end{eqnarray} | \end{eqnarray} |
965 | ||
966 | Substituting | Substituting |
967 | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:09}) and | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:09}) and |
968 | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:10}) into | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:10}) into |
969 | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:08}) yields | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:08}) yields |
970 | ||
971 | \begin{equation} | \begin{equation} |
972 | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:11} | \label{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:11} |
973 | s_{k+1} = \frac{p_{k+1}}{q_{k+1}} | s_{k+1} = \frac{p_{k+1}}{q_{k+1}} |
974 | = \frac{a_{k+1} p_k + p_{k-1}}{a_{k+1} q_k + q_{k-1}} . | = \frac{a_{k+1} p_k + p_{k-1}}{a_{k+1} q_k + q_{k-1}} . |
975 | \end{equation} | \end{equation} |
976 | ||
977 | \noindent{}This completes the inductive step and the proof. | \noindent{}This completes the inductive step and the proof. |
978 | \end{vworktheoremproof} | \end{vworktheoremproof} |
979 | \begin{vworktheoremparsection}{Remarks} | \begin{vworktheoremparsection}{Remarks} |
980 | Note that this algorithm gives a way to convert a continued fraction | Note that this algorithm gives a way to convert a continued fraction |
981 | $[a_0;a_1,\ldots{},a_n]$ to a rational number $a/b$, as the value of | $[a_0;a_1,\ldots{},a_n]$ to a rational number $a/b$, as the value of |
982 | a continued fraction is the value of the final convergent $s_n$. | a continued fraction is the value of the final convergent $s_n$. |
983 | Note also that it is possible to convert a continued fraction to | Note also that it is possible to convert a continued fraction to |
984 | a rational number starting from $a_n$ (i.e. working ``from | a rational number starting from $a_n$ (i.e. working ``from |
985 | the right''), and that starting with $a_n$ is probably the | the right''), and that starting with $a_n$ is probably the |
986 | more intuitive approach. | more intuitive approach. |
987 | \end{vworktheoremparsection} | \end{vworktheoremparsection} |
988 | \vworktheoremfooter{} | \vworktheoremfooter{} |
989 | ||
990 | It is sometimes convenient to consider a convergent of order | It is sometimes convenient to consider a convergent of order |
991 | $-1$ (\cite{bibref:b:KhinchinClassic}, p. 5), and for | $-1$ (\cite{bibref:b:KhinchinClassic}, p. 5), and for |
992 | algebraic convenience to adopt the convention that | algebraic convenience to adopt the convention that |
993 | $p_{-1} = 1$ and $q_{-1} = 0$. If this is done, the recursive | $p_{-1} = 1$ and $q_{-1} = 0$. If this is done, the recursive |
994 | relationships of Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} | relationships of Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} |
995 | apply for $k \geq 1$ rather than for $k \geq 2$. All of the subsequent | apply for $k \geq 1$ rather than for $k \geq 2$. All of the subsequent |
996 | theorems and proofs assume this convention. | theorems and proofs assume this convention. |
997 | ||
998 | We now prove several properties of convergents. | We now prove several properties of convergents. |
999 | ||
1000 | \begin{vworktheoremstatement} | \begin{vworktheoremstatement} |
1001 | \label{thm:ccfr0:scnv0:crossproductminusone} | \label{thm:ccfr0:scnv0:crossproductminusone} |
1002 | For all $k \geq 0$, | For all $k \geq 0$, |
1003 | ||
1004 | \begin{equation} | \begin{equation} |
1005 | \label{eq:ccfr0:scnv0:thm:crossproductminusone:00} | \label{eq:ccfr0:scnv0:thm:crossproductminusone:00} |
1006 | q_k p_{k-1} - p_k q_{k-1} = (-1)^k | q_k p_{k-1} - p_k q_{k-1} = (-1)^k |
1007 | \end{equation} | \end{equation} |
1008 | \end{vworktheoremstatement} | \end{vworktheoremstatement} |
1009 | \begin{vworktheoremproof}\hspace{-0.4em}\footnote{From | \begin{vworktheoremproof}\hspace{-0.4em}\footnote{From |
1010 | \cite{bibref:b:KhinchinClassic}, Theorem 2, p. 5.} | \cite{bibref:b:KhinchinClassic}, Theorem 2, p. 5.} |
1011 | Multiplying (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}) | Multiplying (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}) |
1012 | by $p_{k-1}$, multiplying | by $p_{k-1}$, multiplying |
1013 | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}) by | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}) by |
1014 | $q_{k-1}$, then subtracting the equations yields | $q_{k-1}$, then subtracting the equations yields |
1015 | ||
1016 | \begin{equation} | \begin{equation} |
1017 | \label{eq:ccfr0:scnv0:thm:crossproductminusone:01} | \label{eq:ccfr0:scnv0:thm:crossproductminusone:01} |
1018 | q_k p_{k-1} - p_k q_{k-1} = -(q_{k-1} p_{k-2} - p_{k-1} q_{k-2}) , | q_k p_{k-1} - p_k q_{k-1} = -(q_{k-1} p_{k-2} - p_{k-1} q_{k-2}) , |
1019 | \end{equation} | \end{equation} |
1020 | ||
1021 | and since $q_0 p_{-1} - p_0 q_{-1} = 1$, the theorem is | and since $q_0 p_{-1} - p_0 q_{-1} = 1$, the theorem is |
1022 | proved. | proved. |
1023 | \end{vworktheoremproof} | \end{vworktheoremproof} |
1024 | \begin{vworktheoremparsection}{Corollary I} | \begin{vworktheoremparsection}{Corollary I} |
1025 | For all $k \geq 1$, | For all $k \geq 1$, |
1026 | ||
1027 | \begin{equation} | \begin{equation} |
1028 | \label{eq:ccfr0:scnv0:thm:crossproductminusone:02} | \label{eq:ccfr0:scnv0:thm:crossproductminusone:02} |
1029 | \frac{p_{k-1}}{q_{k-1}} - \frac{p_k}{q_k} = \frac{(-1)^k}{q_k q_{k-1}} . | \frac{p_{k-1}}{q_{k-1}} - \frac{p_k}{q_k} = \frac{(-1)^k}{q_k q_{k-1}} . |
1030 | \end{equation} | \end{equation} |
1031 | \end{vworktheoremparsection} | \end{vworktheoremparsection} |
1032 | \begin{vworktheoremproof} | \begin{vworktheoremproof} |
1033 | (\ref{eq:ccfr0:scnv0:thm:crossproductminusone:02}) can be obtained | (\ref{eq:ccfr0:scnv0:thm:crossproductminusone:02}) can be obtained |
1034 | in a straightforward way by algebraic operations on | in a straightforward way by algebraic operations on |
1035 | (\ref{eq:ccfr0:scnv0:thm:crossproductminusone:00}). | (\ref{eq:ccfr0:scnv0:thm:crossproductminusone:00}). |
1036 | \end{vworktheoremproof} | \end{vworktheoremproof} |
1037 | %\vworktheoremfooter{} | %\vworktheoremfooter{} |
1038 | ||
1039 | \begin{vworktheoremstatement} | \begin{vworktheoremstatement} |
1040 | \label{thm:ccfr0:scnv0:crossproductminusonebacktwo} | \label{thm:ccfr0:scnv0:crossproductminusonebacktwo} |
1041 | For all $k \geq 1$, | For all $k \geq 1$, |
1042 | ||
1043 | \begin{equation} | \begin{equation} |
1044 | q_k p_{k-2} - p_k q_{k-2} = (-1)^{k-1} a_k . | q_k p_{k-2} - p_k q_{k-2} = (-1)^{k-1} a_k . |
1045 | \end{equation} | \end{equation} |
1046 | \end{vworktheoremstatement} | \end{vworktheoremstatement} |
1047 | \begin{vworktheoremproof}\hspace{-0.4em}\footnote{From | \begin{vworktheoremproof}\hspace{-0.4em}\footnote{From |
1048 | \cite{bibref:b:KhinchinClassic}, Theorem 3, p. 6.} | \cite{bibref:b:KhinchinClassic}, Theorem 3, p. 6.} |
1049 | By multiplying (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}) | By multiplying (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:01}) |
1050 | by $q_{k-2}$ and | by $q_{k-2}$ and |
1051 | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}) | (\ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:02}) |
1052 | by $p_{k-2}$ and then subtracting the first from the | by $p_{k-2}$ and then subtracting the first from the |
1053 | second, we obtain, on the basis of Theorem | second, we obtain, on the basis of Theorem |
1054 | \ref{thm:ccfr0:scnv0:crossproductminusone}, | \ref{thm:ccfr0:scnv0:crossproductminusone}, |
1055 | ||
1056 | \begin{equation} | \begin{equation} |
1057 | q_k p_{k-2} - p_k q_{k-2} | q_k p_{k-2} - p_k q_{k-2} |
1058 | = a_k (q_{k-1} p_{k-2} - p_{k-1} q_{k-2}) = (-1)^{k-1} a_k , | = a_k (q_{k-1} p_{k-2} - p_{k-1} q_{k-2}) = (-1)^{k-1} a_k , |
1059 | \end{equation} | \end{equation} |
1060 | which completes the proof. | which completes the proof. |
1061 | \end{vworktheoremproof} | \end{vworktheoremproof} |
1062 | \vworktheoremfooter{} | \vworktheoremfooter{} |
1063 | ||
1064 | The results in Theorems \ref{thm:ccfr0:scnv0:crossproductminusone} | The results in Theorems \ref{thm:ccfr0:scnv0:crossproductminusone} |
1065 | and \ref{thm:ccfr0:scnv0:crossproductminusonebacktwo} allow us | and \ref{thm:ccfr0:scnv0:crossproductminusonebacktwo} allow us |
1066 | to establish the relative ordering of convergents. | to establish the relative ordering of convergents. |
1067 | Theorems \ref{thm:ccfr0:scnv0:crossproductminusone} and | Theorems \ref{thm:ccfr0:scnv0:crossproductminusone} and |
1068 | \ref{thm:ccfr0:scnv0:crossproductminusonebacktwo} demonstrate that | \ref{thm:ccfr0:scnv0:crossproductminusonebacktwo} demonstrate that |
1069 | even-ordered convergents form an increasing sequence and that odd-ordered | even-ordered convergents form an increasing sequence and that odd-ordered |
1070 | convergents form a decreasing sequence, and that every odd-ordered convergent | convergents form a decreasing sequence, and that every odd-ordered convergent |
1071 | is greater than every even-ordered convergent. | is greater than every even-ordered convergent. |
1072 | ||
1073 | \begin{vworktheoremstatement} | \begin{vworktheoremstatement} |
1074 | \label{thm:ccfr0:scnv0:irreducibility} | \label{thm:ccfr0:scnv0:irreducibility} |
1075 | For all $k \geq 0$, $s_k = p_k/q_k$ is | For all $k \geq 0$, $s_k = p_k/q_k$ is |
1076 | irreducible. | irreducible. |
1077 | \end{vworktheoremstatement} | \end{vworktheoremstatement} |
1078 | \begin{vworktheoremproof} | \begin{vworktheoremproof} |
1079 | This proof comes immediately from the form | This proof comes immediately from the form |
1080 | of (\ref{eq:ccfr0:scnv0:thm:crossproductminusone:00}). | of (\ref{eq:ccfr0:scnv0:thm:crossproductminusone:00}). |
1081 | Without coprimality of $p_k$ and $q_k$, the difference | Without coprimality of $p_k$ and $q_k$, the difference |
1082 | of $\pm 1$ is impossible (see | of $\pm 1$ is impossible (see |
1083 | \cprizeroxrefcomma{}Lemma \ref{lem:cpri0:ppn0:000p}). | \cprizeroxrefcomma{}Lemma \ref{lem:cpri0:ppn0:000p}). |
1084 | \end{vworktheoremproof} | \end{vworktheoremproof} |
1085 | %\vworktheoremfooter{} | %\vworktheoremfooter{} |
1086 | ||
1087 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
1088 | \label{ex:ccfr0:scrn0:abreconstruction:01} | \label{ex:ccfr0:scrn0:abreconstruction:01} |
1089 | Find an irreducible rational number $a/b$ corresponding to the | Find an irreducible rational number $a/b$ corresponding to the |
1090 | continued fraction $[2;3,4,2]$. | continued fraction $[2;3,4,2]$. |
1091 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
1092 | \begin{vworkexampleparsection}{Solution} | \begin{vworkexampleparsection}{Solution} |
1093 | Application of Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} | Application of Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} |
1094 | yields the following convergents. The final convergent $s_3$ is the | yields the following convergents. The final convergent $s_3$ is the |
1095 | value of the continued fraction $[2;3,4,2]$ and Theorem | value of the continued fraction $[2;3,4,2]$ and Theorem |
1096 | \ref{thm:ccfr0:scnv0:irreducibility} assures us that each convergent is | \ref{thm:ccfr0:scnv0:irreducibility} assures us that each convergent is |
1097 | irreducible. | irreducible. |
1098 | ||
1099 | \begin{equation} | \begin{equation} |
1100 | \label{eq:ccfr0:scrn0:02a} | \label{eq:ccfr0:scrn0:02a} |
1101 | p_{-1} = 1, \; q_{-1} = 0 | p_{-1} = 1, \; q_{-1} = 0 |
1102 | \end{equation} | \end{equation} |
1103 | ||
1104 | \begin{equation} | \begin{equation} |
1105 | \label{eq:ccfr0:scrn0:02b} | \label{eq:ccfr0:scrn0:02b} |
1106 | s_0 = \frac{p_0}{q_0} = \frac{a_0}{1} = \frac{2}{1} | s_0 = \frac{p_0}{q_0} = \frac{a_0}{1} = \frac{2}{1} |
1107 | \end{equation} | \end{equation} |
1108 | ||
1109 | \begin{equation} | \begin{equation} |
1110 | \label{eq:ccfr0:scrn0:02c} | \label{eq:ccfr0:scrn0:02c} |
1111 | s_1 = \frac{p_1}{q_1} = \frac{a_1 p_0 + p_{-1}}{a_1 q_0 + q_{-1}} | s_1 = \frac{p_1}{q_1} = \frac{a_1 p_0 + p_{-1}}{a_1 q_0 + q_{-1}} |
1112 | = \frac{(3)(2) + (1)}{(3)(1)+(0)} | = \frac{(3)(2) + (1)}{(3)(1)+(0)} |
1113 | = \frac{7}{3} | = \frac{7}{3} |
1114 | \end{equation} | \end{equation} |
1115 | ||
1116 | \begin{equation} | \begin{equation} |
1117 | \label{eq:ccfr0:scrn0:02d} | \label{eq:ccfr0:scrn0:02d} |
1118 | s_2 = \frac{p_2}{q_2} = \frac{a_2 p_1 + p_{0}}{a_2 q_1 + q_{0}} | s_2 = \frac{p_2}{q_2} = \frac{a_2 p_1 + p_{0}}{a_2 q_1 + q_{0}} |
1119 | = \frac{(4)(7) + (2)}{(4)(3)+(1)} | = \frac{(4)(7) + (2)}{(4)(3)+(1)} |
1120 | = \frac{30}{13} | = \frac{30}{13} |
1121 | \end{equation} | \end{equation} |
1122 | ||
1123 | \begin{equation} | \begin{equation} |
1124 | \label{eq:ccfr0:scrn0:02e} | \label{eq:ccfr0:scrn0:02e} |
1125 | s_3 = \frac{p_3}{q_3} = \frac{a_3 p_2 + p_{1}}{a_3 q_2 + q_{1}} | s_3 = \frac{p_3}{q_3} = \frac{a_3 p_2 + p_{1}}{a_3 q_2 + q_{1}} |
1126 | = \frac{(2)(30) + (7)}{(2)(13)+(3)} | = \frac{(2)(30) + (7)}{(2)(13)+(3)} |
1127 | = \frac{67}{29} | = \frac{67}{29} |
1128 | \end{equation} | \end{equation} |
1129 | ||
1130 | Note that this result coincides with | Note that this result coincides with |
1131 | Example \ref{ex:ccfr0:scrn0:01}. | Example \ref{ex:ccfr0:scrn0:01}. |
1132 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
1133 | \vworkexamplefooter{} | \vworkexamplefooter{} |
1134 | ||
1135 | We've shown in Algorithm \ref{alg:ccfr0:scrn0:akgenalg} | We've shown in Algorithm \ref{alg:ccfr0:scrn0:akgenalg} |
1136 | that any rational number can be expressed as a continued fraction, and | that any rational number can be expressed as a continued fraction, and |
1137 | with Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} | with Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} |
1138 | that any finite continued fraction can be converted to a rational | that any finite continued fraction can be converted to a rational |
1139 | number. Although we don't say more until Section \ref{ccfr0:scin0}, | number. Although we don't say more until Section \ref{ccfr0:scin0}, |
1140 | it follows directly that any irrational number results in | it follows directly that any irrational number results in |
1141 | an \emph{infinite} (or non-terminating) continued fraction, and that | an \emph{infinite} (or non-terminating) continued fraction, and that |
1142 | any infinite continued fraction represents an irrational number. | any infinite continued fraction represents an irrational number. |
1143 | In the theorems that follow, we don't treat infinite continued | In the theorems that follow, we don't treat infinite continued |
1144 | fractions with mathematical rigor, because our emphasis is on | fractions with mathematical rigor, because our emphasis is on |
1145 | specific applications of continued fractions. | specific applications of continued fractions. |
1146 | ||
1147 | \begin{vworktheoremstatement} | \begin{vworktheoremstatement} |
1148 | \label{thm:ccfr0:scnv0:evenslessthanoddsgreaterthan} | \label{thm:ccfr0:scnv0:evenslessthanoddsgreaterthan} |
1149 | For a finite continued fraction representation of | For a finite continued fraction representation of |
1150 | the [rational] number $\alpha$, every even-ordered convergent | the [rational] number $\alpha$, every even-ordered convergent |
1151 | is less than $\alpha$ and every odd-ordered convergent is | is less than $\alpha$ and every odd-ordered convergent is |
1152 | greater than $\alpha$, with the exception of the final convergent | greater than $\alpha$, with the exception of the final convergent |
1153 | $s_n$, which is equal to $\alpha$. | $s_n$, which is equal to $\alpha$. |
1154 | For an infinite continued fraction corresponding to the | For an infinite continued fraction corresponding to the |
1155 | [irrational] real | [irrational] real |
1156 | number $\alpha$, every even-ordered convergent is less than | number $\alpha$, every even-ordered convergent is less than |
1157 | $\alpha$, and every odd-ordered convergent is greater than | $\alpha$, and every odd-ordered convergent is greater than |
1158 | $\alpha$. | $\alpha$. |
1159 | \end{vworktheoremstatement} | \end{vworktheoremstatement} |
1160 | \begin{vworktheoremparsection}{Proof (Informal)} | \begin{vworktheoremparsection}{Proof (Informal)} |
1161 | In the case of a finite continued fraction, the proof is obvious | In the case of a finite continued fraction, the proof is obvious |
1162 | and immediate. Since $s_n$, the final convergent, is equal | and immediate. Since $s_n$, the final convergent, is equal |
1163 | to the rational number $\alpha$, Theorems \ref{thm:ccfr0:scnv0:crossproductminusone} | to the rational number $\alpha$, Theorems \ref{thm:ccfr0:scnv0:crossproductminusone} |
1164 | and \ref{thm:ccfr0:scnv0:crossproductminusonebacktwo} demonstrate this | and \ref{thm:ccfr0:scnv0:crossproductminusonebacktwo} demonstrate this |
1165 | unequivocally. | unequivocally. |
1166 | ||
1167 | In the case of an infinite continued fraction, | In the case of an infinite continued fraction, |
1168 | note the form of the proof of Theorem | note the form of the proof of Theorem |
1169 | \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction}, | \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction}, |
1170 | where the substitution of $a_k := a_k + 1/a_{k+1}$ | where the substitution of $a_k := a_k + 1/a_{k+1}$ |
1171 | is made. It can be demonstrated that for any even-ordered | is made. It can be demonstrated that for any even-ordered |
1172 | convergent $s_k$, additional partial quotients (except | convergent $s_k$, additional partial quotients (except |
1173 | $a_{k+1} = a_n = 1$, which isn't allowed in general or even | $a_{k+1} = a_n = 1$, which isn't allowed in general or even |
1174 | possible with an infinite continued fraction) can only | possible with an infinite continued fraction) can only |
1175 | increase the value. It can similarly be demonstrated that | increase the value. It can similarly be demonstrated that |
1176 | additional partial quotients can only decrease the value | additional partial quotients can only decrease the value |
1177 | of an odd-ordered convergent. Because the continued fraction | of an odd-ordered convergent. Because the continued fraction |
1178 | is infinite, any particular even-ordered convergent will be | is infinite, any particular even-ordered convergent will be |
1179 | increased if more partial quotients are allowed, and any particular | increased if more partial quotients are allowed, and any particular |
1180 | odd-ordered convergent will be decreased in value if more | odd-ordered convergent will be decreased in value if more |
1181 | partial quotients are allowed. Thus, we can conclude that | partial quotients are allowed. Thus, we can conclude that |
1182 | all even-ordered convergents are less than the value of | all even-ordered convergents are less than the value of |
1183 | $\alpha$, and all odd-ordered convergents are greater | $\alpha$, and all odd-ordered convergents are greater |
1184 | than the value of $\alpha$.\footnote{To make this proof more | than the value of $\alpha$.\footnote{To make this proof more |
1185 | formal would require the discussion of \emph{remainders}, | formal would require the discussion of \emph{remainders}, |
1186 | which wouldn't contribute to the applications discussed in this | which wouldn't contribute to the applications discussed in this |
1187 | work.} | work.} |
1188 | \end{vworktheoremparsection} | \end{vworktheoremparsection} |
1189 | %\vworktheoremfooter{} | %\vworktheoremfooter{} |
1190 | ||
1191 | \begin{vworktheoremstatement} | \begin{vworktheoremstatement} |
1192 | \label{thm:ccfr0:scnv0:minimumrateofconvergentdenominatorincrease} | \label{thm:ccfr0:scnv0:minimumrateofconvergentdenominatorincrease} |
1193 | For $k \geq 2$, | For $k \geq 2$, |
1194 | ||
1195 | \begin{equation} | \begin{equation} |
1196 | q_k \geq 2^{\frac{k-1}{2}} . | q_k \geq 2^{\frac{k-1}{2}} . |
1197 | \end{equation} | \end{equation} |
1198 | \end{vworktheoremstatement} | \end{vworktheoremstatement} |
1199 | \begin{vworktheoremproof}\hspace{-0.4em}\footnote{From | \begin{vworktheoremproof}\hspace{-0.4em}\footnote{From |
1200 | \cite{bibref:b:KhinchinClassic}, Theorem 12, p. 13.} | \cite{bibref:b:KhinchinClassic}, Theorem 12, p. 13.} |
1201 | For $k \geq 2$, | For $k \geq 2$, |
1202 | ||
1203 | \begin{equation} | \begin{equation} |
1204 | q_k = a_k q_{k-1} + q_{k-2} \geq q_{k-1} + q_{k-2} \geq 2 q_{k-2} . | q_k = a_k q_{k-1} + q_{k-2} \geq q_{k-1} + q_{k-2} \geq 2 q_{k-2} . |
1205 | \end{equation} | \end{equation} |
1206 | ||
1207 | Successive application of this inequality yields | Successive application of this inequality yields |
1208 | ||
1209 | \begin{equation} | \begin{equation} |
1210 | q_{2k} \geq 2^k q_0 = 2^k, \; q_{2k+1} \geq 2^k q_1 \geq 2^k , | q_{2k} \geq 2^k q_0 = 2^k, \; q_{2k+1} \geq 2^k q_1 \geq 2^k , |
1211 | \end{equation} | \end{equation} |
1212 | ||
1213 | which proves the theorem. Thus, the denominators of convergents | which proves the theorem. Thus, the denominators of convergents |
1214 | increase at least as rapidly as the terms of a geometric | increase at least as rapidly as the terms of a geometric |
1215 | progression. | progression. |
1216 | \end{vworktheoremproof} | \end{vworktheoremproof} |
1217 | \begin{vworktheoremparsection}{Remarks} | \begin{vworktheoremparsection}{Remarks} |
1218 | (1) This minimum geometric rate of increase of denominators of convergents is how | (1) This minimum geometric rate of increase of denominators of convergents is how |
1219 | we make the claim that Algorithms | we make the claim that Algorithms |
1220 | \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} and | \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} and |
1221 | \ref{alg:ccfr0:scba0:cffareyneighborfn} are $O(log \; N)$ and | \ref{alg:ccfr0:scba0:cffareyneighborfn} are $O(log \; N)$ and |
1222 | that Algorithms | that Algorithms |
1223 | \ref{alg:ccfr0:scba0:cfenclosingneighborsfab} | \ref{alg:ccfr0:scba0:cfenclosingneighborsfab} |
1224 | and \ref{alg:ccfr0:scba0:cffareyneighborfab} are | and \ref{alg:ccfr0:scba0:cffareyneighborfab} are |
1225 | $O(log \; max(h_{MAX}, k_{MAX}))$. | $O(log \; max(h_{MAX}, k_{MAX}))$. |
1226 | (2) This theorem supplies the \emph{minimum} rate of increase, | (2) This theorem supplies the \emph{minimum} rate of increase, |
1227 | but the demonominators of convergents can increase much faster. To achieve | but the demonominators of convergents can increase much faster. To achieve |
1228 | the minimum rate of increase, every $a_k$ must be 1, which occurs | the minimum rate of increase, every $a_k$ must be 1, which occurs |
1229 | only with the continued fraction representation of | only with the continued fraction representation of |
1230 | $\sqrt{5}/2 + 1/2$ (the famous \index{golden ratio}golden ratio). | $\sqrt{5}/2 + 1/2$ (the famous \index{golden ratio}golden ratio). |
1231 | (See also Exercise \ref{exe:cfr0:sexe0:c01}.) | (See also Exercise \ref{exe:cfr0:sexe0:c01}.) |
1232 | \end{vworktheoremparsection} | \end{vworktheoremparsection} |
1233 | \vworktheoremfooter{} | \vworktheoremfooter{} |
1234 | ||
1235 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1236 | ||
1237 | Since Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} | Since Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} |
1238 | provides a concrete | provides a concrete |
1239 | procedure for going from a continued fraction $[a_0; a_1, \ldots{} , a_n]$ | procedure for going from a continued fraction $[a_0; a_1, \ldots{} , a_n]$ |
1240 | to a rational number $a/b$ that, when Algorithm \ref{alg:ccfr0:scrn0:akgenalg} | to a rational number $a/b$ that, when Algorithm \ref{alg:ccfr0:scrn0:akgenalg} |
1241 | is applied, will again result in $[a_0; a_1, \ldots{} , a_n]$, we have | is applied, will again result in $[a_0; a_1, \ldots{} , a_n]$, we have |
1242 | successfully demonstrated that every continued fraction | successfully demonstrated that every continued fraction |
1243 | $[a_0; a_1, \ldots{} , a_n]$ corresponds to [at least one] | $[a_0; a_1, \ldots{} , a_n]$ corresponds to [at least one] |
1244 | rational number $a/b$. | rational number $a/b$. |
1245 | ||
1246 | The next natural questions to ask are questions of representation | The next natural questions to ask are questions of representation |
1247 | uniqueness and the nature of the mapping between the set of rational numbers | uniqueness and the nature of the mapping between the set of rational numbers |
1248 | and the set of continued fractions. For example, will 32/100 and 64/200 have | and the set of continued fractions. For example, will 32/100 and 64/200 have |
1249 | the same continued fraction representation $[a_0; a_1, \ldots{} , a_n]$? | the same continued fraction representation $[a_0; a_1, \ldots{} , a_n]$? |
1250 | Do two different continued fractions ever correspond | Do two different continued fractions ever correspond |
1251 | to the same rational number? We answer these questions now. | to the same rational number? We answer these questions now. |
1252 | ||
1253 | Algorithm \ref{alg:ccfr0:scrn0:akgenalg} will produce the | Algorithm \ref{alg:ccfr0:scrn0:akgenalg} will produce the |
1254 | same $[a_0; a_1, \ldots{} , a_n]$ for any $ia/ib$, i.e. all rational numbers | same $[a_0; a_1, \ldots{} , a_n]$ for any $ia/ib$, i.e. all rational numbers |
1255 | which are equivalent in value will generate the same continued fraction | which are equivalent in value will generate the same continued fraction |
1256 | representation (see Lemma \ref{lem:ccfr0:scrn0:aoverbneednotbeirreducible}). | representation (see Lemma \ref{lem:ccfr0:scrn0:aoverbneednotbeirreducible}). |
1257 | ||
1258 | It was hinted in the introduction (Section | It was hinted in the introduction (Section |
1259 | \ref{ccfr0:sint0}, Footnote \ref{footnote:ccfr0:sint0:00}) | \ref{ccfr0:sint0}, Footnote \ref{footnote:ccfr0:sint0:00}) |
1260 | that, except in the case of representing an integer, it is | that, except in the case of representing an integer, it is |
1261 | not allowed for the final partial quotient $a_n$ to be 1. | not allowed for the final partial quotient $a_n$ to be 1. |
1262 | We now explain the reasons why this must be disallowed. First, | We now explain the reasons why this must be disallowed. First, |
1263 | if $a_n = 1$, then $a_{n-1}$ can be increased by 1 and | if $a_n = 1$, then $a_{n-1}$ can be increased by 1 and |
1264 | the continued fraction can be reduced in order | the continued fraction can be reduced in order |
1265 | by 1 and while still preserving its value. For example, it | by 1 and while still preserving its value. For example, it |
1266 | can easily be verified that $[1;2,3,3,1]$ and $[1;2,3,4]$ | can easily be verified that $[1;2,3,3,1]$ and $[1;2,3,4]$ |
1267 | represent the same number. However, this observation alone | represent the same number. However, this observation alone |
1268 | is not enough to recommend a canonical form---this observation | is not enough to recommend a canonical form---this observation |
1269 | does not suggest that $[1;2,3,4]$ should be preferred | does not suggest that $[1;2,3,4]$ should be preferred |
1270 | over $[1;2,3,3,1]$. However, what \emph{can} be noted is that | over $[1;2,3,3,1]$. However, what \emph{can} be noted is that |
1271 | that a continued fraction representation with $a_n=1$, $n>0$ | that a continued fraction representation with $a_n=1$, $n>0$ |
1272 | cannot be attained using Algorithm \ref{alg:ccfr0:scrn0:akgenalg} or | cannot be attained using Algorithm \ref{alg:ccfr0:scrn0:akgenalg} or |
1273 | (\ref{eq:ccfr0:scrn0:00a}) through (\ref{eq:ccfr0:scrn0:00e}), | (\ref{eq:ccfr0:scrn0:00a}) through (\ref{eq:ccfr0:scrn0:00e}), |
1274 | because a form with $a_n=1$, $n>0$ violates the assumption that | because a form with $a_n=1$, $n>0$ violates the assumption that |
1275 | successive remainders are ever-decreasing (see Eq. | successive remainders are ever-decreasing (see Eq. |
1276 | \ref{eq:lem:ccfr0:scrn0:alwaysterminates}). The property that | \ref{eq:lem:ccfr0:scrn0:alwaysterminates}). The property that |
1277 | remainders are ever-decreasing is a necessary condition in | remainders are ever-decreasing is a necessary condition in |
1278 | proofs of some important properties, and so requiring | proofs of some important properties, and so requiring |
1279 | that $a_n \neq{} 1$, $n>0$ | that $a_n \neq{} 1$, $n>0$ |
1280 | is the most natural convention for a canonical form. | is the most natural convention for a canonical form. |
1281 | ||
1282 | \begin{vworklemmastatement} | \begin{vworklemmastatement} |
1283 | \label{lem:ccfr0:scrn0:aoverbneednotbeirreducible} | \label{lem:ccfr0:scrn0:aoverbneednotbeirreducible} |
1284 | Algorithm \ref{alg:ccfr0:scrn0:akgenalg} will | Algorithm \ref{alg:ccfr0:scrn0:akgenalg} will |
1285 | produce the same result | produce the same result |
1286 | $[a_0; a_1, \ldots{} , a_n]$ for any | $[a_0; a_1, \ldots{} , a_n]$ for any |
1287 | $ia/ib$, i.e. $a/b$ need not be reduced before the algorithm | $ia/ib$, i.e. $a/b$ need not be reduced before the algorithm |
1288 | is applied. | is applied. |
1289 | \end{vworklemmastatement} | \end{vworklemmastatement} |
1290 | \begin{vworklemmaproof} | \begin{vworklemmaproof} |
1291 | Assume that $a/b$ is irreducible, and that $ia/ib$, | Assume that $a/b$ is irreducible, and that $ia/ib$, |
1292 | $i \in \{ 2, 3, \ldots \}$ is used as input to | $i \in \{ 2, 3, \ldots \}$ is used as input to |
1293 | the algorithm. By definition, any rational number | the algorithm. By definition, any rational number |
1294 | with the same value as $a/b$ is of the form $ia/ib$, | with the same value as $a/b$ is of the form $ia/ib$, |
1295 | $i \in \vworkintsetpos$. | $i \in \vworkintsetpos$. |
1296 | Note that (\ref{eq:ccfr0:scrn0:00a}) through | Note that (\ref{eq:ccfr0:scrn0:00a}) through |
1297 | (\ref{eq:ccfr0:scrn0:00e}) ``scale up'', while still producing | (\ref{eq:ccfr0:scrn0:00e}) ``scale up'', while still producing |
1298 | the same partial quotients $[a_0; a_1, \ldots{} , a_n]$. | the same partial quotients $[a_0; a_1, \ldots{} , a_n]$. |
1299 | Specifically: | Specifically: |
1300 | ||
1301 | \begin{equation} | \begin{equation} |
1302 | \label{eq:ccfr0:scrn0:10a} | \label{eq:ccfr0:scrn0:10a} |
1303 | \frac{ia}{ib} | \frac{ia}{ib} |
1304 | = | = |
1305 | a_0 + \frac{ir_0}{ib} | a_0 + \frac{ir_0}{ib} |
1306 | = | = |
1307 | a_0 + \frac{1}{\frac{ib}{ir_0}} | a_0 + \frac{1}{\frac{ib}{ir_0}} |
1308 | \end{equation} | \end{equation} |
1309 | ||
1310 | \begin{equation} | \begin{equation} |
1311 | \label{eq:ccfr0:scrn0:10b} | \label{eq:ccfr0:scrn0:10b} |
1312 | \frac{ib}{ir_0} | \frac{ib}{ir_0} |
1313 | = | = |
1314 | a_1 + \frac{ir_1}{ir_0} | a_1 + \frac{ir_1}{ir_0} |
1315 | \end{equation} | \end{equation} |
1316 | ||
1317 | \begin{equation} | \begin{equation} |
1318 | \label{eq:ccfr0:scrn0:10c} | \label{eq:ccfr0:scrn0:10c} |
1319 | \frac{ir_0}{ir_1} | \frac{ir_0}{ir_1} |
1320 | = | = |
1321 | a_2 + \frac{ir_2}{ir_1} | a_2 + \frac{ir_2}{ir_1} |
1322 | \end{equation} | \end{equation} |
1323 | ||
1324 | \noindent{}Finally, nearing the termination of | \noindent{}Finally, nearing the termination of |
1325 | Algorithm \ref{alg:ccfr0:scrn0:akgenalg}: | Algorithm \ref{alg:ccfr0:scrn0:akgenalg}: |
1326 | ||
1327 | \begin{equation} | \begin{equation} |
1328 | \label{eq:ccfr0:scrn0:10d} | \label{eq:ccfr0:scrn0:10d} |
1329 | \frac{ir_{n-3}}{ir_{n-2}} | \frac{ir_{n-3}}{ir_{n-2}} |
1330 | = | = |
1331 | a_{n-1} + \frac{ir_{n-1}}{ir_{n-2}} | a_{n-1} + \frac{ir_{n-1}}{ir_{n-2}} |
1332 | \end{equation} | \end{equation} |
1333 | ||
1334 | \begin{equation} | \begin{equation} |
1335 | \label{eq:ccfr0:scrn0:10e} | \label{eq:ccfr0:scrn0:10e} |
1336 | \frac{ir_{n-2}}{ir_{n-1}} | \frac{ir_{n-2}}{ir_{n-1}} |
1337 | = | = |
1338 | a_n | a_n |
1339 | \end{equation} | \end{equation} |
1340 | ||
1341 | Thus, it is easy to show that Algorithm \ref{alg:ccfr0:scrn0:akgenalg} | Thus, it is easy to show that Algorithm \ref{alg:ccfr0:scrn0:akgenalg} |
1342 | will produce the same continued fraction representation regardless of whether | will produce the same continued fraction representation regardless of whether |
1343 | the input to the algorithm is reduced. It is also easy to show that the | the input to the algorithm is reduced. It is also easy to show that the |
1344 | last non-zero remainder as the algorithm is applied ($r_{n-1}$, in | last non-zero remainder as the algorithm is applied ($r_{n-1}$, in |
1345 | Eqns. \ref{eq:ccfr0:scrn0:10d} and \ref{eq:ccfr0:scrn0:10e}) | Eqns. \ref{eq:ccfr0:scrn0:10d} and \ref{eq:ccfr0:scrn0:10e}) |
1346 | is the greatest common divisor of $ia$ and $ib$ (this is done | is the greatest common divisor of $ia$ and $ib$ (this is done |
1347 | in the proof of Algorithm \ref{alg:ccfr0:sega0:euclidsgcdalgorithm}). | in the proof of Algorithm \ref{alg:ccfr0:sega0:euclidsgcdalgorithm}). |
1348 | \end{vworklemmaproof} | \end{vworklemmaproof} |
1349 | %\vworklemmafooter{} | %\vworklemmafooter{} |
1350 | ||
1351 | \begin{vworklemmastatement} | \begin{vworklemmastatement} |
1352 | \label{lem:ccfr0:scrn0:cfrepresentationisunique} | \label{lem:ccfr0:scrn0:cfrepresentationisunique} |
1353 | So long as $a_n \neq 1$, $n>0$, a rational number $a/b$ has only | So long as $a_n \neq 1$, $n>0$, a rational number $a/b$ has only |
1354 | one [unique] continued fraction representation | one [unique] continued fraction representation |
1355 | $[a_0; a_1, \ldots{} , a_n]$. | $[a_0; a_1, \ldots{} , a_n]$. |
1356 | \end{vworklemmastatement} | \end{vworklemmastatement} |
1357 | \begin{vworklemmaproof} | \begin{vworklemmaproof} |
1358 | Assume that two different continued fractions, | Assume that two different continued fractions, |
1359 | $[a_0; a_1, \ldots{} , a_m]$ and | $[a_0; a_1, \ldots{} , a_m]$ and |
1360 | $[\overline{a_0}; \overline{a_1}, \ldots{} , \overline{a_n}]$, | $[\overline{a_0}; \overline{a_1}, \ldots{} , \overline{a_n}]$, |
1361 | correspond to the same rational number $a/b$. By | correspond to the same rational number $a/b$. By |
1362 | \emph{different}, we mean either that $m=n$ but | \emph{different}, we mean either that $m=n$ but |
1363 | $\exists i, a_i \neq \overline{a_i}$, or that $m \neq n$. | $\exists i, a_i \neq \overline{a_i}$, or that $m \neq n$. |
1364 | ||
1365 | Note that Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} | Note that Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} |
1366 | will map from any continued fraction to an irreducible rational | will map from any continued fraction to an irreducible rational |
1367 | number $a/b$. Assume we apply | number $a/b$. Assume we apply |
1368 | Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} to | Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} to |
1369 | $[a_0; a_1, \ldots{} , a_m]$ to produce $a/b$, and | $[a_0; a_1, \ldots{} , a_m]$ to produce $a/b$, and |
1370 | to $[\overline{a_0}; \overline{a_1}, \ldots{} , \overline{a_n}]$ | to $[\overline{a_0}; \overline{a_1}, \ldots{} , \overline{a_n}]$ |
1371 | to produce $\overline{a}/\overline{b}$. Because two irreducible | to produce $\overline{a}/\overline{b}$. Because two irreducible |
1372 | rational numbers are equal iff their components are | rational numbers are equal iff their components are |
1373 | equal, | equal, |
1374 | $[(a/b) = (\overline{a}/\overline{b})] \vworkhimp{} % | $[(a/b) = (\overline{a}/\overline{b})] \vworkhimp{} % |
1375 | [(a = \overline{a}) \wedge (b = \overline{b})]$. Because | [(a = \overline{a}) \wedge (b = \overline{b})]$. Because |
1376 | $a/b = \overline{a}/\overline{b}$, we denote both of these | $a/b = \overline{a}/\overline{b}$, we denote both of these |
1377 | numbers simply as $a/b$. | numbers simply as $a/b$. |
1378 | ||
1379 | By (\ref{eq:ccfr0:scrn0:11a}), | By (\ref{eq:ccfr0:scrn0:11a}), |
1380 | $a = a_0 b + r_0 = \overline{a_0} b + \overline{r_0}$, | $a = a_0 b + r_0 = \overline{a_0} b + \overline{r_0}$, |
1381 | $0 < r_0, \overline{r_0} < b$. Because | $0 < r_0, \overline{r_0} < b$. Because |
1382 | $r_0, \overline{r_0} < b$, there is only one combination | $r_0, \overline{r_0} < b$, there is only one combination |
1383 | of $a_0$ and $r_0$ or of $\overline{a_0}$ and | of $a_0$ and $r_0$ or of $\overline{a_0}$ and |
1384 | $\overline{r_0}$ that can result in $a$. Thus, we can conclude | $\overline{r_0}$ that can result in $a$. Thus, we can conclude |
1385 | that $a_0 = \overline{a_0}$ and | that $a_0 = \overline{a_0}$ and |
1386 | $r_0 = \overline{r_0}$. This type of reasoning, can be | $r_0 = \overline{r_0}$. This type of reasoning, can be |
1387 | carried ``downward'' inductively, each time fixing | carried ``downward'' inductively, each time fixing |
1388 | $a_{i}$ and $r_{i}$. Finally, we must conclude | $a_{i}$ and $r_{i}$. Finally, we must conclude |
1389 | that $(a/b = \overline{a}/\overline{b}) \vworkhimp % | that $(a/b = \overline{a}/\overline{b}) \vworkhimp % |
1390 | [a_0; a_1, \ldots{} , a_m] = % | [a_0; a_1, \ldots{} , a_m] = % |
1391 | [\overline{a_0}; \overline{a_1}, \ldots{} , \overline{a_n}]$ | [\overline{a_0}; \overline{a_1}, \ldots{} , \overline{a_n}]$ |
1392 | and that $m=n$. | and that $m=n$. |
1393 | \end{vworklemmaproof} | \end{vworklemmaproof} |
1394 | \begin{vworklemmaparsection}{Remarks} | \begin{vworklemmaparsection}{Remarks} |
1395 | The case of $a_n=1$, $n>0$ deserves further discussion. | The case of $a_n=1$, $n>0$ deserves further discussion. |
1396 | Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} will produce | Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} will produce |
1397 | an irreducible rational number even if | an irreducible rational number even if |
1398 | $a_n = 1$, $n>0$. How is it that uniqueness of representation can | $a_n = 1$, $n>0$. How is it that uniqueness of representation can |
1399 | be claimed when clearly, for example, $[2;3,4,2]$ and $[2;3,4,1,1]$ | be claimed when clearly, for example, $[2;3,4,2]$ and $[2;3,4,1,1]$ |
1400 | are the same number? The answer is that $a_n = 1$, $n>0$ requires | are the same number? The answer is that $a_n = 1$, $n>0$ requires |
1401 | that $r_{n-2}=r_{n-1}=1$, which violates the ``uniqueness'' assumption | that $r_{n-2}=r_{n-1}=1$, which violates the ``uniqueness'' assumption |
1402 | used in fixing $a_i$ and $r_i$ in the proof above---specifically note | used in fixing $a_i$ and $r_i$ in the proof above---specifically note |
1403 | that the condition $0<r_{n-1}<r_{n-2}$ in (\ref{eq:ccfr0:scrn0:11d}) | that the condition $0<r_{n-1}<r_{n-2}$ in (\ref{eq:ccfr0:scrn0:11d}) |
1404 | is violated. If one is allowed to violate the required | is violated. If one is allowed to violate the required |
1405 | ever-decreasing remainders, then $a_i$ and $r_i$ cannot | ever-decreasing remainders, then $a_i$ and $r_i$ cannot |
1406 | be uniquely fixed at each step of the proof, above. | be uniquely fixed at each step of the proof, above. |
1407 | \end{vworklemmaparsection} | \end{vworklemmaparsection} |
1408 | \vworklemmafooter{} | \vworklemmafooter{} |
1409 | ||
1410 | \index{continued fraction!intermediate fraction} | \index{continued fraction!intermediate fraction} |
1411 | An \emph{intermediate fraction} is a fraction represented | An \emph{intermediate fraction} is a fraction represented |
1412 | by the continued fraction representation of a $k$-th order | by the continued fraction representation of a $k$-th order |
1413 | convergent with the final partial quotient $a_k$ reduced | convergent with the final partial quotient $a_k$ reduced |
1414 | (this can naturally only be done when $a_k > 1$). As Khinchin | (this can naturally only be done when $a_k > 1$). As Khinchin |
1415 | points out (\cite{bibref:b:KhinchinClassic}, p. 14): | points out (\cite{bibref:b:KhinchinClassic}, p. 14): |
1416 | ``\emph{In arithmetic applications, these intermediate | ``\emph{In arithmetic applications, these intermediate |
1417 | fractions play an important role (though not as important | fractions play an important role (though not as important |
1418 | a role as the convergents)}''. | a role as the convergents)}''. |
1419 | ||
1420 | The intermediate fractions (of a $k$-th order convergent) | The intermediate fractions (of a $k$-th order convergent) |
1421 | form a monotonically increasing or decreasing sequence of | form a monotonically increasing or decreasing sequence of |
1422 | fractions (\cite{bibref:b:KhinchinClassic}, p. 13): | fractions (\cite{bibref:b:KhinchinClassic}, p. 13): |
1423 | ||
1424 | \begin{equation} | \begin{equation} |
1425 | \label{eq:ccfr0:scrn0:intermediatefrac01} | \label{eq:ccfr0:scrn0:intermediatefrac01} |
1426 | \frac{p_{k-2}}{q_{k-2}}, | \frac{p_{k-2}}{q_{k-2}}, |
1427 | \frac{p_{k-2} + p_{k-1}}{q_{k-2} + q_{k-1}}, | \frac{p_{k-2} + p_{k-1}}{q_{k-2} + q_{k-1}}, |
1428 | \frac{p_{k-2} + 2 p_{k-1}}{q_{k-2} + 2 q_{k-1}}, | \frac{p_{k-2} + 2 p_{k-1}}{q_{k-2} + 2 q_{k-1}}, |
1429 | \ldots{} , | \ldots{} , |
1430 | \frac{p_{k-2} + a_k p_{k-1}}{q_{k-2} + a_k q_{k-1}} | \frac{p_{k-2} + a_k p_{k-1}}{q_{k-2} + a_k q_{k-1}} |
1431 | = | = |
1432 | \frac{p_k}{q_k} . | \frac{p_k}{q_k} . |
1433 | \end{equation} | \end{equation} |
1434 | ||
1435 | Note in (\ref{eq:ccfr0:scrn0:intermediatefrac01}) that the | Note in (\ref{eq:ccfr0:scrn0:intermediatefrac01}) that the |
1436 | first and last fractions are not intermediate fractions (rather, they are | first and last fractions are not intermediate fractions (rather, they are |
1437 | convergents). | convergents). |
1438 | ||
1439 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1440 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1441 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1442 | \section{Euclid's GCD Algorithm} | \section{Euclid's GCD Algorithm} |
1443 | %Section tag: EGA0 | %Section tag: EGA0 |
1444 | \index{Euclid!GCD algorithm} | \index{Euclid!GCD algorithm} |
1445 | ||
1446 | The apparatus of continued fractions is closely related to Euclid's GCD | The apparatus of continued fractions is closely related to Euclid's GCD |
1447 | algorithm (in fact, historically, Euclid's GCD algorithm is considered | algorithm (in fact, historically, Euclid's GCD algorithm is considered |
1448 | a precursor of continued fractions). It was noted in | a precursor of continued fractions). It was noted in |
1449 | Lemma \ref{lem:ccfr0:scrn0:aoverbneednotbeirreducible} that the last non-zero | Lemma \ref{lem:ccfr0:scrn0:aoverbneednotbeirreducible} that the last non-zero |
1450 | remainder when Algorithm \ref{alg:ccfr0:scrn0:akgenalg} is applied | remainder when Algorithm \ref{alg:ccfr0:scrn0:akgenalg} is applied |
1451 | is the greatest common divisor of $a$ and $b$. In this section, we | is the greatest common divisor of $a$ and $b$. In this section, we |
1452 | present Euclid's algorithm, prove it, and show it similarity to | present Euclid's algorithm, prove it, and show it similarity to |
1453 | Algorithm \ref{alg:ccfr0:scrn0:akgenalg}. | Algorithm \ref{alg:ccfr0:scrn0:akgenalg}. |
1454 | ||
1455 | Knuth (\cite{bibref:b:knuthclassic2ndedvol2}, p. 335) presents some background | Knuth (\cite{bibref:b:knuthclassic2ndedvol2}, p. 335) presents some background |
1456 | information about Euclid's GCD algorithm: | information about Euclid's GCD algorithm: |
1457 | ||
1458 | \begin{quote} | \begin{quote} |
1459 | Euclid's algorithm is found in Book 7, Propositions 1 and 2 of his | Euclid's algorithm is found in Book 7, Propositions 1 and 2 of his |
1460 | \emph{Elements} (c. 300 B.C.), but it probably wasn't his own | \emph{Elements} (c. 300 B.C.), but it probably wasn't his own |
1461 | invention. Some scholars believe that the method was known up to | invention. Some scholars believe that the method was known up to |
1462 | 200 years earlier, at least in its subtractive form, and it | 200 years earlier, at least in its subtractive form, and it |
1463 | was almost certainly known to Eudoxus (c. 375 B.C.); see | was almost certainly known to Eudoxus (c. 375 B.C.); see |
1464 | K. von Fritz, \emph{Ann. Math.} (2) \textbf{46} 1945, 242-264. | K. von Fritz, \emph{Ann. Math.} (2) \textbf{46} 1945, 242-264. |
1465 | Aristotle (c. 330 B.C.) hinted at it in his \emph{Topics}, 158b, | Aristotle (c. 330 B.C.) hinted at it in his \emph{Topics}, 158b, |
1466 | 29-35. However, very little hard evidence about such early history | 29-35. However, very little hard evidence about such early history |
1467 | has survived [see. W. R. Knorr, \emph{The Evolution of the Euclidian | has survived [see. W. R. Knorr, \emph{The Evolution of the Euclidian |
1468 | Elements} (Dordrecht: 1975)]. | Elements} (Dordrecht: 1975)]. |
1469 | ||
1470 | We might call Euclid's method the granddaddy of all algorithms, because | We might call Euclid's method the granddaddy of all algorithms, because |
1471 | it is the oldest nontrivial algorithm that has survived to the present day. | it is the oldest nontrivial algorithm that has survived to the present day. |
1472 | (The chief rival for this honor is perhaps the ancient Egyptian method | (The chief rival for this honor is perhaps the ancient Egyptian method |
1473 | for multiplication, which was based on doubling and adding, and which forms | for multiplication, which was based on doubling and adding, and which forms |
1474 | the basis for efficient calculation of \emph{n}th powers as explained | the basis for efficient calculation of \emph{n}th powers as explained |
1475 | in section 4.6.3. \ldots{}) | in section 4.6.3. \ldots{}) |
1476 | \end{quote} | \end{quote} |
1477 | ||
1478 | \begin{vworkalgorithmstatementpar} | \begin{vworkalgorithmstatementpar} |
1479 | {Euclid's GCD Algorithm For Greatest Common Divisor Of Positive | {Euclid's GCD Algorithm For Greatest Common Divisor Of Positive |
1480 | Integers \mbox{\boldmath $a$} | Integers \mbox{\boldmath $a$} |
1481 | And \mbox{\boldmath $b$}}\hspace{-0.4em}\footnote{Knuth | And \mbox{\boldmath $b$}}\hspace{-0.4em}\footnote{Knuth |
1482 | (\cite{bibref:b:knuthclassic2ndedvol2}, pp. 336-337) distinguishes between the \emph{original} | (\cite{bibref:b:knuthclassic2ndedvol2}, pp. 336-337) distinguishes between the \emph{original} |
1483 | Euclidian algorithm and the \emph{modern} Euclidian algorithm. The algorithm presented here | Euclidian algorithm and the \emph{modern} Euclidian algorithm. The algorithm presented here |
1484 | is more closely patterned after the \emph{modern} Euclidian algorithm.} | is more closely patterned after the \emph{modern} Euclidian algorithm.} |
1485 | \label{alg:ccfr0:sega0:euclidsgcdalgorithm} | \label{alg:ccfr0:sega0:euclidsgcdalgorithm} |
1486 | \begin{alglvl0} | \begin{alglvl0} |
1487 | \item If ($a < b$), swap $a$ and $b$.\footnote{This step isn't strictly necessary, but is usually done | \item If ($a < b$), swap $a$ and $b$.\footnote{This step isn't strictly necessary, but is usually done |
1488 | to save one iteration.} | to save one iteration.} |
1489 | \item Repeat | \item Repeat |
1490 | \begin{alglvl1} | \begin{alglvl1} |
1491 | \item $r := a \; mod \; b$. | \item $r := a \; mod \; b$. |
1492 | \item If ($r = 0$), STOP. | \item If ($r = 0$), STOP. |
1493 | \item $a := b$. | \item $a := b$. |
1494 | \item $b := r$. | \item $b := r$. |
1495 | \end{alglvl1} | \end{alglvl1} |
1496 | \item \textbf{Exit condition:} $b$ will be the g.c.d. of $a$ and $b$. | \item \textbf{Exit condition:} $b$ will be the g.c.d. of $a$ and $b$. |
1497 | \end{alglvl0} | \end{alglvl0} |
1498 | \end{vworkalgorithmstatementpar} | \end{vworkalgorithmstatementpar} |
1499 | \begin{vworkalgorithmproof} | \begin{vworkalgorithmproof} |
1500 | Olds (\cite{bibref:b:OldsClassic}, pp. 16-17) shows the | Olds (\cite{bibref:b:OldsClassic}, pp. 16-17) shows the |
1501 | relationship between Algorithm | relationship between Algorithm |
1502 | \ref{alg:ccfr0:scrn0:akgenalg} and Euclid's algorithm, and presents | \ref{alg:ccfr0:scrn0:akgenalg} and Euclid's algorithm, and presents |
1503 | a proof, which is reproduced nearly verbatim here. | a proof, which is reproduced nearly verbatim here. |
1504 | ||
1505 | First, note that $d$ is the GCD of $a$ and $b$ iff: | First, note that $d$ is the GCD of $a$ and $b$ iff: |
1506 | ||
1507 | \begin{itemize} | \begin{itemize} |
1508 | \item (Necessary Condition I) $d$ divides both $a$ and $b$, and | \item (Necessary Condition I) $d$ divides both $a$ and $b$, and |
1509 | \item (Necessary Condition II) any common divisor $c$ of $a$ and $b$ divides $d$. | \item (Necessary Condition II) any common divisor $c$ of $a$ and $b$ divides $d$. |
1510 | \end{itemize} | \end{itemize} |
1511 | ||
1512 | Essentially, we will prove that the final non-zero remainder | Essentially, we will prove that the final non-zero remainder |
1513 | when the algorithm is applied meets the two criteria above, | when the algorithm is applied meets the two criteria above, |
1514 | and hence must be the GCD of $a$ and $b$. | and hence must be the GCD of $a$ and $b$. |
1515 | ||
1516 | Note that (\ref{eq:ccfr0:scrn0:00a}) through | Note that (\ref{eq:ccfr0:scrn0:00a}) through |
1517 | (\ref{eq:ccfr0:scrn0:00e}) can be rewritten as | (\ref{eq:ccfr0:scrn0:00e}) can be rewritten as |
1518 | (\ref{eq:ccfr0:scrn0:11a}) through | (\ref{eq:ccfr0:scrn0:11a}) through |
1519 | (\ref{eq:ccfr0:scrn0:11e}), which make them | (\ref{eq:ccfr0:scrn0:11e}), which make them |
1520 | consistent with the form Olds' presents. | consistent with the form Olds' presents. |
1521 | ||
1522 | \begin{eqnarray} | \begin{eqnarray} |
1523 | \label{eq:ccfr0:scrn0:11a} | \label{eq:ccfr0:scrn0:11a} |
1524 | a = a_0 b + r_0, && 0 < r_0 < b \\ | a = a_0 b + r_0, && 0 < r_0 < b \\ |
1525 | \label{eq:ccfr0:scrn0:11b} | \label{eq:ccfr0:scrn0:11b} |
1526 | b = a_1 r_0 + r_1, && 0 < r_1 < r_0 \\ | b = a_1 r_0 + r_1, && 0 < r_1 < r_0 \\ |
1527 | \label{eq:ccfr0:scrn0:11c} | \label{eq:ccfr0:scrn0:11c} |
1528 | r_0 = a_2 r_1 + r_2, && 0 < r_2 < r_1 \\ | r_0 = a_2 r_1 + r_2, && 0 < r_2 < r_1 \\ |
1529 | \ldots{}\hspace{-1.67em}&& \nonumber \\ | \ldots{}\hspace{-1.67em}&& \nonumber \\ |
1530 | \label{eq:ccfr0:scrn0:11d} | \label{eq:ccfr0:scrn0:11d} |
1531 | r_{n-3} = a_{n-1} r_{n-2} + r_{n-1}, && 0 < r_{n-1} < r_{n-2} \\ | r_{n-3} = a_{n-1} r_{n-2} + r_{n-1}, && 0 < r_{n-1} < r_{n-2} \\ |
1532 | \label{eq:ccfr0:scrn0:11e} | \label{eq:ccfr0:scrn0:11e} |
1533 | r_{n-2} = a_n r_{n-1} + 0, && 0 = r_n | r_{n-2} = a_n r_{n-1} + 0, && 0 = r_n |
1534 | \end{eqnarray} | \end{eqnarray} |
1535 | ||
1536 | First, we will show that \emph{Necessary Condition I}, above, is met. | First, we will show that \emph{Necessary Condition I}, above, is met. |
1537 | Note from (\ref{eq:ccfr0:scrn0:11e}) that $r_{n-1} \vworkdivides r_{n-2}$, | Note from (\ref{eq:ccfr0:scrn0:11e}) that $r_{n-1} \vworkdivides r_{n-2}$, |
1538 | since $r_{n-2}$ is an integer multiple of $r_{n-1}$. Note from | since $r_{n-2}$ is an integer multiple of $r_{n-1}$. Note from |
1539 | (\ref{eq:ccfr0:scrn0:11d}) that $r_{n-1} \vworkdivides r_{n-3}$, since | (\ref{eq:ccfr0:scrn0:11d}) that $r_{n-1} \vworkdivides r_{n-3}$, since |
1540 | $r_{n-3}$ is also an integer multiple of $r_{n-1}$. This logic can | $r_{n-3}$ is also an integer multiple of $r_{n-1}$. This logic can |
1541 | be carried ``upward'' in the set of equations represented | be carried ``upward'' in the set of equations represented |
1542 | by (\ref{eq:ccfr0:scrn0:11a}) through (\ref{eq:ccfr0:scrn0:11e}), | by (\ref{eq:ccfr0:scrn0:11a}) through (\ref{eq:ccfr0:scrn0:11e}), |
1543 | and we can finally conclude that $r_{n-1} \vworkdivides b$ and | and we can finally conclude that $r_{n-1} \vworkdivides b$ and |
1544 | $r_{n-1} \vworkdivides a$. This proves \emph{Necessary Condition I}. | $r_{n-1} \vworkdivides a$. This proves \emph{Necessary Condition I}. |
1545 | ||
1546 | Second, we will show that \emph{Necessary Condition II}, above, is met. | Second, we will show that \emph{Necessary Condition II}, above, is met. |
1547 | This time, in (\ref{eq:ccfr0:scrn0:11a}) through (\ref{eq:ccfr0:scrn0:11e}), | This time, in (\ref{eq:ccfr0:scrn0:11a}) through (\ref{eq:ccfr0:scrn0:11e}), |
1548 | we work top-down rather than bottom-up. Assume that $c$ is a divisor | we work top-down rather than bottom-up. Assume that $c$ is a divisor |
1549 | of $a$ and a divisor of $b$. Then, from the form of (\ref{eq:ccfr0:scrn0:11a}), | of $a$ and a divisor of $b$. Then, from the form of (\ref{eq:ccfr0:scrn0:11a}), |
1550 | $c$ divides $r_0$.\footnote{This implication may be counterintuitive | $c$ divides $r_0$.\footnote{This implication may be counterintuitive |
1551 | at first glance. It concerns "reachability" of linear combinations | at first glance. It concerns "reachability" of linear combinations |
1552 | of integers with a common divisor. Specifically, if | of integers with a common divisor. Specifically, if |
1553 | $a$ and $b$ have a common divisor $c$, any linear combination | $a$ and $b$ have a common divisor $c$, any linear combination |
1554 | $ia + jb$, ($i,j \in \vworkintset$), can ``reach'' only multiples of $c$. | $ia + jb$, ($i,j \in \vworkintset$), can ``reach'' only multiples of $c$. |
1555 | In (\ref{eq:ccfr0:scrn0:11a}), $(1)(a)+(-a_0)(b)=r_0$, thus | In (\ref{eq:ccfr0:scrn0:11a}), $(1)(a)+(-a_0)(b)=r_0$, thus |
1556 | $r_0$ must be a multiple of $c$. An identical argument applies for | $r_0$ must be a multiple of $c$. An identical argument applies for |
1557 | (\ref{eq:ccfr0:scrn0:11a}) through (\ref{eq:ccfr0:scrn0:11e}).} | (\ref{eq:ccfr0:scrn0:11a}) through (\ref{eq:ccfr0:scrn0:11e}).} |
1558 | Similarly, from the form of (\ref{eq:ccfr0:scrn0:11b}), | Similarly, from the form of (\ref{eq:ccfr0:scrn0:11b}), |
1559 | $c$ divides $r_1$. This rationale can be carried ``downward'' to finally | $c$ divides $r_1$. This rationale can be carried ``downward'' to finally |
1560 | conclude that $c$ divides $r_{n-1}$. Thus, | conclude that $c$ divides $r_{n-1}$. Thus, |
1561 | $(c \vworkdivides a) \wedge (c \vworkdivides b) \vworkhimp (c \vworkdivides r_{n-1})$, | $(c \vworkdivides a) \wedge (c \vworkdivides b) \vworkhimp (c \vworkdivides r_{n-1})$, |
1562 | where $r_{n-1}$ is the last non-zero remainder. This proves | where $r_{n-1}$ is the last non-zero remainder. This proves |
1563 | \emph{Necessary Condition II}. | \emph{Necessary Condition II}. |
1564 | ||
1565 | Thus, $r_{n-1}$ is the GCD of $a$ and $b$. | Thus, $r_{n-1}$ is the GCD of $a$ and $b$. |
1566 | \end{vworkalgorithmproof} | \end{vworkalgorithmproof} |
1567 | \begin{vworkalgorithmparsection}{Remarks} | \begin{vworkalgorithmparsection}{Remarks} |
1568 | It is easy to observe that the only difference between | It is easy to observe that the only difference between |
1569 | Algorithm \ref{alg:ccfr0:scrn0:akgenalg} and | Algorithm \ref{alg:ccfr0:scrn0:akgenalg} and |
1570 | Algorithm \ref{alg:ccfr0:sega0:euclidsgcdalgorithm} is | Algorithm \ref{alg:ccfr0:sega0:euclidsgcdalgorithm} is |
1571 | that Algorithm \ref{alg:ccfr0:scrn0:akgenalg} records the | that Algorithm \ref{alg:ccfr0:scrn0:akgenalg} records the |
1572 | quotient of each division, whereas | quotient of each division, whereas |
1573 | Algorithm \ref{alg:ccfr0:sega0:euclidsgcdalgorithm} | Algorithm \ref{alg:ccfr0:sega0:euclidsgcdalgorithm} |
1574 | does not. | does not. |
1575 | \end{vworkalgorithmparsection} | \end{vworkalgorithmparsection} |
1576 | %\vworkalgorithmfooter{} | %\vworkalgorithmfooter{} |
1577 | ||
1578 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
1579 | \label{ex:ccfr0:sega0:01} | \label{ex:ccfr0:sega0:01} |
1580 | Use Euclid's algorithm to find the greatest common divisor | Use Euclid's algorithm to find the greatest common divisor |
1581 | of 1,736,651 and 26,023. | of 1,736,651 and 26,023. |
1582 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
1583 | \begin{vworkexampleparsection}{Solution} | \begin{vworkexampleparsection}{Solution} |
1584 | \begin{table} | \begin{table} |
1585 | \caption{Euclid's Algorithm Applied To Find Greatest Common Divisor Of 1,736,651 and 26,023 | \caption{Euclid's Algorithm Applied To Find Greatest Common Divisor Of 1,736,651 and 26,023 |
1586 | (Example \ref{ex:ccfr0:sega0:01})} | (Example \ref{ex:ccfr0:sega0:01})} |
1587 | \label{tbl:ex:ccfr0:sega0:01} | \label{tbl:ex:ccfr0:sega0:01} |
1588 | \begin{center} | \begin{center} |
1589 | \begin{tabular}{|c|c|c|c|} | \begin{tabular}{|c|c|c|c|} |
1590 | \hline | \hline |
1591 | \small{Iteration} & \small{$a$} & \small{$b$} & \small{$r : = a \; mod \; b$} \\ | \small{Iteration} & \small{$a$} & \small{$b$} & \small{$r : = a \; mod \; b$} \\ |
1592 | \hline | \hline |
1593 | \hline | \hline |
1594 | \small{1} & \small{1,736,651} & \small{26,023} & \small{19,133} \\ | \small{1} & \small{1,736,651} & \small{26,023} & \small{19,133} \\ |
1595 | \hline | \hline |
1596 | \small{2} & \small{26,023} & \small{19,133} & \small{6,890} \\ | \small{2} & \small{26,023} & \small{19,133} & \small{6,890} \\ |
1597 | \hline | \hline |
1598 | \small{3} & \small{19,133} & \small{6,890} & \small{5,353} \\ | \small{3} & \small{19,133} & \small{6,890} & \small{5,353} \\ |
1599 | \hline | \hline |
1600 | \small{4} & \small{6,890} & \small{5,353} & \small{1,537} \\ | \small{4} & \small{6,890} & \small{5,353} & \small{1,537} \\ |
1601 | \hline | \hline |
1602 | \small{5} & \small{5,353} & \small{1,537} & \small{742} \\ | \small{5} & \small{5,353} & \small{1,537} & \small{742} \\ |
1603 | \hline | \hline |
1604 | \small{6} & \small{1,537} & \small{742} & \small{53} \\ | \small{6} & \small{1,537} & \small{742} & \small{53} \\ |
1605 | \hline | \hline |
1606 | \small{7} & \small{742} & \small{53} & \small{0} \\ | \small{7} & \small{742} & \small{53} & \small{0} \\ |
1607 | \hline | \hline |
1608 | \end{tabular} | \end{tabular} |
1609 | \end{center} | \end{center} |
1610 | \end{table} | \end{table} |
1611 | Table \ref{tbl:ex:ccfr0:sega0:01} shows the application of | Table \ref{tbl:ex:ccfr0:sega0:01} shows the application of |
1612 | Algorithm \ref{alg:ccfr0:sega0:euclidsgcdalgorithm} (Euclid's | Algorithm \ref{alg:ccfr0:sega0:euclidsgcdalgorithm} (Euclid's |
1613 | GCD algorithm) to find the greatest common divisor | GCD algorithm) to find the greatest common divisor |
1614 | of 1,736,651 and 26,023. The last non-zero remainder (and hence | of 1,736,651 and 26,023. The last non-zero remainder (and hence |
1615 | the greatest common divisor) is 53. | the greatest common divisor) is 53. |
1616 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
1617 | \begin{vworkexampleparsection}{Remarks} | \begin{vworkexampleparsection}{Remarks} |
1618 | The prime factorization of 1,736,651 is $151 \times 53 \times 31 \times 7$, and | The prime factorization of 1,736,651 is $151 \times 53 \times 31 \times 7$, and |
1619 | the prime factorization of 26,023 is $491 \times 53$, which is consistent | the prime factorization of 26,023 is $491 \times 53$, which is consistent |
1620 | with the result in Table \ref{tbl:ex:ccfr0:sega0:01}. | with the result in Table \ref{tbl:ex:ccfr0:sega0:01}. |
1621 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
1622 | \vworkexamplefooter{} | \vworkexamplefooter{} |
1623 | ||
1624 | ||
1625 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1626 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1627 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1628 | \section[CF Representation Of Irrationals] | \section[CF Representation Of Irrationals] |
1629 | {Continued Fraction Representation Of Irrational Numbers} | {Continued Fraction Representation Of Irrational Numbers} |
1630 | %Section tag: CIN0 | %Section tag: CIN0 |
1631 | \label{ccfr0:scin0} | \label{ccfr0:scin0} |
1632 | ||
1633 | \index{continued fraction!irrational numbers@of irrational numbers} | \index{continued fraction!irrational numbers@of irrational numbers} |
1634 | \index{irrational number!continued fraction representation of}Irrational | \index{irrational number!continued fraction representation of}Irrational |
1635 | numbers (such as $\sqrt{2}$ or $\pi$) necessarily have infinite continued | numbers (such as $\sqrt{2}$ or $\pi$) necessarily have infinite continued |
1636 | fraction representations (i.e. the representations do not terminate). This is clear, since | fraction representations (i.e. the representations do not terminate). This is clear, since |
1637 | Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} gives a concrete procedure | Theorem \ref{thm:ccfr0:scnv0:canonicalconvergentconstruction} gives a concrete procedure |
1638 | for constructing a rational number from any \emph{finite} continued fraction; | for constructing a rational number from any \emph{finite} continued fraction; |
1639 | therefore a continued fraction corresponding to an irrational number | therefore a continued fraction corresponding to an irrational number |
1640 | cannot be finite. | cannot be finite. |
1641 | ||
1642 | The algorithm for determining the partial quotients of an irrational number is | The algorithm for determining the partial quotients of an irrational number is |
1643 | awkward, because it is a symbolic (rather than a numerical) algorithm. | awkward, because it is a symbolic (rather than a numerical) algorithm. |
1644 | We present the algorithm here for perspective and completeness, although it | We present the algorithm here for perspective and completeness, although it |
1645 | is not often useful in practical engineering work. In practical work, an | is not often useful in practical engineering work. In practical work, an |
1646 | ordinary handheld calculator will supply a real number to far more precision | ordinary handheld calculator will supply a real number to far more precision |
1647 | than necessary, and the displayed real number can be converted to a rational | than necessary, and the displayed real number can be converted to a rational |
1648 | number for the application of Algorithm \ref{alg:ccfr0:scrn0:akgenalg}. | number for the application of Algorithm \ref{alg:ccfr0:scrn0:akgenalg}. |
1649 | For practical work, it is rarely necessary to apply a Algorithm | For practical work, it is rarely necessary to apply a Algorithm |
1650 | \ref{alg:ccfr0:scin0:symboliccfalgorithm}. | \ref{alg:ccfr0:scin0:symboliccfalgorithm}. |
1651 | ||
1652 | The symbolic algorithm for determining the continued fraction partial quotients | The symbolic algorithm for determining the continued fraction partial quotients |
1653 | of a real number is a recursive process very similar to the algorithm for | of a real number is a recursive process very similar to the algorithm for |
1654 | determining the continued fraction partial quotients of a rational number. | determining the continued fraction partial quotients of a rational number. |
1655 | The essential activity is choosing the largest possible integer $a_i$ in each | The essential activity is choosing the largest possible integer $a_i$ in each |
1656 | iteration. | iteration. |
1657 | ||
1658 | Algorithm \ref{alg:ccfr0:scin0:symboliccfalgorithm} begins by choosing | Algorithm \ref{alg:ccfr0:scin0:symboliccfalgorithm} begins by choosing |
1659 | the largest integer $a_0$ not larger than $x$, then expressing $x$ as | the largest integer $a_0$ not larger than $x$, then expressing $x$ as |
1660 | ||
1661 | \begin{equation} | \begin{equation} |
1662 | x = a_0 + \frac{1}{x_1} . | x = a_0 + \frac{1}{x_1} . |
1663 | \end{equation} | \end{equation} |
1664 | ||
1665 | \noindent{}With $a_0$ chosen, $x_1$ can then be expressed as | \noindent{}With $a_0$ chosen, $x_1$ can then be expressed as |
1666 | ||
1667 | \begin{equation} | \begin{equation} |
1668 | x_1 = \frac{1}{x - a_0} . | x_1 = \frac{1}{x - a_0} . |
1669 | \end{equation} | \end{equation} |
1670 | ||
1671 | \noindent{}$x_1$ can then be expressed as | \noindent{}$x_1$ can then be expressed as |
1672 | ||
1673 | \begin{equation} | \begin{equation} |
1674 | x_1 = a_1 + \frac{1}{x_2} , | x_1 = a_1 + \frac{1}{x_2} , |
1675 | \end{equation} | \end{equation} |
1676 | ||
1677 | \noindent{}and $a_1$, the largest integer not larger than $x_1$, | \noindent{}and $a_1$, the largest integer not larger than $x_1$, |
1678 | can be chosen. | can be chosen. |
1679 | This process can be continued indefinitely (with an irrational $x$, it won't terminate) | This process can be continued indefinitely (with an irrational $x$, it won't terminate) |
1680 | to determine as many partial quotients as desired. | to determine as many partial quotients as desired. |
1681 | ||
1682 | \begin{vworkalgorithmstatementpar} | \begin{vworkalgorithmstatementpar} |
1683 | {Symbolic Algorithm For Obtaining | {Symbolic Algorithm For Obtaining |
1684 | Continued Fraction Representation Of An Irrational Number | Continued Fraction Representation Of An Irrational Number |
1685 | \mbox{\boldmath $x$}} | \mbox{\boldmath $x$}} |
1686 | \label{alg:ccfr0:scin0:symboliccfalgorithm} | \label{alg:ccfr0:scin0:symboliccfalgorithm} |
1687 | \begin{alglvl0} | \begin{alglvl0} |
1688 | \item $x_0 := x$ (the real number whose partial quotients are desired). | \item $x_0 := x$ (the real number whose partial quotients are desired). |
1689 | \item $k := -1$. | \item $k := -1$. |
1690 | \item Repeat | \item Repeat |
1691 | \begin{alglvl1} | \begin{alglvl1} |
1692 | \item $k := k + 1$. | \item $k := k + 1$. |
1693 | \item $a_k := \lfloor x_k \rfloor$. | \item $a_k := \lfloor x_k \rfloor$. |
1694 | \item $x_{k+1} := \displaystyle{\frac{1}{x_k - a_k}}$. | \item $x_{k+1} := \displaystyle{\frac{1}{x_k - a_k}}$. |
1695 | \end{alglvl1} | \end{alglvl1} |
1696 | \item Until (as many partial quotients as desired are obtained). | \item Until (as many partial quotients as desired are obtained). |
1697 | \end{alglvl0} | \end{alglvl0} |
1698 | \end{vworkalgorithmstatementpar} | \end{vworkalgorithmstatementpar} |
1699 | %\vworkalgorithmfooter{} | %\vworkalgorithmfooter{} |
1700 | ||
1701 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
1702 | \label{ex:ccfr0:scin0:symboliccfalgorithmexample} | \label{ex:ccfr0:scin0:symboliccfalgorithmexample} |
1703 | Find the first several continued fraction partial quotients of $\sqrt{3}$. | Find the first several continued fraction partial quotients of $\sqrt{3}$. |
1704 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
1705 | \begin{vworkexampleparsection}{Solution} | \begin{vworkexampleparsection}{Solution} |
1706 | Applying Algorithm \ref{alg:ccfr0:scin0:symboliccfalgorithm}: | Applying Algorithm \ref{alg:ccfr0:scin0:symboliccfalgorithm}: |
1707 | ||
1708 | \begin{equation} | \begin{equation} |
1709 | x_0 := x = \sqrt{3} | x_0 := x = \sqrt{3} |
1710 | \end{equation} | \end{equation} |
1711 | ||
1712 | \begin{equation} | \begin{equation} |
1713 | k := -1 | k := -1 |
1714 | \end{equation} | \end{equation} |
1715 | ||
1716 | \begin{equation} | \begin{equation} |
1717 | k := k+1 = 0 | k := k+1 = 0 |
1718 | \end{equation} | \end{equation} |
1719 | ||
1720 | \begin{equation} | \begin{equation} |
1721 | a_0 := \lfloor x_0 \rfloor = \lfloor \sqrt{3} \rfloor = 1 | a_0 := \lfloor x_0 \rfloor = \lfloor \sqrt{3} \rfloor = 1 |
1722 | \end{equation} | \end{equation} |
1723 | ||
1724 | \begin{equation} | \begin{equation} |
1725 | x_1 := \frac{1}{x_0 - a_0} | x_1 := \frac{1}{x_0 - a_0} |
1726 | = \frac{1}{\sqrt{3}-1} | = \frac{1}{\sqrt{3}-1} |
1727 | = \frac{\sqrt{3}+1}{2} | = \frac{\sqrt{3}+1}{2} |
1728 | \end{equation} | \end{equation} |
1729 | ||
1730 | \begin{equation} | \begin{equation} |
1731 | k := k + 1 = 1 | k := k + 1 = 1 |
1732 | \end{equation} | \end{equation} |
1733 | ||
1734 | \begin{equation} | \begin{equation} |
1735 | a_1 := \lfloor x_1 \rfloor = | a_1 := \lfloor x_1 \rfloor = |
1736 | \left\lfloor {\frac{\sqrt{3}+1}{2}} \right\rfloor = 1 | \left\lfloor {\frac{\sqrt{3}+1}{2}} \right\rfloor = 1 |
1737 | \end{equation} | \end{equation} |
1738 | ||
1739 | \begin{equation} | \begin{equation} |
1740 | x_2 := \frac{1}{x_1 - 1} | x_2 := \frac{1}{x_1 - 1} |
1741 | = \frac{1}{\frac{\sqrt{3}+1}{2}-1} | = \frac{1}{\frac{\sqrt{3}+1}{2}-1} |
1742 | = \sqrt{3}+1 | = \sqrt{3}+1 |
1743 | \end{equation} | \end{equation} |
1744 | ||
1745 | \begin{equation} | \begin{equation} |
1746 | k := k + 1 = 2 | k := k + 1 = 2 |
1747 | \end{equation} | \end{equation} |
1748 | ||
1749 | \begin{equation} | \begin{equation} |
1750 | a_2 := \lfloor x_2 \rfloor = \lfloor \sqrt{3}+1 \rfloor = 2 | a_2 := \lfloor x_2 \rfloor = \lfloor \sqrt{3}+1 \rfloor = 2 |
1751 | \end{equation} | \end{equation} |
1752 | ||
1753 | \begin{equation} | \begin{equation} |
1754 | x_3 := \frac{1}{(\sqrt{3}+1)-2} = \frac{\sqrt{3}+1}{2} | x_3 := \frac{1}{(\sqrt{3}+1)-2} = \frac{\sqrt{3}+1}{2} |
1755 | \end{equation} | \end{equation} |
1756 | ||
1757 | Note that $x_3 = x_1$, so the algorithm will repeat with | Note that $x_3 = x_1$, so the algorithm will repeat with |
1758 | $a_3=1$, $a_4=2$, $a_5=1$, $a_6=2$, etc. Thus, the continued | $a_3=1$, $a_4=2$, $a_5=1$, $a_6=2$, etc. Thus, the continued |
1759 | fraction representation of $\sqrt{3}$ is | fraction representation of $\sqrt{3}$ is |
1760 | $[1;1,2,1,2,1,2, \ldots{}]$ = $[1;\overline{1,2}]$. | $[1;1,2,1,2,1,2, \ldots{}]$ = $[1;\overline{1,2}]$. |
1761 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
1762 | \begin{vworkexampleparsection}{Remarks} | \begin{vworkexampleparsection}{Remarks} |
1763 | \index{continued fraction!repeating} | \index{continued fraction!repeating} |
1764 | It can be proved that all continued fractions that repeat or repeat | It can be proved that all continued fractions that repeat or repeat |
1765 | from some point onward | from some point onward |
1766 | represent real numbers of the form $\frac{P \pm \sqrt{D}}{Q}$, | represent real numbers of the form $\frac{P \pm \sqrt{D}}{Q}$, |
1767 | where $D \in \vworkintsetpos$ is not the square of an integer. | where $D \in \vworkintsetpos$ is not the square of an integer. |
1768 | It can also be shown | It can also be shown |
1769 | that all numbers of this form result in continued fractions that | that all numbers of this form result in continued fractions that |
1770 | repeat or repeat from some point onward. (See Olds | repeat or repeat from some point onward. (See Olds |
1771 | \cite{bibref:b:OldsClassic}, Chapter 4.) It is beyond the scope | \cite{bibref:b:OldsClassic}, Chapter 4.) It is beyond the scope |
1772 | of our interest in continued fractions to develop these properties. | of our interest in continued fractions to develop these properties. |
1773 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
1774 | \vworkexamplefooter{} | \vworkexamplefooter{} |
1775 | ||
1776 | ||
1777 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1778 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1779 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1780 | \section{Convergents As Best Approximations} | \section{Convergents As Best Approximations} |
1781 | %Section tag: CBA0 | %Section tag: CBA0 |
1782 | ||
1783 | Up until this point, we've presented general properties of continued | Up until this point, we've presented general properties of continued |
1784 | fractions and convergents without regard for practical applications. | fractions and convergents without regard for practical applications. |
1785 | In this section, we present results and algorithms to use the | In this section, we present results and algorithms to use the |
1786 | apparatus of continued fractions to obtain best rational approximations. | apparatus of continued fractions to obtain best rational approximations. |
1787 | ||
1788 | Although we don't dwell on other algorithms for locating best | Although we don't dwell on other algorithms for locating best |
1789 | rational approximations (we present only the single best | rational approximations (we present only the single best |
1790 | algorithm), it is worth noting that there are many naive algorithms | algorithm), it is worth noting that there are many naive algorithms |
1791 | for locating best rational approximations. These include: | for locating best rational approximations. These include: |
1792 | ||
1793 | \begin{itemize} | \begin{itemize} |
1794 | ||
1795 | \item Exhaustive search of the integer lattice | \item Exhaustive search of the integer lattice |
1796 | [$O(h_{MAX} k_{MAX})$]. | [$O(h_{MAX} k_{MAX})$]. |
1797 | ||
1798 | \item Building the Farey series starting at an | \item Building the Farey series starting at an |
1799 | integer [$O(max(h_{MAX}, k_{MAX})^2)$] | integer [$O(max(h_{MAX}, k_{MAX})^2)$] |
1800 | (see Algorithm \cfryzeroxrefhyphen{}\ref{alg:cfry0:sgfs0:02}). | (see Algorithm \cfryzeroxrefhyphen{}\ref{alg:cfry0:sgfs0:02}). |
1801 | ||
1802 | \item Building the Farey series starting at a rational number with | \item Building the Farey series starting at a rational number with |
1803 | a large prime denominator [$O(max(h_{MAX}, k_{MAX}))$]. | a large prime denominator [$O(max(h_{MAX}, k_{MAX}))$]. |
1804 | ||
1805 | \item Building the Stern-Brocot tree (see Section \ref{ccfr0:ssbt0}) | \item Building the Stern-Brocot tree (see Section \ref{ccfr0:ssbt0}) |
1806 | [$O(max(h_{MAX}, k_{MAX}))$]. | [$O(max(h_{MAX}, k_{MAX}))$]. |
1807 | ||
1808 | \end{itemize} | \end{itemize} |
1809 | ||
1810 | Although we don't justify it formally, | Although we don't justify it formally, |
1811 | the continued fraction algorithms presented here are | the continued fraction algorithms presented here are |
1812 | $O(log \; max(h_{MAX}, k_{MAX}))$.\footnote{Well, | $O(log \; max(h_{MAX}, k_{MAX}))$.\footnote{Well, |
1813 | not exactly. In the classical computer science | not exactly. In the classical computer science |
1814 | sense (speaking only in terms of number of operations), | sense (speaking only in terms of number of operations), |
1815 | the algorithms are $O(log \; max(h_{MAX}, k_{MAX}))$. However, | the algorithms are $O(log \; max(h_{MAX}, k_{MAX}))$. However, |
1816 | if $h_{MAX}$ and $k_{MAX}$ are increased beyond the sizes | if $h_{MAX}$ and $k_{MAX}$ are increased beyond the sizes |
1817 | of integers that a computer can inherently accomodate, one must | of integers that a computer can inherently accomodate, one must |
1818 | use long integer calculation software, which requires more time for | use long integer calculation software, which requires more time for |
1819 | each integer operation than is required for machine native | each integer operation than is required for machine native |
1820 | integer sizes. As $h_{MAX}$ and $k_{MAX}$ are increased far | integer sizes. As $h_{MAX}$ and $k_{MAX}$ are increased far |
1821 | beyond integer sizes accomodated inherently by the computer, | beyond integer sizes accomodated inherently by the computer, |
1822 | the relationship surely is not $O(log \; max(h_{MAX}, k_{MAX}))$.} | the relationship surely is not $O(log \; max(h_{MAX}, k_{MAX}))$.} |
1823 | The basis on which we | The basis on which we |
1824 | make that assertion is the geometric rate of | make that assertion is the geometric rate of |
1825 | increase of convergents (see Theorem | increase of convergents (see Theorem |
1826 | \ref{thm:ccfr0:scnv0:minimumrateofconvergentdenominatorincrease}), | \ref{thm:ccfr0:scnv0:minimumrateofconvergentdenominatorincrease}), |
1827 | which means that the number of steps required is tied to the | which means that the number of steps required is tied to the |
1828 | logarithm of the maximum denominator involved, as it is | logarithm of the maximum denominator involved, as it is |
1829 | necessary to obtain partial quotients and convergents only until | necessary to obtain partial quotients and convergents only until |
1830 | $q_k \geq max(h_{MAX},k_{MAX})$. | $q_k \geq max(h_{MAX},k_{MAX})$. |
1831 | ||
1832 | \begin{vworktheoremstatement} | \begin{vworktheoremstatement} |
1833 | \label{thm:ccfr0:scba0:convergentcloseness} | \label{thm:ccfr0:scba0:convergentcloseness} |
1834 | In the case of an infinite continued fraction for $k \geq 0$ or in the | In the case of an infinite continued fraction for $k \geq 0$ or in the |
1835 | case of a finite continued fraction for $0 \leq k < n-1$, a convergent | case of a finite continued fraction for $0 \leq k < n-1$, a convergent |
1836 | $s_k = p_k/q_k$ to a [rational or irrational] number | $s_k = p_k/q_k$ to a [rational or irrational] number |
1837 | $\alpha \in \vworkrealsetnonneg$ satisfies | $\alpha \in \vworkrealsetnonneg$ satisfies |
1838 | ||
1839 | \begin{equation} | \begin{equation} |
1840 | \left| {\alpha - \frac{p_k}{q_k}} \right| | \left| {\alpha - \frac{p_k}{q_k}} \right| |
1841 | < | < |
1842 | \frac{1}{q_k q_{k+1}} . | \frac{1}{q_k q_{k+1}} . |
1843 | \end{equation} | \end{equation} |
1844 | ||
1845 | In the case of a finite continued fraction with $k = n-1$, | In the case of a finite continued fraction with $k = n-1$, |
1846 | ||
1847 | \begin{equation} | \begin{equation} |
1848 | \left| {\alpha - \frac{p_k}{q_k}} \right| | \left| {\alpha - \frac{p_k}{q_k}} \right| |
1849 | = | = |
1850 | \frac{1}{q_k q_{k+1}} . | \frac{1}{q_k q_{k+1}} . |
1851 | \end{equation} | \end{equation} |
1852 | \end{vworktheoremstatement} | \end{vworktheoremstatement} |
1853 | \begin{vworktheoremproof} | \begin{vworktheoremproof} |
1854 | The proof comes directly from Theorem | The proof comes directly from Theorem |
1855 | \ref{thm:ccfr0:scnv0:crossproductminusone} (Corollary I) | \ref{thm:ccfr0:scnv0:crossproductminusone} (Corollary I) |
1856 | and Theorem | and Theorem |
1857 | \ref{thm:ccfr0:scnv0:evenslessthanoddsgreaterthan}. | \ref{thm:ccfr0:scnv0:evenslessthanoddsgreaterthan}. |
1858 | \end{vworktheoremproof} | \end{vworktheoremproof} |
1859 | \begin{vworktheoremparsection}{Remarks} | \begin{vworktheoremparsection}{Remarks} |
1860 | Khinchin describes this result (\cite{bibref:b:KhinchinClassic}, p. 9) | Khinchin describes this result (\cite{bibref:b:KhinchinClassic}, p. 9) |
1861 | as playing a basic role in the arithmetic | as playing a basic role in the arithmetic |
1862 | applications of continued fractions. In fact, this theorem is used | applications of continued fractions. In fact, this theorem is used |
1863 | in the proof of Theorem | in the proof of Theorem |
1864 | \ref{thm:ccfr0:scba0:cfenclosingneighbors}. | \ref{thm:ccfr0:scba0:cfenclosingneighbors}. |
1865 | \end{vworktheoremparsection} | \end{vworktheoremparsection} |
1866 | \vworktheoremfooter{} | \vworktheoremfooter{} |
1867 | ||
1868 | ||
1869 | We now present and prove the fundamental theorem of this chapter, which gives an | We now present and prove the fundamental theorem of this chapter, which gives an |
1870 | $O(log \; N)$ algorithm for finding the enclosing neighbors in $F_N$ to an arbitrary | $O(log \; N)$ algorithm for finding the enclosing neighbors in $F_N$ to an arbitrary |
1871 | rational number $a/b$.\footnote{\label{footnote:ccfr0:scba0:rationalitynotrequired}Theorem | rational number $a/b$.\footnote{\label{footnote:ccfr0:scba0:rationalitynotrequired}Theorem |
1872 | \ref{thm:ccfr0:scba0:cfenclosingneighbors} | \ref{thm:ccfr0:scba0:cfenclosingneighbors} |
1873 | applies to irrational numbers as well, | applies to irrational numbers as well, |
1874 | so long as one can obtain enough partial quotients, but we don't highlight this | so long as one can obtain enough partial quotients, but we don't highlight this |
1875 | fact because it is rare in engineering applications that one uses symbolic methods | fact because it is rare in engineering applications that one uses symbolic methods |
1876 | to obtain best rational approximations. | to obtain best rational approximations. |
1877 | As emphasized by (\ref{eq:ccfr0:spar0:01}), (\ref{eq:ccfr0:spar0:02}), and | As emphasized by (\ref{eq:ccfr0:spar0:01}), (\ref{eq:ccfr0:spar0:02}), and |
1878 | (\ref{eq:ccfr0:spar0:03}), | (\ref{eq:ccfr0:spar0:03}), |
1879 | the process of obtaining partial quotients is essentially a process of determining in which | the process of obtaining partial quotients is essentially a process of determining in which |
1880 | partition a number lies. All numbers in the same partition---rational or | partition a number lies. All numbers in the same partition---rational or |
1881 | irrational---have the same Farey neighbors in all Farey series up to a certain order. | irrational---have the same Farey neighbors in all Farey series up to a certain order. |
1882 | If the partial quotients of | If the partial quotients of |
1883 | an irrational number can be obtained up through $a_k$ s.t. $s_k = p_k/q_k$ is the | an irrational number can be obtained up through $a_k$ s.t. $s_k = p_k/q_k$ is the |
1884 | highest-order convergent with $q_k \leq N$, then this theorem can be applied. | highest-order convergent with $q_k \leq N$, then this theorem can be applied. |
1885 | Knowledge of all $a_0, \ldots{} , a_k$ is equivalent | Knowledge of all $a_0, \ldots{} , a_k$ is equivalent |
1886 | to the knowledge that the number is in a partition where all numbers in that partition have | to the knowledge that the number is in a partition where all numbers in that partition have |
1887 | the same Farey neighbors in all Farey series up through at least order $q_k$.} | the same Farey neighbors in all Farey series up through at least order $q_k$.} |
1888 | ||
1889 | ||
1890 | \begin{vworktheoremstatementpar}{Enclosing Neighbors Of \mbox{\boldmath $x \notin F_N$} | \begin{vworktheoremstatementpar}{Enclosing Neighbors Of \mbox{\boldmath $x \notin F_N$} |
1891 | In \mbox{\boldmath $F_N$}} | In \mbox{\boldmath $F_N$}} |
1892 | \label{thm:ccfr0:scba0:cfenclosingneighbors} | \label{thm:ccfr0:scba0:cfenclosingneighbors} |
1893 | For a non-negative rational | For a non-negative rational |
1894 | number $a/b$ not in | number $a/b$ not in |
1895 | $F_N$ which has a | $F_N$ which has a |
1896 | continued fraction representation | continued fraction representation |
1897 | $[a_0;a_1,a_2,\ldots{} ,a_n]$, the | $[a_0;a_1,a_2,\ldots{} ,a_n]$, the |
1898 | highest-order convergent $s_k = p_k/q_k$ with $q_k \leq N$ is one | highest-order convergent $s_k = p_k/q_k$ with $q_k \leq N$ is one |
1899 | neighbor\footnote{By neighbors in $F_N$ we mean the rational numbers | neighbor\footnote{By neighbors in $F_N$ we mean the rational numbers |
1900 | in $F_N$ immediately to the left and immediately to the right | in $F_N$ immediately to the left and immediately to the right |
1901 | of $a/b$.} | of $a/b$.} |
1902 | to $a/b$ in $F_N$, and the other neighbor in | to $a/b$ in $F_N$, and the other neighbor in |
1903 | $F_N$ is\footnote{Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} | $F_N$ is\footnote{Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} |
1904 | is a somewhat stronger statement about best approximations | is a somewhat stronger statement about best approximations |
1905 | than Khinchin makes in \cite{bibref:b:KhinchinClassic}, Theorem 15. | than Khinchin makes in \cite{bibref:b:KhinchinClassic}, Theorem 15. |
1906 | We were not able to locate | We were not able to locate |
1907 | this theorem or a proof in print, | this theorem or a proof in print, |
1908 | but this theorem is understood within the number theory community. | but this theorem is understood within the number theory community. |
1909 | It appears on the Web | It appears on the Web |
1910 | page of David Eppstein \cite{bibref:i:davideppstein} in the form of a | page of David Eppstein \cite{bibref:i:davideppstein} in the form of a |
1911 | `C'-language computer program, | `C'-language computer program, |
1912 | \texttt{http://www.ics.uci.edu/\~{}{}eppstein/numth/frap.c}. | \texttt{http://www.ics.uci.edu/\~{}{}eppstein/numth/frap.c}. |
1913 | Although | Although |
1914 | Dr. Eppstein phrases the solution in terms of modifying | Dr. Eppstein phrases the solution in terms of modifying |
1915 | a partial quotient, his approach is equivalent to | a partial quotient, his approach is equivalent to |
1916 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}).} | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}).} |
1917 | ||
1918 | \begin{equation} | \begin{equation} |
1919 | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:01} | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:01} |
1920 | \frac{{\displaystyle{\left\lfloor {\frac{{N - q_{k - 1} }}{{q_k }}} \right\rfloor} | \frac{{\displaystyle{\left\lfloor {\frac{{N - q_{k - 1} }}{{q_k }}} \right\rfloor} |
1921 | p_k + p_{k - 1} }}{{\displaystyle{\left\lfloor {\frac{{N - q_{k - 1} }}{{q_k }}} | p_k + p_{k - 1} }}{{\displaystyle{\left\lfloor {\frac{{N - q_{k - 1} }}{{q_k }}} |
1922 | \right\rfloor} q_k + q_{k - 1} }}. | \right\rfloor} q_k + q_{k - 1} }}. |
1923 | \end{equation} | \end{equation} |
1924 | \end{vworktheoremstatementpar} | \end{vworktheoremstatementpar} |
1925 | \begin{vworktheoremproof} | \begin{vworktheoremproof} |
1926 | First, it is proved that the highest-order | First, it is proved that the highest-order |
1927 | convergent $s_k = p_k/q_k$ with $q_k \leq N$ is one of the two | convergent $s_k = p_k/q_k$ with $q_k \leq N$ is one of the two |
1928 | neighbors to $a/b$ in $F_N$. $s_k \in F_N$, since $q_k \leq N$. | neighbors to $a/b$ in $F_N$. $s_k \in F_N$, since $q_k \leq N$. |
1929 | By Theorem \ref{thm:ccfr0:scba0:convergentcloseness}, the upper bound on the | By Theorem \ref{thm:ccfr0:scba0:convergentcloseness}, the upper bound on the |
1930 | difference between $a/b$ and arbitrary $s_k$ is given by | difference between $a/b$ and arbitrary $s_k$ is given by |
1931 | ||
1932 | \begin{equation} | \begin{equation} |
1933 | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:02} | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:02} |
1934 | \left| {\frac{a}{b} - \frac{{p_k }}{{q_k }}} \right| < \frac{1}{{q_k q_{k + 1} }}. | \left| {\frac{a}{b} - \frac{{p_k }}{{q_k }}} \right| < \frac{1}{{q_k q_{k + 1} }}. |
1935 | \end{equation} | \end{equation} |
1936 | ||
1937 | For two consecutive terms in $F_N$, $Kh-Hk=1$ | For two consecutive terms in $F_N$, $Kh-Hk=1$ |
1938 | (\cfryzeroxrefcomma{}Theorem \ref{thm:cfry0:spfs:02}). | (\cfryzeroxrefcomma{}Theorem \ref{thm:cfry0:spfs:02}). |
1939 | For a Farey neighbor $H/K$ to $s_k$ in $F_N$, | For a Farey neighbor $H/K$ to $s_k$ in $F_N$, |
1940 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:03}) must hold. | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:03}) must hold. |
1941 | ||
1942 | \begin{equation} | \begin{equation} |
1943 | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:03} | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:03} |
1944 | \frac{1}{q_k N} \leq \left| {\frac{H}{K} - \frac{p_k}{q_k}} \right| | \frac{1}{q_k N} \leq \left| {\frac{H}{K} - \frac{p_k}{q_k}} \right| |
1945 | \end{equation} | \end{equation} |
1946 | ||
1947 | $q_{k+1}>N$, because $q_{k+1}>q_k$ and $p_k/q_k$ was chosen to be the | $q_{k+1}>N$, because $q_{k+1}>q_k$ and $p_k/q_k$ was chosen to be the |
1948 | highest-order convergent with $q_k\leq N$. Using this knowledge and | highest-order convergent with $q_k\leq N$. Using this knowledge and |
1949 | combining (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:02}) and | combining (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:02}) and |
1950 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:03}) leads to | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:03}) leads to |
1951 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:04}). | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:04}). |
1952 | ||
1953 | \begin{equation} | \begin{equation} |
1954 | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:04} | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:04} |
1955 | \left| {\frac{a}{b} - \frac{{p_k }}{{q_k }}} \right| < \frac{1}{{q_k q_{k + 1} }} | \left| {\frac{a}{b} - \frac{{p_k }}{{q_k }}} \right| < \frac{1}{{q_k q_{k + 1} }} |
1956 | < | < |
1957 | \frac{1}{q_k N} \leq \left| {\frac{H}{K} - \frac{p_k}{q_k}} \right| | \frac{1}{q_k N} \leq \left| {\frac{H}{K} - \frac{p_k}{q_k}} \right| |
1958 | \end{equation} | \end{equation} |
1959 | ||
1960 | This proves that $s_k$ is one neighbor to $a/b$ in $F_N$. | This proves that $s_k$ is one neighbor to $a/b$ in $F_N$. |
1961 | The apparatus of continued fractions ensures that the | The apparatus of continued fractions ensures that the |
1962 | highest order convergent $s_k$ with $q_k\leq N$ is closer to $a/b$ than | highest order convergent $s_k$ with $q_k\leq N$ is closer to $a/b$ than |
1963 | to any neighboring term in $F_N$. Thus, there is | to any neighboring term in $F_N$. Thus, there is |
1964 | no intervening term of $F_N$ between $s_k$ and $a/b$. | no intervening term of $F_N$ between $s_k$ and $a/b$. |
1965 | If $k$ is even, $s_k<a/b$, and if $k$ is | If $k$ is even, $s_k<a/b$, and if $k$ is |
1966 | odd, $s_k>a/b$. | odd, $s_k>a/b$. |
1967 | ||
1968 | It must be proved that (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) is the other Farey | It must be proved that (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) is the other Farey |
1969 | neighbor. For brevity, only the case of $k$ even is proved: the | neighbor. For brevity, only the case of $k$ even is proved: the |
1970 | case of $k$ odd is symmetrical. (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | case of $k$ odd is symmetrical. (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
1971 | is of the form (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}), | is of the form (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}), |
1972 | where $i \in \vworkintsetnonneg$. | where $i \in \vworkintsetnonneg$. |
1973 | ||
1974 | \begin{equation} | \begin{equation} |
1975 | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:05} | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:05} |
1976 | \frac{{ip_k + p_{k - 1} }}{{iq_k + q_{k - 1} }} | \frac{{ip_k + p_{k - 1} }}{{iq_k + q_{k - 1} }} |
1977 | \end{equation} | \end{equation} |
1978 | ||
1979 | $k$ is even, $s_k < a/b$, and the two Farey terms enclosing $a/b$, in | $k$ is even, $s_k < a/b$, and the two Farey terms enclosing $a/b$, in |
1980 | order, are | order, are |
1981 | ||
1982 | \begin{equation} | \begin{equation} |
1983 | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:06} | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:06} |
1984 | \frac{p_k }{q_k },\frac{ip_k + p_{k - 1} }{iq_k + q_{k - 1} }. | \frac{p_k }{q_k },\frac{ip_k + p_{k - 1} }{iq_k + q_{k - 1} }. |
1985 | \end{equation} | \end{equation} |
1986 | ||
1987 | Applying the $Kh - Hk = 1$ test, (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:07}), \ | Applying the $Kh - Hk = 1$ test, (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:07}), \ |
1988 | gives the | gives the |
1989 | result of 1, since by Theorem | result of 1, since by Theorem |
1990 | \ref{thm:ccfr0:scnv0:crossproductminusone}, | \ref{thm:ccfr0:scnv0:crossproductminusone}, |
1991 | $q_kp_{k-1}-p_kq_{k-1}=(-1)^k$. | $q_kp_{k-1}-p_kq_{k-1}=(-1)^k$. |
1992 | ||
1993 | \begin{equation} | \begin{equation} |
1994 | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:07} | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:07} |
1995 | \begin{array}{*{20}c} | \begin{array}{*{20}c} |
1996 | {(q_k )(ip_k + p_{k - 1} ) - (p_k )(iq_k + q_{k - 1} ) = 1} | {(q_k )(ip_k + p_{k - 1} ) - (p_k )(iq_k + q_{k - 1} ) = 1} |
1997 | \end{array} | \end{array} |
1998 | \end{equation} | \end{equation} |
1999 | ||
2000 | Thus, every potential Farey neighbor of the form (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) | Thus, every potential Farey neighbor of the form (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) |
2001 | meets the $Kh - Hk = 1$ test. It is also straightforward | meets the $Kh - Hk = 1$ test. It is also straightforward |
2002 | to show that \emph{only} potential Farey neighbors of the | to show that \emph{only} potential Farey neighbors of the |
2003 | form (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) can meet the $Kh-Hk=1$ test, using the | form (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) can meet the $Kh-Hk=1$ test, using the |
2004 | property that $p_k$ and $q_k$ are coprime. | property that $p_k$ and $q_k$ are coprime. |
2005 | ||
2006 | It must be established that a | It must be established that a |
2007 | rational number of the form (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) | rational number of the form (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) |
2008 | is irreducible. This result comes directly from | is irreducible. This result comes directly from |
2009 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:07}), | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:07}), |
2010 | since if the numerator | since if the numerator |
2011 | and denominator of (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) or | and denominator of (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) or |
2012 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) are not coprime, the difference of 1 is | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) are not coprime, the difference of 1 is |
2013 | not possible. | not possible. |
2014 | ||
2015 | The denominator of (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | The denominator of (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
2016 | can be rewritten as | can be rewritten as |
2017 | ||
2018 | \begin{equation} | \begin{equation} |
2019 | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:08} | \label{eq:ccfr0:scba0:thm:cfenclosingneighbors:08} |
2020 | N - \left[ {\left( {N - q_{k - 1} } \right)\bmod q_k } \right] \in \left\{ {N - q_k + 1,...,N} \right\}. | N - \left[ {\left( {N - q_{k - 1} } \right)\bmod q_k } \right] \in \left\{ {N - q_k + 1,...,N} \right\}. |
2021 | \end{equation} | \end{equation} |
2022 | ||
2023 | It must be shown that if one irreducible | It must be shown that if one irreducible |
2024 | rational number---namely, the rational number given by | rational number---namely, the rational number given by |
2025 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01})---with a denominator | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01})---with a denominator |
2026 | $\in \{ N-q_k+1,\ldots{} ,N \}$ meets the $Kh - Hk = 1$ | $\in \{ N-q_k+1,\ldots{} ,N \}$ meets the $Kh - Hk = 1$ |
2027 | test, there can be no other irreducible rational number | test, there can be no other irreducible rational number |
2028 | in $F_N$ with a larger | in $F_N$ with a larger |
2029 | denominator which also meets this test. | denominator which also meets this test. |
2030 | ||
2031 | Given (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:07}), | Given (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:07}), |
2032 | and given that \emph{only} rational numbers of the form | and given that \emph{only} rational numbers of the form |
2033 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) |
2034 | can meet the $Kh-Hk=1$ test, and given that any number of the | can meet the $Kh-Hk=1$ test, and given that any number of the |
2035 | form (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) | form (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:05}) |
2036 | is irreducible, the irreducible number meeting the | is irreducible, the irreducible number meeting the |
2037 | $Kh-Hk=1$ test with the next larger denominator after the denominator of | $Kh-Hk=1$ test with the next larger denominator after the denominator of |
2038 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
2039 | will have a denominator $\in \{ N+1, \ldots, N+q_k \}$. Thus, | will have a denominator $\in \{ N+1, \ldots, N+q_k \}$. Thus, |
2040 | no other irreducible rational number in $F_N$ besides that given | no other irreducible rational number in $F_N$ besides that given |
2041 | by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) with a larger denominator | by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) with a larger denominator |
2042 | $\leq N$ and which meets the $Kh - Hk = 1$ test can exist; therefore | $\leq N$ and which meets the $Kh - Hk = 1$ test can exist; therefore |
2043 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) is the other enclosing Farey neighbor to | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) is the other enclosing Farey neighbor to |
2044 | $a/b$ in $F_N$. | $a/b$ in $F_N$. |
2045 | \end{vworktheoremproof} | \end{vworktheoremproof} |
2046 | \vworktheoremfooter{} | \vworktheoremfooter{} |
2047 | ||
2048 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} establishes that | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} establishes that |
2049 | the two neighbors in $F_N$ to a rational number $a/b$ will be the highest-order | the two neighbors in $F_N$ to a rational number $a/b$ will be the highest-order |
2050 | convergent with a denominator not exceeding $N$, and the number | convergent with a denominator not exceeding $N$, and the number |
2051 | specified by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}). | specified by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}). |
2052 | An interesting and worthwhile question to ask about | An interesting and worthwhile question to ask about |
2053 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} is which of the | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} is which of the |
2054 | two neighbors will be \emph{closer} to $a/b$---the convergent or the | two neighbors will be \emph{closer} to $a/b$---the convergent or the |
2055 | number specified by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01})? | number specified by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01})? |
2056 | Can we make any strong, simple, and easy-to-remember statements | Can we make any strong, simple, and easy-to-remember statements |
2057 | about the relative distance? We answer this question and some related | about the relative distance? We answer this question and some related |
2058 | questions now. | questions now. |
2059 | ||
2060 | We are not aware of any rules that decisively predict which of the two | We are not aware of any rules that decisively predict which of the two |
2061 | Farey neighbors in Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} | Farey neighbors in Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} |
2062 | will be closer to $a/b$\footnote{We should qualify this by saying that | will be closer to $a/b$\footnote{We should qualify this by saying that |
2063 | we mean a rule which uses only partial quotients up through | we mean a rule which uses only partial quotients up through |
2064 | $a_k$ or at most $a_{k+1}$, which is the same information used | $a_k$ or at most $a_{k+1}$, which is the same information used |
2065 | by the theorem. We should also add that although Theorem | by the theorem. We should also add that although Theorem |
2066 | \ref{thm:ccfr0:scba0:cfenclosingneighbors} is worded to only consider | \ref{thm:ccfr0:scba0:cfenclosingneighbors} is worded to only consider |
2067 | non-negative rational numbers, the theorem and the results here | non-negative rational numbers, the theorem and the results here |
2068 | apply to non-negative irrational numbers as well, so long as enough partial | apply to non-negative irrational numbers as well, so long as enough partial |
2069 | quotients can be obtained.}, although Lemma | quotients can be obtained.}, although Lemma |
2070 | \ref{lem:ccfr0:scba0:enclosingneighborstheoremfurtherresult} is able | \ref{lem:ccfr0:scba0:enclosingneighborstheoremfurtherresult} is able |
2071 | to predict that the highest-ordered convergent $s_k$ with a denominator | to predict that the highest-ordered convergent $s_k$ with a denominator |
2072 | not exceeding $N$ will be closer in many cases. | not exceeding $N$ will be closer in many cases. |
2073 | In general, either neighbor may be closer. | In general, either neighbor may be closer. |
2074 | The most straightforward approach that we | The most straightforward approach that we |
2075 | are aware of is to calculate both Farey neighbors and to calculate | are aware of is to calculate both Farey neighbors and to calculate |
2076 | their respective distances from $a/b$. The difficulty in devising a simple rule | their respective distances from $a/b$. The difficulty in devising a simple rule |
2077 | to predict which neighbor | to predict which neighbor |
2078 | is closer is compounded by that fact that knowledge of | is closer is compounded by that fact that knowledge of |
2079 | $[a_0; a_1, \ldots{} , a_k]$ such that $s_k$ is the highest-ordered | $[a_0; a_1, \ldots{} , a_k]$ such that $s_k$ is the highest-ordered |
2080 | convergent with $q_k \leq N$ is incomplete knowledge of $a/b$ and can | convergent with $q_k \leq N$ is incomplete knowledge of $a/b$ and can |
2081 | only confine $a/b$ to an inequality in the sense suggested by | only confine $a/b$ to an inequality in the sense suggested by |
2082 | (\ref{eq:ccfr0:spar0:01}) through | (\ref{eq:ccfr0:spar0:01}) through |
2083 | (\ref{eq:ccfr0:spar0:03}). Note that the | (\ref{eq:ccfr0:spar0:03}). Note that the |
2084 | value specified by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | value specified by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
2085 | is an intermediate fraction, and that the statement of Theorem | is an intermediate fraction, and that the statement of Theorem |
2086 | \ref{thm:ccfr0:scba0:cfenclosingneighbors} | \ref{thm:ccfr0:scba0:cfenclosingneighbors} |
2087 | coincides with Khinchin's Theorem 15 | coincides with Khinchin's Theorem 15 |
2088 | (\cite{bibref:b:KhinchinClassic}, p. 22). | (\cite{bibref:b:KhinchinClassic}, p. 22). |
2089 | ||
2090 | However, even in the absence of a rule to decisively | However, even in the absence of a rule to decisively |
2091 | predict which of the | predict which of the |
2092 | two Farey neighbors specified by Theorem | two Farey neighbors specified by Theorem |
2093 | \ref{thm:ccfr0:scba0:cfenclosingneighbors} is closer to $a/b$, | \ref{thm:ccfr0:scba0:cfenclosingneighbors} is closer to $a/b$, |
2094 | there are other useful properties of convergents | there are other useful properties of convergents |
2095 | as best approximations which we present | as best approximations which we present |
2096 | now. | now. |
2097 | ||
2098 | It has been stated earlier that even-ordered convergents form an | It has been stated earlier that even-ordered convergents form an |
2099 | increasing sequence and that odd-ordered convergents also form a | increasing sequence and that odd-ordered convergents also form a |
2100 | decreasing sequence. However, up to this point, we have not made | decreasing sequence. However, up to this point, we have not made |
2101 | a statement about the relationship between even- and odd-ordered | a statement about the relationship between even- and odd-ordered |
2102 | convergents. We present such a statement as Lemma | convergents. We present such a statement as Lemma |
2103 | \ref{lem:ccfr0:scba0:convergenterrordecreases}, | \ref{lem:ccfr0:scba0:convergenterrordecreases}, |
2104 | below. | below. |
2105 | ||
2106 | \begin{vworklemmastatement} | \begin{vworklemmastatement} |
2107 | \label{lem:ccfr0:scba0:convergenterrordecreases} | \label{lem:ccfr0:scba0:convergenterrordecreases} |
2108 | In the case of a finite or infinite continued fraction representation | In the case of a finite or infinite continued fraction representation |
2109 | of a non-negative rational or irrational | of a non-negative rational or irrational |
2110 | number $\alpha \in \vworkrealsetnonneg$, for all $k$, | number $\alpha \in \vworkrealsetnonneg$, for all $k$, |
2111 | ||
2112 | \begin{equation} | \begin{equation} |
2113 | \label{eq:lem:ccfr0:scba0:convergenterrordecreases:01} | \label{eq:lem:ccfr0:scba0:convergenterrordecreases:01} |
2114 | \left| {\alpha - \frac{p_k}{q_k}} \right| < | \left| {\alpha - \frac{p_k}{q_k}} \right| < |
2115 | \left| {\alpha - \frac{p_{k-1}}{q_{k-1}}} \right| . | \left| {\alpha - \frac{p_{k-1}}{q_{k-1}}} \right| . |
2116 | \end{equation} | \end{equation} |
2117 | In other words, convergents get ever-closer to $\alpha$, without | In other words, convergents get ever-closer to $\alpha$, without |
2118 | respect to whether they are even- or odd-ordered convergents. | respect to whether they are even- or odd-ordered convergents. |
2119 | \end{vworklemmastatement} | \end{vworklemmastatement} |
2120 | \begin{vworklemmaproof} | \begin{vworklemmaproof} |
2121 | In this proof, we show that for all $k$, | In this proof, we show that for all $k$, |
2122 | ||
2123 | \begin{equation} | \begin{equation} |
2124 | \label{eq:lem:ccfr0:scba0:convergenterrordecreases:02} | \label{eq:lem:ccfr0:scba0:convergenterrordecreases:02} |
2125 | | s_{k-2} - s_{k-1} | > 2 | s_{k-1} - s_k | . | | s_{k-2} - s_{k-1} | > 2 | s_{k-1} - s_k | . |
2126 | \end{equation} | \end{equation} |
2127 | ||
2128 | To understand why the proof is valid, consider the case of $k$ even, | To understand why the proof is valid, consider the case of $k$ even, |
2129 | in which case $s_k < \alpha$, so that $s_{k-1} - \alpha < s_{k-1} - s_k$. | in which case $s_k < \alpha$, so that $s_{k-1} - \alpha < s_{k-1} - s_k$. |
2130 | If $s_{k-1} - s_{k-2} > 2 (s_{k-1} - s_k)$, then | If $s_{k-1} - s_{k-2} > 2 (s_{k-1} - s_k)$, then |
2131 | $s_{k-2}$ is further to the left of $\alpha$ than $s_{k-1}$ is to the | $s_{k-2}$ is further to the left of $\alpha$ than $s_{k-1}$ is to the |
2132 | right of $\alpha$; thus (\ref{eq:lem:ccfr0:scba0:convergenterrordecreases:01}) | right of $\alpha$; thus (\ref{eq:lem:ccfr0:scba0:convergenterrordecreases:01}) |
2133 | applies. A symmetrical argument holds for $k$ odd. | applies. A symmetrical argument holds for $k$ odd. |
2134 | ||
2135 | By Theorem \ref{thm:ccfr0:scnv0:crossproductminusone}, | By Theorem \ref{thm:ccfr0:scnv0:crossproductminusone}, |
2136 | ||
2137 | \begin{equation} | \begin{equation} |
2138 | \label{eq:lem:ccfr0:scba0:convergenterrordecreases:03} | \label{eq:lem:ccfr0:scba0:convergenterrordecreases:03} |
2139 | s_{k-2} - s_{k-1} | s_{k-2} - s_{k-1} |
2140 | = \frac{p_{k-2}}{q_{k-2}} | = \frac{p_{k-2}}{q_{k-2}} |
2141 | - \frac{p_{k-1}}{q_{k-1}} | - \frac{p_{k-1}}{q_{k-1}} |
2142 | = \frac{(-1)^{k-1}}{q_{k-2} q_{k-1}} , | = \frac{(-1)^{k-1}}{q_{k-2} q_{k-1}} , |
2143 | \end{equation} | \end{equation} |
2144 | ||
2145 | and similarly | and similarly |
2146 | ||
2147 | \begin{equation} | \begin{equation} |
2148 | \label{eq:lem:ccfr0:scba0:convergenterrordecreases:04} | \label{eq:lem:ccfr0:scba0:convergenterrordecreases:04} |
2149 | s_{k-1} - s_{k} | s_{k-1} - s_{k} |
2150 | = \frac{p_{k-1}}{q_{k-1}} | = \frac{p_{k-1}}{q_{k-1}} |
2151 | - \frac{p_{k}}{q_{k}} | - \frac{p_{k}}{q_{k}} |
2152 | = \frac{(-1)^{k}}{q_{k-1} q_{k}} . | = \frac{(-1)^{k}}{q_{k-1} q_{k}} . |
2153 | \end{equation} | \end{equation} |
2154 | ||
2155 | In order for (\ref{eq:lem:ccfr0:scba0:convergenterrordecreases:02}) | In order for (\ref{eq:lem:ccfr0:scba0:convergenterrordecreases:02}) |
2156 | to be met, it must be true that | to be met, it must be true that |
2157 | $2 q_{k-2} q_{k-1} < q_{k-1} q_k$, or equivalently that | $2 q_{k-2} q_{k-1} < q_{k-1} q_k$, or equivalently that |
2158 | $2 q_{k-2} < q_k$. Since canonically $q_k = a_k q_{k-1} + q_{k-2}$ | $2 q_{k-2} < q_k$. Since canonically $q_k = a_k q_{k-1} + q_{k-2}$ |
2159 | (Eq. \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:10}), | (Eq. \ref{eq:thm:ccfr0:scnv0:canonicalconvergentconstruction:10}), |
2160 | the requirement is that $2 q_{k-2} < a_k q_{k-1} + q_{k-2}$. | the requirement is that $2 q_{k-2} < a_k q_{k-1} + q_{k-2}$. |
2161 | Since $a_k \geq 1$ and convergents are ever-increasing, | Since $a_k \geq 1$ and convergents are ever-increasing, |
2162 | (\ref{eq:lem:ccfr0:scba0:convergenterrordecreases:02}) is met and the | (\ref{eq:lem:ccfr0:scba0:convergenterrordecreases:02}) is met and the |
2163 | lemma is proved. | lemma is proved. |
2164 | \end{vworklemmaproof} | \end{vworklemmaproof} |
2165 | \vworklemmafooter{} | \vworklemmafooter{} |
2166 | ||
2167 | Theorem \ref{thm:ccfr0:scba0:convergentcloseness} establishes a | Theorem \ref{thm:ccfr0:scba0:convergentcloseness} establishes a |
2168 | maximum distance from a number $\alpha$ that we wish to approximate | maximum distance from a number $\alpha$ that we wish to approximate |
2169 | to a convergent. We now provide a second result that establishes | to a convergent. We now provide a second result that establishes |
2170 | a \emph{minimum} distance. (This result is Theorem 13, p. | a \emph{minimum} distance. (This result is Theorem 13, p. |
2171 | 15, from \cite{bibref:b:KhinchinClassic}.) | 15, from \cite{bibref:b:KhinchinClassic}.) |
2172 | ||
2173 | \begin{vworktheoremstatement} | \begin{vworktheoremstatement} |
2174 | \label{thm:ccfr0:scba0:convergentfarness} | \label{thm:ccfr0:scba0:convergentfarness} |
2175 | In the case of an infinite continued fraction representation | In the case of an infinite continued fraction representation |
2176 | $[a_0; a_1, a_2, \ldots]$ of | $[a_0; a_1, a_2, \ldots]$ of |
2177 | a non-negative irrational number $\alpha \in \vworkrealsetnonneg$, | a non-negative irrational number $\alpha \in \vworkrealsetnonneg$, |
2178 | for all $k \geq 0$; or in the case of a [necessarily finite] | for all $k \geq 0$; or in the case of a [necessarily finite] |
2179 | continued fraction representation $[a_0; a_1, a_2, \ldots , a_n]$ | continued fraction representation $[a_0; a_1, a_2, \ldots , a_n]$ |
2180 | of a non-negative rational number | of a non-negative rational number |
2181 | $\alpha \in \vworkrealsetnonneg$, for all $0 \leq k \leq n-1$, | $\alpha \in \vworkrealsetnonneg$, for all $0 \leq k \leq n-1$, |
2182 | ||
2183 | \begin{equation} | \begin{equation} |
2184 | \label{eq:thm:ccfr0:scba0:convergentfarness:01} | \label{eq:thm:ccfr0:scba0:convergentfarness:01} |
2185 | \left| {\alpha - \frac{p_k}{q_k}} \right| > \frac{1}{q_k(q_{k+1}+q_k)} . | \left| {\alpha - \frac{p_k}{q_k}} \right| > \frac{1}{q_k(q_{k+1}+q_k)} . |
2186 | \end{equation} | \end{equation} |
2187 | \end{vworktheoremstatement} | \end{vworktheoremstatement} |
2188 | \begin{vworktheoremproof} | \begin{vworktheoremproof} |
2189 | We've already established (Lemma \ref{lem:ccfr0:scba0:convergenterrordecreases}) | We've already established (Lemma \ref{lem:ccfr0:scba0:convergenterrordecreases}) |
2190 | that each convergent $s_{k+1}$ is nearer to | that each convergent $s_{k+1}$ is nearer to |
2191 | a number $\alpha$ to be approximated than the previous | a number $\alpha$ to be approximated than the previous |
2192 | convergent, $s_k$, i.e. for all $k$, | convergent, $s_k$, i.e. for all $k$, |
2193 | ||
2194 | \begin{equation} | \begin{equation} |
2195 | \label{eq:thm:ccfr0:scba0:convergentfarness:02} | \label{eq:thm:ccfr0:scba0:convergentfarness:02} |
2196 | \left| {\alpha - \frac{p_{k+1}}{q_{k+1}}} \right| < | \left| {\alpha - \frac{p_{k+1}}{q_{k+1}}} \right| < |
2197 | \left| {\alpha - \frac{p_{k}}{q_{k}}} \right| . | \left| {\alpha - \frac{p_{k}}{q_{k}}} \right| . |
2198 | \end{equation} | \end{equation} |
2199 | ||
2200 | Since the mediant of two fractions always lies between | Since the mediant of two fractions always lies between |
2201 | them (Lemma \cfryzeroxrefhyphen{}\ref{lem:cfry0:spfs:02c}), it | them (Lemma \cfryzeroxrefhyphen{}\ref{lem:cfry0:spfs:02c}), it |
2202 | follows directly that | follows directly that |
2203 | ||
2204 | \begin{equation} | \begin{equation} |
2205 | \label{eq:thm:ccfr0:scba0:convergentfarness:03} | \label{eq:thm:ccfr0:scba0:convergentfarness:03} |
2206 | \left| {\alpha - \frac{p_{k}}{q_{k}}} \right| > | \left| {\alpha - \frac{p_{k}}{q_{k}}} \right| > |
2207 | \left| {\frac{p_k + p_{k+1}}{q_k + q_{k+1}} - \frac{p_k}{q_k}} \right| = | \left| {\frac{p_k + p_{k+1}}{q_k + q_{k+1}} - \frac{p_k}{q_k}} \right| = |
2208 | \frac{1}{q_k ( q_k + q_{k+1})} . | \frac{1}{q_k ( q_k + q_{k+1})} . |
2209 | \end{equation} | \end{equation} |
2210 | \end{vworktheoremproof} | \end{vworktheoremproof} |
2211 | \begin{vworktheoremparsection}{Remark I} | \begin{vworktheoremparsection}{Remark I} |
2212 | This theorem can be combined with Theorem \ref{thm:ccfr0:scba0:convergentcloseness} | This theorem can be combined with Theorem \ref{thm:ccfr0:scba0:convergentcloseness} |
2213 | to give the following combined inequality: | to give the following combined inequality: |
2214 | ||
2215 | \begin{equation} | \begin{equation} |
2216 | \label{eq:thm:ccfr0:scba0:convergentfarness:04} | \label{eq:thm:ccfr0:scba0:convergentfarness:04} |
2217 | \frac{1}{q_k ( q_k + q_{k+1})} < | \frac{1}{q_k ( q_k + q_{k+1})} < |
2218 | \left| {\alpha - \frac{p_{k}}{q_{k}}} \right| \leq | \left| {\alpha - \frac{p_{k}}{q_{k}}} \right| \leq |
2219 | \frac{1}{q_k q_{k+1}} . | \frac{1}{q_k q_{k+1}} . |
2220 | \end{equation} | \end{equation} |
2221 | \end{vworktheoremparsection} | \end{vworktheoremparsection} |
2222 | \vworktheoremfooter{} | \vworktheoremfooter{} |
2223 | ||
2224 | We now supply an interesting and sometimes useful property of | We now supply an interesting and sometimes useful property of |
2225 | convergents used as best approximations. Note that we later show that | convergents used as best approximations. Note that we later show that |
2226 | Lemma \ref{lem:ccfr0:scba0:convergentbetterappthanlesserdenominator} | Lemma \ref{lem:ccfr0:scba0:convergentbetterappthanlesserdenominator} |
2227 | is a weak statement (a stronger statement can be made, Lemma | is a weak statement (a stronger statement can be made, Lemma |
2228 | \ref{lem:ccfr0:scba0:morecomplexconvergentbapprule}), but this lemma | \ref{lem:ccfr0:scba0:morecomplexconvergentbapprule}), but this lemma |
2229 | has the advantage of being extremely easy to remember. | has the advantage of being extremely easy to remember. |
2230 | ||
2231 | \begin{vworklemmastatement} | \begin{vworklemmastatement} |
2232 | \label{lem:ccfr0:scba0:convergentbetterappthanlesserdenominator} | \label{lem:ccfr0:scba0:convergentbetterappthanlesserdenominator} |
2233 | A convergent $s_k = p_k/q_k$ to a non-negative [rational or irrational] number | A convergent $s_k = p_k/q_k$ to a non-negative [rational or irrational] number |
2234 | $\alpha \in \vworkrealsetnonneg$ is closer to $\alpha$ | $\alpha \in \vworkrealsetnonneg$ is closer to $\alpha$ |
2235 | than any other rational number with the same or a smaller | than any other rational number with the same or a smaller |
2236 | denominator. | denominator. |
2237 | \end{vworklemmastatement} | \end{vworklemmastatement} |
2238 | \begin{vworklemmaproof} | \begin{vworklemmaproof} |
2239 | Let $\alpha$ be the non-negative real number, rational or irrational, | Let $\alpha$ be the non-negative real number, rational or irrational, |
2240 | that we wish to approximate. | that we wish to approximate. |
2241 | ||
2242 | If there is a number (let's call it $c/d$) | If there is a number (let's call it $c/d$) |
2243 | closer to $\alpha$ than $s_k = p_k / q_k$, with the same or a smaller denominator | closer to $\alpha$ than $s_k = p_k / q_k$, with the same or a smaller denominator |
2244 | than $s_k$, then by definition it must be in the Farey series of order | than $s_k$, then by definition it must be in the Farey series of order |
2245 | $q_k$, which we denote $F_{q_k}$. | $q_k$, which we denote $F_{q_k}$. |
2246 | ||
2247 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} |
2248 | assures us that the two Farey neighbors to $\alpha$ in $F_{q_k}$ will be | assures us that the two Farey neighbors to $\alpha$ in $F_{q_k}$ will be |
2249 | $s_k$ and the number given by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}). | $s_k$ and the number given by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}). |
2250 | Note that Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} | Note that Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} |
2251 | applies to irrational numbers as well (although the theorem statement | applies to irrational numbers as well (although the theorem statement |
2252 | does not indicate this), so we interpret | does not indicate this), so we interpret |
2253 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} |
2254 | in that sense. | in that sense. |
2255 | ||
2256 | Note in (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) that the | Note in (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) that the |
2257 | expression involving the $floor(\cdot{})$ function will evaluate | expression involving the $floor(\cdot{})$ function will evaluate |
2258 | to be zero, since $N=q_k$. Thus, the other Farey neighbor to | to be zero, since $N=q_k$. Thus, the other Farey neighbor to |
2259 | $\alpha$ in $F_{q_k}$ will be $s_{k-1} = p_{k-1}/q_{k-1}$. | $\alpha$ in $F_{q_k}$ will be $s_{k-1} = p_{k-1}/q_{k-1}$. |
2260 | ||
2261 | We have already shown in Lemma \ref{lem:ccfr0:scba0:convergenterrordecreases} | We have already shown in Lemma \ref{lem:ccfr0:scba0:convergenterrordecreases} |
2262 | that $|\alpha - s_{k-1}| > |\alpha - s_{k}|$, therefore | that $|\alpha - s_{k-1}| > |\alpha - s_{k}|$, therefore |
2263 | $s_k$ is closer to $\alpha$ than the other Farey neighbor given | $s_k$ is closer to $\alpha$ than the other Farey neighbor given |
2264 | by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}). | by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}). |
2265 | Furthermore, because any $c/d$ which is closer to $\alpha$ than | Furthermore, because any $c/d$ which is closer to $\alpha$ than |
2266 | $s_k$ must be present in $F_{q_k}$, such a $c/d$ does not exist. | $s_k$ must be present in $F_{q_k}$, such a $c/d$ does not exist. |
2267 | \end{vworklemmaproof} | \end{vworklemmaproof} |
2268 | \begin{vworklemmaparsection}{Remark I} | \begin{vworklemmaparsection}{Remark I} |
2269 | In practice, this lemma is little more than parlor trivia | In practice, this lemma is little more than parlor trivia |
2270 | (it is not mathematically significant), but it is useful | (it is not mathematically significant), but it is useful |
2271 | information and very easy to remember. For example, $355/113$ is a convergent to | information and very easy to remember. For example, $355/113$ is a convergent to |
2272 | $\pi$, and it is sometimes useful to know that no better rational approximation | $\pi$, and it is sometimes useful to know that no better rational approximation |
2273 | can exist with the same or a smaller denominator. | can exist with the same or a smaller denominator. |
2274 | \end{vworklemmaparsection} | \end{vworklemmaparsection} |
2275 | \begin{vworklemmaparsection}{Remark II} | \begin{vworklemmaparsection}{Remark II} |
2276 | A stronger statement can be made (see | A stronger statement can be made (see |
2277 | Lemma \ref{lem:ccfr0:scba0:morecomplexconvergentbapprule}). | Lemma \ref{lem:ccfr0:scba0:morecomplexconvergentbapprule}). |
2278 | \end{vworklemmaparsection} | \end{vworklemmaparsection} |
2279 | \vworklemmafooter{} | \vworklemmafooter{} |
2280 | ||
2281 | We now | We now |
2282 | present a stronger statement about convergents as best approximations that | present a stronger statement about convergents as best approximations that |
2283 | is not as easy to remember as | is not as easy to remember as |
2284 | Lemma \ref{lem:ccfr0:scba0:convergentbetterappthanlesserdenominator}. | Lemma \ref{lem:ccfr0:scba0:convergentbetterappthanlesserdenominator}. |
2285 | ||
2286 | \begin{vworklemmastatement} | \begin{vworklemmastatement} |
2287 | \label{lem:ccfr0:scba0:morecomplexconvergentbapprule} | \label{lem:ccfr0:scba0:morecomplexconvergentbapprule} |
2288 | A convergent $s_k = p_k/q_k$ to a non-negative | A convergent $s_k = p_k/q_k$ to a non-negative |
2289 | [rational or irrational] number | [rational or irrational] number |
2290 | $\alpha \in \vworkrealsetnonneg$ is closer to $\alpha$ | $\alpha \in \vworkrealsetnonneg$ is closer to $\alpha$ |
2291 | than any other rational number with a denominator | than any other rational number with a denominator |
2292 | less than $q_k + q_{k-1}$. | less than $q_k + q_{k-1}$. |
2293 | \end{vworklemmastatement} | \end{vworklemmastatement} |
2294 | \begin{vworklemmaproof} | \begin{vworklemmaproof} |
2295 | Let $N$ be the denominator of a rational number which is | Let $N$ be the denominator of a rational number which is |
2296 | potentially closer to $\alpha$ than $s_k$. If | potentially closer to $\alpha$ than $s_k$. If |
2297 | $N < q_k + q_{k+1}$, then | $N < q_k + q_{k+1}$, then |
2298 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
2299 | evaluates to $s_{k-1}$, and Lemma | evaluates to $s_{k-1}$, and Lemma |
2300 | \ref{lem:ccfr0:scba0:convergenterrordecreases} has established that | \ref{lem:ccfr0:scba0:convergenterrordecreases} has established that |
2301 | $s_k$ is closer to $\alpha$ than $s_{k-1}$. If, on the other | $s_k$ is closer to $\alpha$ than $s_{k-1}$. If, on the other |
2302 | hand, $N \geq q_k + q_{k+1}$, then the intermediate fraction specified by | hand, $N \geq q_k + q_{k+1}$, then the intermediate fraction specified by |
2303 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) \emph{may} be | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) \emph{may} be |
2304 | closer to $\alpha$ than $s_k$. | closer to $\alpha$ than $s_k$. |
2305 | \end{vworklemmaproof} | \end{vworklemmaproof} |
2306 | \begin{vworklemmaparsection}{Remark I} | \begin{vworklemmaparsection}{Remark I} |
2307 | This statement is harder to remember, but a stronger statement | This statement is harder to remember, but a stronger statement |
2308 | than Lemma \ref{lem:ccfr0:scba0:convergentbetterappthanlesserdenominator}. | than Lemma \ref{lem:ccfr0:scba0:convergentbetterappthanlesserdenominator}. |
2309 | \end{vworklemmaparsection} | \end{vworklemmaparsection} |
2310 | \begin{vworklemmaparsection}{Remark II} | \begin{vworklemmaparsection}{Remark II} |
2311 | Note that the only valid implication is $N < q_k + q_{k+1} \rightarrow$ | Note that the only valid implication is $N < q_k + q_{k+1} \rightarrow$ |
2312 | (convergent is closer). Note that | (convergent is closer). Note that |
2313 | $N \geq q_k + q_{k+1} \nrightarrow$ (intermediate fraction is closer)! | $N \geq q_k + q_{k+1} \nrightarrow$ (intermediate fraction is closer)! |
2314 | If $N \geq q_k + q_{k+1}$, either the convergent or the intermediate fraction | If $N \geq q_k + q_{k+1}$, either the convergent or the intermediate fraction |
2315 | may be closer. | may be closer. |
2316 | This statement is harder to remember, but a stronger statement | This statement is harder to remember, but a stronger statement |
2317 | than Lemma \ref{lem:ccfr0:scba0:convergentbetterappthanlesserdenominator}. | than Lemma \ref{lem:ccfr0:scba0:convergentbetterappthanlesserdenominator}. |
2318 | \end{vworklemmaparsection} | \end{vworklemmaparsection} |
2319 | \vworklemmafooter{} | \vworklemmafooter{} |
2320 | ||
2321 | Finally, we present a result about Theorem | Finally, we present a result about Theorem |
2322 | \ref{thm:ccfr0:scba0:cfenclosingneighbors} that will predict | \ref{thm:ccfr0:scba0:cfenclosingneighbors} that will predict |
2323 | in some circumstances that the highest-ordered convergent | in some circumstances that the highest-ordered convergent |
2324 | $s_k$ with a denominator not exceeding $N$ must be closer | $s_k$ with a denominator not exceeding $N$ must be closer |
2325 | to $a/b$ than the intermediate fraction specified by | to $a/b$ than the intermediate fraction specified by |
2326 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}). | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}). |
2327 | ||
2328 | \begin{vworklemmastatement} | \begin{vworklemmastatement} |
2329 | \label{lem:ccfr0:scba0:enclosingneighborstheoremfurtherresult} | \label{lem:ccfr0:scba0:enclosingneighborstheoremfurtherresult} |
2330 | In Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}, | In Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}, |
2331 | if $N < q_k + q_{k-1}$, the highest ordered convergent | if $N < q_k + q_{k-1}$, the highest ordered convergent |
2332 | $s_k$ with a denominator not exceeding $N$ is closer to | $s_k$ with a denominator not exceeding $N$ is closer to |
2333 | $a/b$\footnote{Note that this result is also valid for | $a/b$\footnote{Note that this result is also valid for |
2334 | convergents to an irrational number, although | convergents to an irrational number, although |
2335 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} is | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} is |
2336 | not worded in this way.} then the intermediate fraction specified by | not worded in this way.} then the intermediate fraction specified by |
2337 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}). | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}). |
2338 | If $N \geq q_k + q_{k-1}$, either $s_k$ or the intermediate | If $N \geq q_k + q_{k-1}$, either $s_k$ or the intermediate |
2339 | fraction specified by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | fraction specified by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
2340 | may be closer. | may be closer. |
2341 | \end{vworklemmastatement} | \end{vworklemmastatement} |
2342 | \begin{vworklemmaproof} | \begin{vworklemmaproof} |
2343 | See the proof of Lemma | See the proof of Lemma |
2344 | \ref{lem:ccfr0:scba0:morecomplexconvergentbapprule}. | \ref{lem:ccfr0:scba0:morecomplexconvergentbapprule}. |
2345 | \end{vworklemmaproof} | \end{vworklemmaproof} |
2346 | \vworklemmafooter{} | \vworklemmafooter{} |
2347 | ||
2348 | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} immediately suggests | Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors} immediately suggests |
2349 | an algorithm for obtaining the enclosing rational numbers in $F_N$ | an algorithm for obtaining the enclosing rational numbers in $F_N$ |
2350 | to a rational number $a/b \notin F_N$, which we present as | to a rational number $a/b \notin F_N$, which we present as |
2351 | Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn}. Although | Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn}. Although |
2352 | we don't formally show it, the algorithm is $O(log \; N)$, due to | we don't formally show it, the algorithm is $O(log \; N)$, due to |
2353 | the minimum geometric rate of increase of convergents | the minimum geometric rate of increase of convergents |
2354 | (Theorem \ref{thm:ccfr0:scnv0:minimumrateofconvergentdenominatorincrease}). | (Theorem \ref{thm:ccfr0:scnv0:minimumrateofconvergentdenominatorincrease}). |
2355 | Note that the algorithm will proceed only until $q_k > N$, not necessarily | Note that the algorithm will proceed only until $q_k > N$, not necessarily |
2356 | until all partial quotients of $a/b$ are obtained. Note also that the | until all partial quotients of $a/b$ are obtained. Note also that the |
2357 | algorithm can be applied to irrational numbers with minor | algorithm can be applied to irrational numbers with minor |
2358 | modification (all that matters is that we can obtain enough | modification (all that matters is that we can obtain enough |
2359 | partial quotients). | partial quotients). |
2360 | ||
2361 | \begin{vworkalgorithmstatementpar}{Enclosing Neighbors Of \mbox{\boldmath $a/b \notin F_N$} | \begin{vworkalgorithmstatementpar}{Enclosing Neighbors Of \mbox{\boldmath $a/b \notin F_N$} |
2362 | In \mbox{\boldmath $F_N$}} | In \mbox{\boldmath $F_N$}} |
2363 | \label{alg:ccfr0:scba0:cfenclosingneighborsfn} | \label{alg:ccfr0:scba0:cfenclosingneighborsfn} |
2364 | \begin{alglvl0} | \begin{alglvl0} |
2365 | \item $k := -1$. | \item $k := -1$. |
2366 | \item $divisor_{-1} := a$. | \item $divisor_{-1} := a$. |
2367 | \item $remainder_{-1} := b$. | \item $remainder_{-1} := b$. |
2368 | \item $p_{-1} := 1$. | \item $p_{-1} := 1$. |
2369 | \item $q_{-1} := 0$. | \item $q_{-1} := 0$. |
2370 | ||
2371 | \item Repeat | \item Repeat |
2372 | ||
2373 | \begin{alglvl1} | \begin{alglvl1} |
2374 | \item $k := k + 1$. | \item $k := k + 1$. |
2375 | \item $dividend_k := divisor_{k-1}$. | \item $dividend_k := divisor_{k-1}$. |
2376 | \item $divisor_k := remainder_{k-1}$. | \item $divisor_k := remainder_{k-1}$. |
2377 | \item $a_k := dividend_k \; div \; divisor_k$. | \item $a_k := dividend_k \; div \; divisor_k$. |
2378 | \item $remainder_k := dividend_k \; mod \; divisor_k$. | \item $remainder_k := dividend_k \; mod \; divisor_k$. |
2379 | \item If $k=0$ then $p_k := a_k$ else $p_k := a_k p_{k-1} + p_{k-2}$. | \item If $k=0$ then $p_k := a_k$ else $p_k := a_k p_{k-1} + p_{k-2}$. |
2380 | \item If $k=0$ then $q_k := 1$ else $q_k := a_k q_{k-1} + q_{k-2}$. | \item If $k=0$ then $q_k := 1$ else $q_k := a_k q_{k-1} + q_{k-2}$. |
2381 | \end{alglvl1} | \end{alglvl1} |
2382 | ||
2383 | \item Until ($q_k > k_{MAX}$). | \item Until ($q_k > k_{MAX}$). |
2384 | \item $s_{k-1} = p_{k-1}/q_{k-1}$ will be one Farey neighbor to $a/b$ in $F_{k_{MAX}}$. | \item $s_{k-1} = p_{k-1}/q_{k-1}$ will be one Farey neighbor to $a/b$ in $F_{k_{MAX}}$. |
2385 | Apply (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | Apply (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
2386 | to obtain the other Farey neighbor. | to obtain the other Farey neighbor. |
2387 | \end{alglvl0} | \end{alglvl0} |
2388 | \end{vworkalgorithmstatementpar} | \end{vworkalgorithmstatementpar} |
2389 | %\vworkalgorithmfooter{} | %\vworkalgorithmfooter{} |
2390 | ||
2391 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
2392 | \label{ex:ccfr0:scba0:bestrapapptoratnum} | \label{ex:ccfr0:scba0:bestrapapptoratnum} |
2393 | Find the members of $F_{100}$ which enclose the conversion factor | Find the members of $F_{100}$ which enclose the conversion factor |
2394 | from kilometers-per-hour to miles-per-hour. Assume that | from kilometers-per-hour to miles-per-hour. Assume that |
2395 | one mile is 1.6093 kilometers (exactly). | one mile is 1.6093 kilometers (exactly). |
2396 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
2397 | \begin{vworkexampleparsection}{Solution} | \begin{vworkexampleparsection}{Solution} |
2398 | The conversion factor from KPH to MPH is the reciprocal of 1.6093. As a rational | The conversion factor from KPH to MPH is the reciprocal of 1.6093. As a rational |
2399 | number, 1.6093 is 16,093/10,000, so 10,000/16,093 is its exact reciprocal. | number, 1.6093 is 16,093/10,000, so 10,000/16,093 is its exact reciprocal. |
2400 | Applying Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} | Applying Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} |
2401 | with $a/b = 10,000/16,093$ and $k_{MAX} = 100$ yields Table | with $a/b = 10,000/16,093$ and $k_{MAX} = 100$ yields Table |
2402 | \ref{tbl:ex:ccfr0:scba0:bestrapapptoratnum}. | \ref{tbl:ex:ccfr0:scba0:bestrapapptoratnum}. |
2403 | ||
2404 | \begin{table} | \begin{table} |
2405 | \caption{Application Of Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} | \caption{Application Of Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} |
2406 | To Find Members Of $F_{100}$ Which Enclose 10,000/16,093 | To Find Members Of $F_{100}$ Which Enclose 10,000/16,093 |
2407 | (Example \ref{ex:ccfr0:scba0:bestrapapptoratnum})} | (Example \ref{ex:ccfr0:scba0:bestrapapptoratnum})} |
2408 | \label{tbl:ex:ccfr0:scba0:bestrapapptoratnum} | \label{tbl:ex:ccfr0:scba0:bestrapapptoratnum} |
2409 | \begin{center} | \begin{center} |
2410 | \begin{tabular}{|c|c|c|c|c|c|c|} | \begin{tabular}{|c|c|c|c|c|c|c|} |
2411 | \hline | \hline |
2412 | \small{Index} & \small{$dividend_k$} & \small{$divisor_k$} & \small{$a_k$} & \small{$remainder_k$} & \small{$p_k$} & \small{$q_k$} \\ | \small{Index} & \small{$dividend_k$} & \small{$divisor_k$} & \small{$a_k$} & \small{$remainder_k$} & \small{$p_k$} & \small{$q_k$} \\ |
2413 | \small{($k$)} & & & & & & \\ | \small{($k$)} & & & & & & \\ |
2414 | \hline | \hline |
2415 | \hline | \hline |
2416 | \small{-1} & \small{N/A} & \small{10,000} & \small{N/A} & \small{16,093} & \small{1} & \small{0} \\ | \small{-1} & \small{N/A} & \small{10,000} & \small{N/A} & \small{16,093} & \small{1} & \small{0} \\ |
2417 | \hline | \hline |
2418 | \small{0} & \small{10,000} & \small{16,093} & \small{0} & \small{10,000} & \small{0} & \small{1} \\ | \small{0} & \small{10,000} & \small{16,093} & \small{0} & \small{10,000} & \small{0} & \small{1} \\ |
2419 | \hline | \hline |
2420 | \small{1} & \small{16,093} & \small{10,000} & \small{1} & \small{6,093} & \small{1} & \small{1} \\ | \small{1} & \small{16,093} & \small{10,000} & \small{1} & \small{6,093} & \small{1} & \small{1} \\ |
2421 | \hline | \hline |
2422 | \small{2} & \small{10,000} & \small{6,093} & \small{1} & \small{3,907} & \small{1} & \small{2} \\ | \small{2} & \small{10,000} & \small{6,093} & \small{1} & \small{3,907} & \small{1} & \small{2} \\ |
2423 | \hline | \hline |
2424 | \small{3} & \small{6,093} & \small{3,907} & \small{1} & \small{2,186} & \small{2} & \small{3} \\ | \small{3} & \small{6,093} & \small{3,907} & \small{1} & \small{2,186} & \small{2} & \small{3} \\ |
2425 | \hline | \hline |
2426 | \small{4} & \small{3,907} & \small{2,186} & \small{1} & \small{1,721} & \small{3} & \small{5} \\ | \small{4} & \small{3,907} & \small{2,186} & \small{1} & \small{1,721} & \small{3} & \small{5} \\ |
2427 | \hline | \hline |
2428 | \small{5} & \small{2,186} & \small{1,721} & \small{1} & \small{465} & \small{5} & \small{8} \\ | \small{5} & \small{2,186} & \small{1,721} & \small{1} & \small{465} & \small{5} & \small{8} \\ |
2429 | \hline | \hline |
2430 | \small{6} & \small{1,721} & \small{465} & \small{3} & \small{326} & \small{18} & \small{29} \\ | \small{6} & \small{1,721} & \small{465} & \small{3} & \small{326} & \small{18} & \small{29} \\ |
2431 | \hline | \hline |
2432 | \small{7} & \small{465} & \small{326} & \small{1} & \small{139} & \small{23} & \small{37} \\ | \small{7} & \small{465} & \small{326} & \small{1} & \small{139} & \small{23} & \small{37} \\ |
2433 | \hline | \hline |
2434 | \small{8} & \small{326} & \small{139} & \small{2} & \small{48} & \small{64} & \small{103} \\ | \small{8} & \small{326} & \small{139} & \small{2} & \small{48} & \small{64} & \small{103} \\ |
2435 | \hline | \hline |
2436 | \end{tabular} | \end{tabular} |
2437 | \end{center} | \end{center} |
2438 | \end{table} | \end{table} |
2439 | ||
2440 | Note from Table \ref{tbl:ex:ccfr0:scba0:bestrapapptoratnum} | Note from Table \ref{tbl:ex:ccfr0:scba0:bestrapapptoratnum} |
2441 | that the 7-th order convergent, $s_7 = 23/37$, | that the 7-th order convergent, $s_7 = 23/37$, |
2442 | is the highest-ordered convergent with $q_k \leq 100$, so | is the highest-ordered convergent with $q_k \leq 100$, so |
2443 | by Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}, 23/37 | by Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}, 23/37 |
2444 | is one neighbor in $F_{100}$ to 10,000/16,093. Because $s_7$ | is one neighbor in $F_{100}$ to 10,000/16,093. Because $s_7$ |
2445 | is an odd-ordered convergent, it will be the right | is an odd-ordered convergent, it will be the right |
2446 | Farey neighbor. | Farey neighbor. |
2447 | By (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}), the | By (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}), the |
2448 | other Farey neighbor is 41/66, and it will be the left | other Farey neighbor is 41/66, and it will be the left |
2449 | Farey neighbor. | Farey neighbor. |
2450 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
2451 | %\vworkexamplefooter{} | %\vworkexamplefooter{} |
2452 | ||
2453 | ||
2454 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
2455 | \label{ex:ccfr0:scba0:bestrapapptoirratnum} | \label{ex:ccfr0:scba0:bestrapapptoirratnum} |
2456 | Find the members of $F_{200}$ which | Find the members of $F_{200}$ which |
2457 | enclose $\sqrt{3}$. | enclose $\sqrt{3}$. |
2458 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
2459 | \begin{vworkexampleparsection}{Solution} | \begin{vworkexampleparsection}{Solution} |
2460 | We demonstrated in Example | We demonstrated in Example |
2461 | \ref{ex:ccfr0:scin0:symboliccfalgorithmexample} | \ref{ex:ccfr0:scin0:symboliccfalgorithmexample} |
2462 | that the continued fraction representation of | that the continued fraction representation of |
2463 | $\sqrt{3}$ is $[1;\overline{1,2}]$. | $\sqrt{3}$ is $[1;\overline{1,2}]$. |
2464 | As is highlighted in Footnote | As is highlighted in Footnote |
2465 | \ref{footnote:ccfr0:scba0:rationalitynotrequired}, it isn't required | \ref{footnote:ccfr0:scba0:rationalitynotrequired}, it isn't required |
2466 | that a number be rational to apply Theorem | that a number be rational to apply Theorem |
2467 | \ref{thm:ccfr0:scba0:cfenclosingneighbors}, so long as | \ref{thm:ccfr0:scba0:cfenclosingneighbors}, so long as |
2468 | enough partial quotients can be obtained. | enough partial quotients can be obtained. |
2469 | Using knowledge of the partial quotients of | Using knowledge of the partial quotients of |
2470 | $\sqrt{3}$ and applying Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} | $\sqrt{3}$ and applying Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} |
2471 | yields Table \ref{tbl:ex:ccfr0:scba0:bestrapapptoirratnum} (note that it isn't necessary | yields Table \ref{tbl:ex:ccfr0:scba0:bestrapapptoirratnum} (note that it isn't necessary |
2472 | to track remainders, as we already have all of the partial quotients for | to track remainders, as we already have all of the partial quotients for |
2473 | $\sqrt{3}$). | $\sqrt{3}$). |
2474 | ||
2475 | \begin{table} | \begin{table} |
2476 | \caption{Application Of Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} | \caption{Application Of Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} |
2477 | To Find Members Of $F_{100}$ Which Enclose $\sqrt{3}$ | To Find Members Of $F_{100}$ Which Enclose $\sqrt{3}$ |
2478 | (Example \ref{ex:ccfr0:scba0:bestrapapptoirratnum})} | (Example \ref{ex:ccfr0:scba0:bestrapapptoirratnum})} |
2479 | \label{tbl:ex:ccfr0:scba0:bestrapapptoirratnum} | \label{tbl:ex:ccfr0:scba0:bestrapapptoirratnum} |
2480 | \begin{center} | \begin{center} |
2481 | \begin{tabular}{|c|c|c|c|} | \begin{tabular}{|c|c|c|c|} |
2482 | \hline | \hline |
2483 | \hspace{0.15in}\small{Index ($k$)}\hspace{0.15in} & \hspace{0.15in}\small{$a_k$}\hspace{0.15in} & | \hspace{0.15in}\small{Index ($k$)}\hspace{0.15in} & \hspace{0.15in}\small{$a_k$}\hspace{0.15in} & |
2484 | \hspace{0.15in}\small{$p_k$}\hspace{0.15in} & \hspace{0.15in}\small{$q_k$}\hspace{0.15in} \\ | \hspace{0.15in}\small{$p_k$}\hspace{0.15in} & \hspace{0.15in}\small{$q_k$}\hspace{0.15in} \\ |
2485 | \hline | \hline |
2486 | \hline | \hline |
2487 | \small{-1} & \small{N/A} & \small{1} & \small{0} \\ | \small{-1} & \small{N/A} & \small{1} & \small{0} \\ |
2488 | \hline | \hline |
2489 | \small{0} & \small{1} & \small{1} & \small{1} \\ | \small{0} & \small{1} & \small{1} & \small{1} \\ |
2490 | \hline | \hline |
2491 | \small{1} & \small{1} & \small{2} & \small{1} \\ | \small{1} & \small{1} & \small{2} & \small{1} \\ |
2492 | \hline | \hline |
2493 | \small{2} & \small{2} & \small{5} & \small{3} \\ | \small{2} & \small{2} & \small{5} & \small{3} \\ |
2494 | \hline | \hline |
2495 | \small{3} & \small{1} & \small{7} & \small{4} \\ | \small{3} & \small{1} & \small{7} & \small{4} \\ |
2496 | \hline | \hline |
2497 | \small{4} & \small{2} & \small{19} & \small{11} \\ | \small{4} & \small{2} & \small{19} & \small{11} \\ |
2498 | \hline | \hline |
2499 | \small{5} & \small{1} & \small{26} & \small{15} \\ | \small{5} & \small{1} & \small{26} & \small{15} \\ |
2500 | \hline | \hline |
2501 | \small{6} & \small{2} & \small{71} & \small{41} \\ | \small{6} & \small{2} & \small{71} & \small{41} \\ |
2502 | \hline | \hline |
2503 | \small{7} & \small{1} & \small{97} & \small{56} \\ | \small{7} & \small{1} & \small{97} & \small{56} \\ |
2504 | \hline | \hline |
2505 | \small{8} & \small{2} & \small{265} & \small{153} \\ | \small{8} & \small{2} & \small{265} & \small{153} \\ |
2506 | \hline | \hline |
2507 | \small{9} & \small{1} & \small{362} & \small{209} \\ | \small{9} & \small{1} & \small{362} & \small{209} \\ |
2508 | \hline | \hline |
2509 | \end{tabular} | \end{tabular} |
2510 | \end{center} | \end{center} |
2511 | \end{table} | \end{table} |
2512 | ||
2513 | Note from Table \ref{tbl:ex:ccfr0:scba0:bestrapapptoirratnum} | Note from Table \ref{tbl:ex:ccfr0:scba0:bestrapapptoirratnum} |
2514 | that the 8-th order convergent, $s_8 = 265/153$, | that the 8-th order convergent, $s_8 = 265/153$, |
2515 | is the highest-ordered convergent with $q_k \leq 200$, so | is the highest-ordered convergent with $q_k \leq 200$, so |
2516 | by Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}, 265/153 | by Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}, 265/153 |
2517 | is one neighbor in $F_{100}$ to $\sqrt{3}$. Because $s_8$ | is one neighbor in $F_{100}$ to $\sqrt{3}$. Because $s_8$ |
2518 | is an even-ordered convergent, it will be the left | is an even-ordered convergent, it will be the left |
2519 | Farey neighbor. | Farey neighbor. |
2520 | By (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}), the | By (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}), the |
2521 | other Farey neighbor is 97/56, and it will be the right | other Farey neighbor is 97/56, and it will be the right |
2522 | Farey neighbor. | Farey neighbor. |
2523 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
2524 | \vworkexamplefooter{} | \vworkexamplefooter{} |
2525 | ||
2526 | ||
2527 | It is clear that Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} | It is clear that Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} |
2528 | can be trivially modified to find enclosing neighbors | can be trivially modified to find enclosing neighbors |
2529 | in $F_{k_{MAX},\overline{h_{MAX}}}$, and we present this | in $F_{k_{MAX},\overline{h_{MAX}}}$, and we present this |
2530 | trivial modification as Algorithm | trivial modification as Algorithm |
2531 | \ref{alg:ccfr0:scba0:cfenclosingneighborsfab}. | \ref{alg:ccfr0:scba0:cfenclosingneighborsfab}. |
2532 | ||
2533 | \begin{vworkalgorithmstatementpar}{Enclosing Neighbors Of | \begin{vworkalgorithmstatementpar}{Enclosing Neighbors Of |
2534 | \mbox{\boldmath $x \notin F_{k_{MAX},\overline{h_{MAX}}}$} | \mbox{\boldmath $x \notin F_{k_{MAX},\overline{h_{MAX}}}$} |
2535 | In \mbox{\boldmath $F_{k_{MAX},\overline{h_{MAX}}}$}} | In \mbox{\boldmath $F_{k_{MAX},\overline{h_{MAX}}}$}} |
2536 | \label{alg:ccfr0:scba0:cfenclosingneighborsfab} | \label{alg:ccfr0:scba0:cfenclosingneighborsfab} |
2537 | \begin{alglvl0} | \begin{alglvl0} |
2538 | \item If $a/b < h_{MAX}/k_{MAX}$, apply Algorithm | \item If $a/b < h_{MAX}/k_{MAX}$, apply Algorithm |
2539 | \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} directly; | \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} directly; |
2540 | ||
2541 | \item Else if $a/b > h_{MAX}/k_{MAX}$, apply Algorithm | \item Else if $a/b > h_{MAX}/k_{MAX}$, apply Algorithm |
2542 | \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} using $b/a$ rather | \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} using $b/a$ rather |
2543 | than $a/b$ as the input to the algorithm, using $h_{MAX}$ | than $a/b$ as the input to the algorithm, using $h_{MAX}$ |
2544 | rather than $k_{MAX}$ as $N$, and | rather than $k_{MAX}$ as $N$, and |
2545 | using the reciprocals of the results of the algorithm.\footnote{The | using the reciprocals of the results of the algorithm.\footnote{The |
2546 | basis for taking the reciprocals of input and output and | basis for taking the reciprocals of input and output and |
2547 | using $h_{MAX}$ rather than $k_{MAX}$ are explained | using $h_{MAX}$ rather than $k_{MAX}$ are explained |
2548 | in \cfryzeroxrefcomma{}Section \ref{cfry0:schk0}.} | in \cfryzeroxrefcomma{}Section \ref{cfry0:schk0}.} |
2549 | ||
2550 | \end{alglvl0} | \end{alglvl0} |
2551 | \end{vworkalgorithmstatementpar} | \end{vworkalgorithmstatementpar} |
2552 | %\vworkalgorithmfooter{} | %\vworkalgorithmfooter{} |
2553 | ||
2554 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
2555 | \label{ex:ccfr0:scba0:bestratapptoirratnum2d} | \label{ex:ccfr0:scba0:bestratapptoirratnum2d} |
2556 | Find the members of $F_{200,\overline{100}}$ which | Find the members of $F_{200,\overline{100}}$ which |
2557 | enclose $\sqrt{3}$. | enclose $\sqrt{3}$. |
2558 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
2559 | \begin{vworkexampleparsection}{Solution} | \begin{vworkexampleparsection}{Solution} |
2560 | It was shown in Example \ref{ex:ccfr0:scba0:bestrapapptoirratnum} | It was shown in Example \ref{ex:ccfr0:scba0:bestrapapptoirratnum} |
2561 | that the two enclosing neighbors to $\sqrt{3}$ in | that the two enclosing neighbors to $\sqrt{3}$ in |
2562 | $F_{200}$ are 265/153 and 97/56. Note that the first of | $F_{200}$ are 265/153 and 97/56. Note that the first of |
2563 | these neighbors, 265/153, violates the constraint on the | these neighbors, 265/153, violates the constraint on the |
2564 | numerator. | numerator. |
2565 | As explained in Section \cfryzeroxrefhyphen{}\ref{cfry0:schk0}, | As explained in Section \cfryzeroxrefhyphen{}\ref{cfry0:schk0}, |
2566 | because $\sqrt{3} > 100/200$, the constraint on the | because $\sqrt{3} > 100/200$, the constraint on the |
2567 | numerator is the dominant constraint, and the necessary | numerator is the dominant constraint, and the necessary |
2568 | approach is to find the neighbors of $1/\sqrt{3}$ in | approach is to find the neighbors of $1/\sqrt{3}$ in |
2569 | $F_{100}$, then invert the results. | $F_{100}$, then invert the results. |
2570 | Although we don't explain it in this work, | Although we don't explain it in this work, |
2571 | the reciprocal of a continued fraction can be formed by | the reciprocal of a continued fraction can be formed by |
2572 | ``right-shifting'' or ``left-shifting'' the continued fraction one position. | ``right-shifting'' or ``left-shifting'' the continued fraction one position. |
2573 | Thus, if $[1;1,2,1,2,1,2, \ldots{}]$ = $[1;\overline{1,2}]$ | Thus, if $[1;1,2,1,2,1,2, \ldots{}]$ = $[1;\overline{1,2}]$ |
2574 | is the continued fraction representation of $\sqrt{3}$, then | is the continued fraction representation of $\sqrt{3}$, then |
2575 | $[0;1,1,2,1,2,1, \ldots{}]$ = $[0;1,\overline{1,2}]$ is the | $[0;1,1,2,1,2,1, \ldots{}]$ = $[0;1,\overline{1,2}]$ is the |
2576 | continued fraction representation of $1/\sqrt{3}$. Using | continued fraction representation of $1/\sqrt{3}$. Using |
2577 | this result and constructing the convergents until | this result and constructing the convergents until |
2578 | $q_k \geq 100$ yields Table \ref{tbl:ex:ccfr0:scba0:bestratapptoirratnum2d}. | $q_k \geq 100$ yields Table \ref{tbl:ex:ccfr0:scba0:bestratapptoirratnum2d}. |
2579 | ||
2580 | \begin{table} | \begin{table} |
2581 | \caption{Application Of Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfab} | \caption{Application Of Algorithm \ref{alg:ccfr0:scba0:cfenclosingneighborsfab} |
2582 | To Find Members Of $F_{100}$ Which Enclose $1/\sqrt{3}$ | To Find Members Of $F_{100}$ Which Enclose $1/\sqrt{3}$ |
2583 | (Example \ref{ex:ccfr0:scba0:bestratapptoirratnum2d})} | (Example \ref{ex:ccfr0:scba0:bestratapptoirratnum2d})} |
2584 | \label{tbl:ex:ccfr0:scba0:bestratapptoirratnum2d} | \label{tbl:ex:ccfr0:scba0:bestratapptoirratnum2d} |
2585 | \begin{center} | \begin{center} |
2586 | \begin{tabular}{|c|c|c|c|} | \begin{tabular}{|c|c|c|c|} |
2587 | \hline | \hline |
2588 | \hspace{0.15in}\small{Index ($k$)}\hspace{0.15in} & \hspace{0.15in}\small{$a_k$}\hspace{0.15in} & | \hspace{0.15in}\small{Index ($k$)}\hspace{0.15in} & \hspace{0.15in}\small{$a_k$}\hspace{0.15in} & |
2589 | \hspace{0.15in}\small{$p_k$}\hspace{0.15in} & \hspace{0.15in}\small{$q_k$}\hspace{0.15in} \\ | \hspace{0.15in}\small{$p_k$}\hspace{0.15in} & \hspace{0.15in}\small{$q_k$}\hspace{0.15in} \\ |
2590 | \hline | \hline |
2591 | \hline | \hline |
2592 | \small{-1} & \small{N/A} & \small{1} & \small{0} \\ | \small{-1} & \small{N/A} & \small{1} & \small{0} \\ |
2593 | \hline | \hline |
2594 | \small{0} & \small{0} & \small{0} & \small{1} \\ | \small{0} & \small{0} & \small{0} & \small{1} \\ |
2595 | \hline | \hline |
2596 | \small{1} & \small{1} & \small{1} & \small{1} \\ | \small{1} & \small{1} & \small{1} & \small{1} \\ |
2597 | \hline | \hline |
2598 | \small{2} & \small{1} & \small{1} & \small{2} \\ | \small{2} & \small{1} & \small{1} & \small{2} \\ |
2599 | \hline | \hline |
2600 | \small{3} & \small{2} & \small{3} & \small{5} \\ | \small{3} & \small{2} & \small{3} & \small{5} \\ |
2601 | \hline | \hline |
2602 | \small{4} & \small{1} & \small{4} & \small{7} \\ | \small{4} & \small{1} & \small{4} & \small{7} \\ |
2603 | \hline | \hline |
2604 | \small{5} & \small{2} & \small{11} & \small{19} \\ | \small{5} & \small{2} & \small{11} & \small{19} \\ |
2605 | \hline | \hline |
2606 | \small{6} & \small{1} & \small{15} & \small{26} \\ | \small{6} & \small{1} & \small{15} & \small{26} \\ |
2607 | \hline | \hline |
2608 | \small{7} & \small{2} & \small{41} & \small{71} \\ | \small{7} & \small{2} & \small{41} & \small{71} \\ |
2609 | \hline | \hline |
2610 | \small{8} & \small{1} & \small{56} & \small{97} \\ | \small{8} & \small{1} & \small{56} & \small{97} \\ |
2611 | \hline | \hline |
2612 | \small{9} & \small{2} & \small{153} & \small{265} \\ | \small{9} & \small{2} & \small{153} & \small{265} \\ |
2613 | \hline | \hline |
2614 | \end{tabular} | \end{tabular} |
2615 | \end{center} | \end{center} |
2616 | \end{table} | \end{table} |
2617 | ||
2618 | Note from Table \ref{tbl:ex:ccfr0:scba0:bestratapptoirratnum2d} | Note from Table \ref{tbl:ex:ccfr0:scba0:bestratapptoirratnum2d} |
2619 | that the 8-th order convergent, $s_8 = 56/97$, | that the 8-th order convergent, $s_8 = 56/97$, |
2620 | is the highest-ordered convergent with $q_k \leq 100$, so | is the highest-ordered convergent with $q_k \leq 100$, so |
2621 | by Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}, 56/97 | by Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}, 56/97 |
2622 | is one neighbor in $F_{100}$ to $1/\sqrt{3}$. Because $s_8$ | is one neighbor in $F_{100}$ to $1/\sqrt{3}$. Because $s_8$ |
2623 | is an even-ordered convergent, it will be the left | is an even-ordered convergent, it will be the left |
2624 | Farey neighbor. | Farey neighbor. |
2625 | By (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}), the | By (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}), the |
2626 | other Farey neighbor is 41/71, and it will be the right | other Farey neighbor is 41/71, and it will be the right |
2627 | Farey neighbor. Taking the reciprocal of these neighbors (and | Farey neighbor. Taking the reciprocal of these neighbors (and |
2628 | reversing their order) yields $97/56 < \sqrt{3} < 41/71$ | reversing their order) yields $97/56 < \sqrt{3} < 41/71$ |
2629 | as the two members of $F_{200, \overline{100}}$ which enclose | as the two members of $F_{200, \overline{100}}$ which enclose |
2630 | $\sqrt{3}$. | $\sqrt{3}$. |
2631 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
2632 | \vworkexamplefooter{} | \vworkexamplefooter{} |
2633 | ||
2634 | ||
2635 | A natural question to ask is whether, given only a \emph{single} | A natural question to ask is whether, given only a \emph{single} |
2636 | rational number $a/b \in F_N$, the apparatus of continued fractions | rational number $a/b \in F_N$, the apparatus of continued fractions |
2637 | can be used to economically find its neighbors in $F_N$. Examining | can be used to economically find its neighbors in $F_N$. Examining |
2638 | the proof of Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}, | the proof of Theorem \ref{thm:ccfr0:scba0:cfenclosingneighbors}, |
2639 | we see that the entire proof applies even if the denominator of the | we see that the entire proof applies even if the denominator of the |
2640 | highest-order convergent, $q_n$, is less than or equal to $N$---that is, | highest-order convergent, $q_n$, is less than or equal to $N$---that is, |
2641 | the number specified by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | the number specified by (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
2642 | is a left or right Farey neighbor in $F_N$ of $a/b$. If $n$ is even, | is a left or right Farey neighbor in $F_N$ of $a/b$. If $n$ is even, |
2643 | $s_{n-1} > s_n$, and the number specified by | $s_{n-1} > s_n$, and the number specified by |
2644 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) will be the right Farey neighbor | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) will be the right Farey neighbor |
2645 | of $s_n$, and | of $s_n$, and |
2646 | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq03}) and | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq03}) and |
2647 | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq04}) | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq04}) |
2648 | can be used to find the left Farey neighbor. | can be used to find the left Farey neighbor. |
2649 | On the other hand if $n$ odd, $s_{n-1} < s_n$, (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | On the other hand if $n$ odd, $s_{n-1} < s_n$, (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
2650 | will be the left Farey neighbor of $s_n$, and | will be the left Farey neighbor of $s_n$, and |
2651 | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq01}) | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq01}) |
2652 | and | and |
2653 | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq02}) | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq02}) |
2654 | can be used to find the | can be used to find the |
2655 | right Farey neighbor. We summarize these observations as Algorithm | right Farey neighbor. We summarize these observations as Algorithm |
2656 | \ref{alg:ccfr0:scba0:cffareyneighborfn}. | \ref{alg:ccfr0:scba0:cffareyneighborfn}. |
2657 | ||
2658 | \begin{vworkalgorithmstatementpar}{Neighbors Of | \begin{vworkalgorithmstatementpar}{Neighbors Of |
2659 | \mbox{\boldmath $a/b \in F_N$} | \mbox{\boldmath $a/b \in F_N$} |
2660 | In \mbox{\boldmath $F_N$}} | In \mbox{\boldmath $F_N$}} |
2661 | \label{alg:ccfr0:scba0:cffareyneighborfn} | \label{alg:ccfr0:scba0:cffareyneighborfn} |
2662 | \end{vworkalgorithmstatementpar} | \end{vworkalgorithmstatementpar} |
2663 | \begin{alglvl0} | \begin{alglvl0} |
2664 | ||
2665 | \item Apply the first part of Algorithm | \item Apply the first part of Algorithm |
2666 | \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} to obtain | \ref{alg:ccfr0:scba0:cfenclosingneighborsfn} to obtain |
2667 | all of the partial quotients and convergents of $a/b$. | all of the partial quotients and convergents of $a/b$. |
2668 | The final convergent, $s_n = p_n/q_n$, will be | The final convergent, $s_n = p_n/q_n$, will be |
2669 | $a/b$ in reduced form. | $a/b$ in reduced form. |
2670 | ||
2671 | \item Use (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | \item Use (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
2672 | (with $k = n$) to obtain the right Farey neighbor | (with $k = n$) to obtain the right Farey neighbor |
2673 | (if $n$ is even) or the left Farey neighbor (if $n$ | (if $n$ is even) or the left Farey neighbor (if $n$ |
2674 | is odd). | is odd). |
2675 | ||
2676 | \item If $n$ is even, $s_{n-1} > s_n$, and the number specified by | \item If $n$ is even, $s_{n-1} > s_n$, and the number specified by |
2677 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) will be | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) will be |
2678 | the right Farey neighbor of $s_n$. Use | the right Farey neighbor of $s_n$. Use |
2679 | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq03}) and | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq03}) and |
2680 | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq04}) | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq04}) |
2681 | to find the left Farey neighbor. | to find the left Farey neighbor. |
2682 | On the other hand if $n$ is odd, $s_{n-1} < s_n$, | On the other hand if $n$ is odd, $s_{n-1} < s_n$, |
2683 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) |
2684 | will be the left Farey neighbor of $s_n$. Use | will be the left Farey neighbor of $s_n$. Use |
2685 | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq01}) | (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq01}) |
2686 | and (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq02}) | and (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq02}) |
2687 | to find the right Farey neighbor. | to find the right Farey neighbor. |
2688 | ||
2689 | \end{alglvl0} | \end{alglvl0} |
2690 | %\vworkalgorithmfooter{} | %\vworkalgorithmfooter{} |
2691 | ||
2692 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
2693 | \label{ex:ccfr0:scba0:fnneighborsaoverbinfn} | \label{ex:ccfr0:scba0:fnneighborsaoverbinfn} |
2694 | Find the neighbors of 5/7 in $F_{1,000,000}$. | Find the neighbors of 5/7 in $F_{1,000,000}$. |
2695 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
2696 | \begin{vworkexampleparsection}{Solution} | \begin{vworkexampleparsection}{Solution} |
2697 | As per Algorithm \ref{alg:ccfr0:scba0:cffareyneighborfn}, the first | As per Algorithm \ref{alg:ccfr0:scba0:cffareyneighborfn}, the first |
2698 | step is to obtain the partial quotients and convergents of 5/7 | step is to obtain the partial quotients and convergents of 5/7 |
2699 | (these partial quotients and convergents are shown in | (these partial quotients and convergents are shown in |
2700 | Table \ref{tbl:ex:ccfr0:scba0:fnneighborsaoverbinfn}). | Table \ref{tbl:ex:ccfr0:scba0:fnneighborsaoverbinfn}). |
2701 | ||
2702 | \begin{table} | \begin{table} |
2703 | \caption{Partial Quotients And Convergents Of 5/7 | \caption{Partial Quotients And Convergents Of 5/7 |
2704 | (Example \ref{ex:ccfr0:scba0:fnneighborsaoverbinfn})} | (Example \ref{ex:ccfr0:scba0:fnneighborsaoverbinfn})} |
2705 | \label{tbl:ex:ccfr0:scba0:fnneighborsaoverbinfn} | \label{tbl:ex:ccfr0:scba0:fnneighborsaoverbinfn} |
2706 | \begin{center} | \begin{center} |
2707 | \begin{tabular}{|c|c|c|c|c|c|c|} | \begin{tabular}{|c|c|c|c|c|c|c|} |
2708 | \hline | \hline |
2709 | \small{Index ($k$)} & \small{$dividend_k$} & \small{$divisor_k$} & \small{$a_k$} & \small{$remainder_k$} & \small{$p_k$} & \small{$q_k$} \\ | \small{Index ($k$)} & \small{$dividend_k$} & \small{$divisor_k$} & \small{$a_k$} & \small{$remainder_k$} & \small{$p_k$} & \small{$q_k$} \\ |
2710 | \hline | \hline |
2711 | \hline | \hline |
2712 | \small{-1} & \small{N/A} & \small{5} & \small{N/A} & \small{7} & \small{1} & \small{0} \\ | \small{-1} & \small{N/A} & \small{5} & \small{N/A} & \small{7} & \small{1} & \small{0} \\ |
2713 | \hline | \hline |
2714 | \small{0} & \small{5} & \small{7} & \small{0} & \small{5} & \small{0} & \small{1} \\ | \small{0} & \small{5} & \small{7} & \small{0} & \small{5} & \small{0} & \small{1} \\ |
2715 | \hline | \hline |
2716 | \small{1} & \small{7} & \small{5} & \small{1} & \small{2} & \small{1} & \small{1} \\ | \small{1} & \small{7} & \small{5} & \small{1} & \small{2} & \small{1} & \small{1} \\ |
2717 | \hline | \hline |
2718 | \small{2} & \small{5} & \small{2} & \small{2} & \small{1} & \small{2} & \small{3} \\ | \small{2} & \small{5} & \small{2} & \small{2} & \small{1} & \small{2} & \small{3} \\ |
2719 | \hline | \hline |
2720 | \small{3} & \small{2} & \small{1} & \small{2} & \small{0} & \small{5} & \small{7} \\ | \small{3} & \small{2} & \small{1} & \small{2} & \small{0} & \small{5} & \small{7} \\ |
2721 | \hline | \hline |
2722 | \end{tabular} | \end{tabular} |
2723 | \end{center} | \end{center} |
2724 | \end{table} | \end{table} |
2725 | ||
2726 | Since the final convergent, $s_{3}$, is an odd-ordered convergent, $s_{k-1} < s_k$, and | Since the final convergent, $s_{3}$, is an odd-ordered convergent, $s_{k-1} < s_k$, and |
2727 | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) will supply the left Farey neighbor | (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) will supply the left Farey neighbor |
2728 | of 5/7. Applying (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) with | of 5/7. Applying (\ref{eq:ccfr0:scba0:thm:cfenclosingneighbors:01}) with |
2729 | $N=1,000,000$, $k=3$, $k-1=2$, $p_k = 5$, $q_k = 7$, $p_{k-1}=2$, and $q_{k-1}=3$ | $N=1,000,000$, $k=3$, $k-1=2$, $p_k = 5$, $q_k = 7$, $p_{k-1}=2$, and $q_{k-1}=3$ |
2730 | yields $\frac{714,282}{999,995}$ as the left Farey neighbor of 5/7 in $F_{1,000,000}$. | yields $\frac{714,282}{999,995}$ as the left Farey neighbor of 5/7 in $F_{1,000,000}$. |
2731 | Application of (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq01}) | Application of (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq01}) |
2732 | and (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq02}) | and (\cfryzeroxrefhyphen{}\ref{eq:cfry0:sgfs0:thm:01:eq02}) |
2733 | yields $\frac{714,283}{999,996}$ as the right Farey neighbor. | yields $\frac{714,283}{999,996}$ as the right Farey neighbor. |
2734 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
2735 | %\vworkexamplefooter{} | %\vworkexamplefooter{} |
2736 | ||
2737 | ||
2738 | \begin{vworkalgorithmstatementpar}{Neighbors Of | \begin{vworkalgorithmstatementpar}{Neighbors Of |
2739 | \mbox{\boldmath $x \in F_{k_{MAX},\overline{h_{MAX}}}$} | \mbox{\boldmath $x \in F_{k_{MAX},\overline{h_{MAX}}}$} |
2740 | In \mbox{\boldmath $F_{k_{MAX},\overline{h_{MAX}}}$}} | In \mbox{\boldmath $F_{k_{MAX},\overline{h_{MAX}}}$}} |
2741 | \label{alg:ccfr0:scba0:cffareyneighborfab} | \label{alg:ccfr0:scba0:cffareyneighborfab} |
2742 | \begin{alglvl0} | \begin{alglvl0} |
2743 | \item If $a/b < h_{MAX}/k_{MAX}$, apply Algorithm | \item If $a/b < h_{MAX}/k_{MAX}$, apply Algorithm |
2744 | \ref{alg:ccfr0:scba0:cffareyneighborfn} directly; | \ref{alg:ccfr0:scba0:cffareyneighborfn} directly; |
2745 | ||
2746 | \item Else if $a/b > h_{MAX}/k_{MAX}$, apply Algorithm | \item Else if $a/b > h_{MAX}/k_{MAX}$, apply Algorithm |
2747 | \ref{alg:ccfr0:scba0:cffareyneighborfn} using $b/a$ rather | \ref{alg:ccfr0:scba0:cffareyneighborfn} using $b/a$ rather |
2748 | than $a/b$ as the input to the algorithm, using $h_{MAX}$ | than $a/b$ as the input to the algorithm, using $h_{MAX}$ |
2749 | rather than $k_{MAX}$ as $N$, and | rather than $k_{MAX}$ as $N$, and |
2750 | using the reciprocals of the results of the algorithm.\footnote{The | using the reciprocals of the results of the algorithm.\footnote{The |
2751 | basis for taking the reciprocals of input and output and | basis for taking the reciprocals of input and output and |
2752 | using $h_{MAX}$ rather than $k_{MAX}$ are explained | using $h_{MAX}$ rather than $k_{MAX}$ are explained |
2753 | in \cfryzeroxrefcomma{}Section \ref{cfry0:schk0}.} | in \cfryzeroxrefcomma{}Section \ref{cfry0:schk0}.} |
2754 | \end{alglvl0} | \end{alglvl0} |
2755 | \end{vworkalgorithmstatementpar} | \end{vworkalgorithmstatementpar} |
2756 | \vworkalgorithmfooter{} | \vworkalgorithmfooter{} |
2757 | ||
2758 | ||
2759 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
2760 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
2761 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
2762 | \section{The Stern-Brocot Tree} | \section{The Stern-Brocot Tree} |
2763 | %Section tag: SBT0 | %Section tag: SBT0 |
2764 | \label{ccfr0:ssbt0} | \label{ccfr0:ssbt0} |
2765 | ||
2766 | In this chapter, we've developed continued fraction techniques of best | In this chapter, we've developed continued fraction techniques of best |
2767 | rational approximation without reference to any other models or theory. | rational approximation without reference to any other models or theory. |
2768 | Because the algorithms presented in this chapter are $O(log \; N)$, | Because the algorithms presented in this chapter are $O(log \; N)$, |
2769 | the results so far are completely satisfactory and usable in practice. | the results so far are completely satisfactory and usable in practice. |
2770 | It is not necessary to go further or to present additional information. | It is not necessary to go further or to present additional information. |
2771 | ||
2772 | However, there is a second model of best rational approximation (and a second | However, there is a second model of best rational approximation (and a second |
2773 | set of algorithms), involving | set of algorithms), involving |
2774 | the Stern-Brocot tree. In fact, when reviewing the material in this | the Stern-Brocot tree. In fact, when reviewing the material in this |
2775 | chapter, some readers have inquired why the Stern-Brocot tree was not | chapter, some readers have inquired why the Stern-Brocot tree was not |
2776 | used.\footnote{In brief, the Stern-Brocot tree was not used because | used.\footnote{In brief, the Stern-Brocot tree was not used because |
2777 | the resulting algorithms are $O(N)$, and so will introduce practical | the resulting algorithms are $O(N)$, and so will introduce practical |
2778 | computational difficulties when used with large integers.} In this | computational difficulties when used with large integers.} In this |
2779 | section, we introduce the Stern-Brocot tree, demonstrate how to construct it, | section, we introduce the Stern-Brocot tree, demonstrate how to construct it, |
2780 | mention its major properties, show its correspondence with the apparatus of | mention its major properties, show its correspondence with the apparatus of |
2781 | continued fractions, and finally show why we \emph{must} use the apparatus | continued fractions, and finally show why we \emph{must} use the apparatus |
2782 | of continued fractions to find best rational approximations. | of continued fractions to find best rational approximations. |
2783 | ||
2784 | ||
2785 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
2786 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
2787 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
2788 | \subsection{Definition And Properties Of The Stern-Brocot Tree} | \subsection{Definition And Properties Of The Stern-Brocot Tree} |
2789 | %Section tag: DPT0 | %Section tag: DPT0 |
2790 | \label{ccfr0:ssbt0:sdpt0} | \label{ccfr0:ssbt0:sdpt0} |
2791 | ||
2792 | \index{Stern-Brocot tree}The Stern-Brocot tree | \index{Stern-Brocot tree}The Stern-Brocot tree |
2793 | (Figure \ref{fig:ccfr0:ssbt0:sdpt0:00}), is | (Figure \ref{fig:ccfr0:ssbt0:sdpt0:00}), is |
2794 | an infinite binary tree which contains all positive rational numbers. | an infinite binary tree which contains all positive rational numbers. |
2795 | ||
2796 | \begin{figure} | \begin{figure} |
2797 | \centering | \centering |
2798 | \includegraphics[width=4.6in]{c_cfr0/sbtdpt01.eps} | \includegraphics[width=4.6in]{c_cfr0/sbtdpt01.eps} |
2799 | \caption{The Stern-Brocot Tree} | \caption{The Stern-Brocot Tree} |
2800 | \label{fig:ccfr0:ssbt0:sdpt0:00} | \label{fig:ccfr0:ssbt0:sdpt0:00} |
2801 | \end{figure} | \end{figure} |
2802 | ||
2803 | To construct the tree, one begins with the two fractions $\frac{0}{1}$ | To construct the tree, one begins with the two fractions $\frac{0}{1}$ |
2804 | and $\frac{1}{0}$, and forms the mediant (see Definition | and $\frac{1}{0}$, and forms the mediant (see Definition |
2805 | \cfryzeroxrefhyphen{}\ref{def:cfry0:spfs:02}) | \cfryzeroxrefhyphen{}\ref{def:cfry0:spfs:02}) |
2806 | of two adjacent fractions as many | of two adjacent fractions as many |
2807 | times as desired to generate additional fractions. | times as desired to generate additional fractions. |
2808 | Figure \ref{fig:ccfr0:ssbt0:sdpt0:00} illustrates the construction process. | Figure \ref{fig:ccfr0:ssbt0:sdpt0:00} illustrates the construction process. |
2809 | Note in Figure \ref{fig:ccfr0:ssbt0:sdpt0:00} that the adjacent fractions | Note in Figure \ref{fig:ccfr0:ssbt0:sdpt0:00} that the adjacent fractions |
2810 | are always above and to the left and above and to the right of the fraction | are always above and to the left and above and to the right of the fraction |
2811 | being constructed, and that in the construction of the Stern-Brocot | being constructed, and that in the construction of the Stern-Brocot |
2812 | tree, one of the adjacent fractions can be many levels upwards in the tree | tree, one of the adjacent fractions can be many levels upwards in the tree |
2813 | from the fraction being constructed. For example, in | from the fraction being constructed. For example, in |
2814 | Figure \ref{fig:ccfr0:ssbt0:sdpt0:00}, when constructing the fraction | Figure \ref{fig:ccfr0:ssbt0:sdpt0:00}, when constructing the fraction |
2815 | 4/5, the left adjacent fraction (3/4) is nearby in the figure, but | 4/5, the left adjacent fraction (3/4) is nearby in the figure, but |
2816 | the right adjacent fraction (1/1) is three levels up to the left and | the right adjacent fraction (1/1) is three levels up to the left and |
2817 | one level up to the right. Note when constructing the fraction 4/5 that | one level up to the right. Note when constructing the fraction 4/5 that |
2818 | its right adjacent fraction is \emph{not} 4/3. | its right adjacent fraction is \emph{not} 4/3. |
2819 | ||
2820 | Note that it is also possible to maintain the Stern-Brocot tree as an | Note that it is also possible to maintain the Stern-Brocot tree as an |
2821 | ordered list, rather than a tree, starting with the list | ordered list, rather than a tree, starting with the list |
2822 | $\{0/1, 1/0\}$. An additional element may be inserted | $\{0/1, 1/0\}$. An additional element may be inserted |
2823 | between any two existing elements in the list by forming their mediant, | between any two existing elements in the list by forming their mediant, |
2824 | and this process may be repeated indefinitely. Note also that two | and this process may be repeated indefinitely. Note also that two |
2825 | elements $s_L$ and $s_R$ are Farey neighbors to any number $\alpha$ | elements $s_L$ and $s_R$ are Farey neighbors to any number $\alpha$ |
2826 | if $s_L < \alpha < s_R$ and the mediant of $s_L$ and $s_R$ has a | if $s_L < \alpha < s_R$ and the mediant of $s_L$ and $s_R$ has a |
2827 | denominator larger than the order of the Farey series. This gives a convenient | denominator larger than the order of the Farey series. This gives a convenient |
2828 | procedure for forming best rational approximations using only the Stern-Brocot | procedure for forming best rational approximations using only the Stern-Brocot |
2829 | tree, as the following example shows. | tree, as the following example shows. |
2830 | ||
2831 | \begin{vworkexamplestatement} | \begin{vworkexamplestatement} |
2832 | \label{ex:ccfr0:ssbt0:sdpt0:01} | \label{ex:ccfr0:ssbt0:sdpt0:01} |
2833 | Find the members of $F_{10}$ which enclose $\pi$. | Find the members of $F_{10}$ which enclose $\pi$. |
2834 | \end{vworkexamplestatement} | \end{vworkexamplestatement} |
2835 | \begin{vworkexampleparsection}{Solution} | \begin{vworkexampleparsection}{Solution} |
2836 | By repeatedly calculating mediants, terms can be added to the list | By repeatedly calculating mediants, terms can be added to the list |
2837 | $\{\frac{0}{1}, \frac{1}{0} \}$ until $\pi$ is enclosed and it is not | $\{\frac{0}{1}, \frac{1}{0} \}$ until $\pi$ is enclosed and it is not |
2838 | possible to generate additional enclosing terms whose denominator does not | possible to generate additional enclosing terms whose denominator does not |
2839 | exceed 10. This process is carried out below. | exceed 10. This process is carried out below. |
2840 | ||
2841 | \begin{equation} | \begin{equation} |
2842 | \left\{ {\frac{0}{1}, \frac{1}{0} } \right\}, | \left\{ {\frac{0}{1}, \frac{1}{0} } \right\}, |
2843 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{1}{0} } \right\}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{1}{0} } \right\}, |
2844 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{1}{0} } \right\}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{1}{0} } \right\}, |
2845 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{3}{1}, \frac{1}{0} } \right\}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{3}{1}, \frac{1}{0} } \right\}, |
2846 | \end{equation} | \end{equation} |
2847 | ||
2848 | \begin{equation} | \begin{equation} |
2849 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{3}{1}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{3}{1}, |
2850 | \frac{4}{1}, \frac{1}{0} } \right\}, | \frac{4}{1}, \frac{1}{0} } \right\}, |
2851 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{3}{1}, \frac{7}{2}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{3}{1}, \frac{7}{2}, |
2852 | \frac{4}{1}, \frac{1}{0} } \right\}, | \frac{4}{1}, \frac{1}{0} } \right\}, |
2853 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{3}{1}, \frac{10}{3}, \frac{7}{2}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{3}{1}, \frac{10}{3}, \frac{7}{2}, |
2854 | \frac{4}{1}, \frac{1}{0} } \right\}, | \frac{4}{1}, \frac{1}{0} } \right\}, |
2855 | \end{equation} | \end{equation} |
2856 | ||
2857 | \begin{equation} | \begin{equation} |
2858 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, |
2859 | \frac{3}{1}, \frac{13}{4}, \frac{10}{3}, \frac{7}{2}, | \frac{3}{1}, \frac{13}{4}, \frac{10}{3}, \frac{7}{2}, |
2860 | \frac{4}{1}, \frac{1}{0} } \right\}, | \frac{4}{1}, \frac{1}{0} } \right\}, |
2861 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, |
2862 | \frac{3}{1}, \frac{16}{5}, \frac{13}{4}, \frac{10}{3}, \frac{7}{2}, | \frac{3}{1}, \frac{16}{5}, \frac{13}{4}, \frac{10}{3}, \frac{7}{2}, |
2863 | \frac{4}{1}, \frac{1}{0} } \right\}, | \frac{4}{1}, \frac{1}{0} } \right\}, |
2864 | \end{equation} | \end{equation} |
2865 | ||
2866 | \begin{equation} | \begin{equation} |
2867 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, |
2868 | \frac{3}{1}, \frac{19}{6}, \frac{16}{5}, \frac{13}{4}, \frac{10}{3}, \frac{7}{2}, | \frac{3}{1}, \frac{19}{6}, \frac{16}{5}, \frac{13}{4}, \frac{10}{3}, \frac{7}{2}, |
2869 | \frac{4}{1}, \frac{1}{0} } \right\}, | \frac{4}{1}, \frac{1}{0} } \right\}, |
2870 | \end{equation} | \end{equation} |
2871 | ||
2872 | \begin{equation} | \begin{equation} |
2873 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, |
2874 | \frac{3}{1}, \frac{22}{7}, \frac{19}{6}, | \frac{3}{1}, \frac{22}{7}, \frac{19}{6}, |
2875 | \frac{16}{5}, \frac{13}{4}, \frac{10}{3}, \frac{7}{2}, | \frac{16}{5}, \frac{13}{4}, \frac{10}{3}, \frac{7}{2}, |
2876 | \frac{4}{1}, \frac{1}{0} } \right\}, | \frac{4}{1}, \frac{1}{0} } \right\}, |
2877 | \end{equation} | \end{equation} |
2878 | ||
2879 | \begin{equation} | \begin{equation} |
2880 | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, | \left\{ {\frac{0}{1}, \frac{1}{1}, \frac{2}{1}, |
2881 | \frac{3}{1}, \frac{25}{8}, \frac{22}{7}, \frac{19}{6}, | \frac{3}{1}, \frac{25}{8}, \frac{22}{7}, \frac{19}{6}, |
2882 | \frac{16}{5}, \frac{13}{4}, \frac{10}{3}, \frac{7}{2}, | \frac{16}{5}, \frac{13}{4}, \frac{10}{3}, \frac{7}{2}, |
2883 | \frac{4}{1}, \frac{1}{0} } \right\}. | \frac{4}{1}, \frac{1}{0} } \right\}. |
2884 | \end{equation} | \end{equation} |
2885 | ||
2886 | Note that 25/8 and 22/7 are the left and right neighbors to | Note that 25/8 and 22/7 are the left and right neighbors to |
2887 | $\pi$ in $F_{10}$, since $25/8 < \pi < 22/7$, and since the | $\pi$ in $F_{10}$, since $25/8 < \pi < 22/7$, and since the |
2888 | mediant of 25/8 and 22/7 (49/15) has a denominator which is | mediant of 25/8 and 22/7 (49/15) has a denominator which is |
2889 | too large for the Farey series being considered. | too large for the Farey series being considered. |
2890 | ||
2891 | Note also that the construction process above could be | Note also that the construction process above could be |
2892 | trivially amended to treat the case of a constrained numerator | trivially amended to treat the case of a constrained numerator |
2893 | rather than a constrained denominator. | rather than a constrained denominator. |
2894 | \end{vworkexampleparsection} | \end{vworkexampleparsection} |
2895 | \vworkexamplefooter{} | \vworkexamplefooter{} |
2896 | ||
2897 | The Stern-Brocot tree has many remarkable properties (especially in view of the | The Stern-Brocot tree has many remarkable properties (especially in view of the |
2898 | simplicity of its construction). We mention the following properties | simplicity of its construction). We mention the following properties |
2899 | without proof. | without proof. |
2900 | ||
2901 | \begin{itemize} | \begin{itemize} |
2902 | \item Each rational number in the tree is irreducible. | \item Each rational number in the tree is irreducible. |
2903 |