Continued fraction

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In mathematics, continued or complex fractions are expressed as follows:

$$x=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots }}}}$$

Here $a_0$is some integer, and all other numbers $a_n$Both are positive integers, and longer expressions can be defined accordingly. If the partial numerator and partial denominator are allowed to assume arbitrary values, including functions in some contexts, the resulting expression is a Generalized continued fraction . When it is necessary to distinguish the above standard forms from generalized continued fractions, it may be called simple or normal continued fractions, or is said to be in canonical form.

1. Common applications

Continued fractions are often used to approximate irrational numbers, for example:

$$\sqrt{2}=1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\ddots }}}}}}$$

This gives $\sqrt{2}$The asymptotic fraction of :

$$\frac{1}{1},\frac{3}{2},\frac{7}{5},\frac{17}{12}、\dots$$
$$\frac{\sqrt{5}+1}{2}=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots }}}}}}$$

This gives the asymptotic fraction of the golden section:

$$\frac{1}{1},\frac{2}{1},\frac{3}{2},\frac{5}{3},\frac{8}{5},\frac{13}{8}、\dots$$

Note that the denominators of the above series are $1,1,2,3,\cdots$ The same numbers can be arranged in order to get the Fibonacci sequence.

$$Pi=3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\frac{1}{1+\frac{1}{1+\ddots }}}}}}$$

From this we get the asymptotic fraction of pi $\frac{3}{1},\frac{22}{7}$(about rate), $\frac{333}{106},\frac{355}{113}$(density rate), $\frac{103993}{33102}\frac{103993}{33102}$、$\cdots$

It can be shown mathematically that an asymptotic fraction obtained from a (narrowly defined) continued fraction has a value that is the closest approximation to the exact value in a fraction whose numerator or denominator is smaller than the next asymptotic fraction.

2. Research motivation

The motivation for studying continued fractions stems from the desire to have a "mathematically pure" representation of real numbers.

Most people are familiar with the decimal representation of real numbers:

$$r=\sum _{i=0}^{\infty }{a}_{i}10^{-i}$$

Here $a_0$Can be any integer, other $a_i$All are { 0 , 1 , 2 , … , 9 } \{0,1,2,\ldots ,9\}{ 0,1,2,…,9 } an element. In this representation, for example the number$\pi$ is represented as a sequence of integers{ 3 , 1 , 4 , 1 , 5 , 9 , 2 , … } \{3,1,4,1,5,9,2,\ldots \}{ 3,1,4,1,5,9,2,…}。

This kind of decimal means something is wrong. For example, the constant 10 is used in this case because we used $Decimal$ system. We can also use octal or binary systems. Another problem is that many rational numbers lack finite representation in this system. For example, the number$\frac{1}{3}$is represented as an infinite sequence { 0 , 3 , 3 , 3 , 3 , … } \{0,3,3,3,3,\ldots \}{ 0,3,3,3,3,…}。

Continued fraction notation avoids these two problems of real number representation. Let us consider how to describe a number such as $\frac{415}{93}$, about $4.4624$ . approximately$4$ , which is actually better than$4$ is a little more, about$4+\frac{1}{2}$. but the 2 2 in the denominator$2$ is inaccurate; a more accurate denominator is$2$ a little more, about$2+\frac{1}{6}$, so $\frac{415}{93}$Approximately $4+\frac{1}{2+\frac{1}{6}}$. but the 6 in the denominator 6$6$ is inaccurate; a more accurate denominator is 6 than$A\; little\; more\; than\; 6$ , actually$6+\frac{1}{7}$. So $\frac{415}{93}$Actually $4+\frac{1}{2+\frac{1}{6+\frac{1}{7}}}$. This is accurate.

Remove the expression $4+\frac{1}{2+\frac{1}{6+\frac{1}{7}}}$The redundant part in can get short notation $[4;2,6,7]$。

The continued fraction representation of real numbers can be defined in this way. It has some desirable properties:

The continued fraction representation of a rational number is finite.

• Continued fraction representations of "simple" rational numbers are short.
• The continued fraction representation of any rational number is unique if it has no trailing 1s. ( $[a_0;a_1,\dots ,a_n,1]=[a_0;a_1,\dots ,a_n+1]$）
• The continued fraction representation of irrational numbers is unique.
• The term of a continued fraction loops if and only if it is a continued fraction representation of a quadratic irrational number (i.e., a real solution to a quadratic equation with integer coefficients).
• number xxThe truncated continued fraction of$x$$The\; "best\; possible"\; rational\; approximation\; of\; x$ in a specific sense (see Corollary 1 of Theorem 5 below).

This last property is very important, and cannot be the case with traditional decimal point representation. A truncated decimal representation of a number yields a rational approximation to the number, but usually not a very good approximation. For example, truncation $\frac{1}{7}=0.142857...$ produces approximation ratios at various positions such as$\frac{142}{1000}$、$\frac{14}{100}$and $\frac{1}{10}$. But the obvious best rational approximation is " $\frac{1}{7}$" itself. $The\; truncated\; decimal\; representation\; of\; \pi$ yields approximation ratios such as$\frac{31415}{10000}$and $\frac{314}{100}$。$The\; continued\; fraction\; representation\; of\; \pi$ starts at$[3;7,15,1,292,...]$ . Truncating this representation yields an excellent rational approximation to$3$、$\frac{22}{7}$、$\frac{333}{106}$、$\frac{355}{113}$、$\frac{103993}{33102}$、$\cdots$。$\frac{314}{100}$and $\frac{333}{106}$The denominators of are quite close, but the approximate value $\frac{314}{100}$The error is much higher than $\frac{333}{106}$19 $19$ times. as pairπ \piApproximation of$\pi$$[3;7,15,1]$ than$3.1416$ Exact$100$ times.

3. Algorithm for representation of continued fractions

Consider the real number $r$ . ii$i$ is$the\; integer\; part\; of\; r$ , while$f$ is its fractional part. thenrrThe continued fraction representation of$r$$[i;\dots ]$ , where “$\dots$ " is$\frac{1}{f}$The continued fraction representation of . It is customary to replace the first comma with a semicolon.

To calculate the real number $The\; continued\; fraction\; representation\; of\; r$ , first write$The\; integer\; part\; of\; r$ (floored), then from$r$ subtracts this integer part. If the difference is$0$ to stop; otherwise find the inverse of this difference and repeat. The process will terminate if and only if$r$ is a rational number.

find 3.245 3.245The continued fraction of$3.245$
$3$	$3.245-3$	= 0.245	$\frac{1}{0.245}$	$=\mathrm{4.082...}$
$4$	$\mathrm{4.082...}-4$	$=\mathrm{0.082...}$	$\frac{1}{\mathrm{0.082...}}$	$=12.250$
$12$	$12.250-12$	$=0.250$	$\frac{1}{0.250}$	$=4.000$
$4$	$4.000-4$	$=0.000$	stop
The continued fraction of 3.245 is $[3;4,12,4]$
$3.245=3+\frac{1}{4+\frac{1}{12+\frac{1}{4}}}$

Number $3.245$ can also be expressed as a continued fraction expansion$[3;4,12,3,1]$ ; see finite continued fractions below.

This algorithm works well for real numbers, but can lead to numerical disaster if implemented with floating point numbers. Instead, any floating-point number is an exact rational number (on modern computers the denominator is usually $power\; of\; 2$ , usually 10 10on electronic calculators$powers\; of\; 10$ ), so variants of Euclid's algorithm can be used to give exact results.

4. Representation of continued fractions

Continued fractions can be abbreviated as:

$$x=[a_0;a_1,a_2,a_3]$$

Or, written in Pringsheim's notation:

$$x=a_0+\frac{1\mid }{\mid a_1}+\frac{1|}{|a_2}+\frac{1|}{|a_3}$$

There is also a related notation:

$$x=a_0+\frac{1}{a_1+}\frac{1}{a_2+}\frac{1}{a_3+}$$

Angle brackets are sometimes used, as in:

$$x=⟨a_0;a_1,a_2,a_3⟩$$

Semicolons are optional when using angle brackets.

It is also possible to define infinite simple continued fractions as limits:

$$[a_0;a_1,a_2,a_3,\dots ]=\underset{n\to \infty }{\lim }[a_0;a_1,a_2,\dots ,a_n]$$

For positive integers $a_1,a_2,a_{3}Any\; choice\; of\; ...$ has this limit.

Or you can use Gaussian notation

$$x=a_0+\underset{i=1}{\stackrel{3}{\mathrm{K}}}\frac{1}{a_i}$$

5. Finite continued fractions

All finite continued fractions represent a rational number, and all rational numbers can be represented as finite continued fractions in two different ways. Both representations are identical except for the final term. expressed in longer continued fractions whose final term is $1$ ; the shorter means the last$1$ , and add 1to the new terminal term$1$ . The final term in the short representation is thus greater than$1$ if short means at least two terms. Its symbol means:

$$[a_0;a_1,a_2,a_3,\dots ,a_n,1]=[a_0;a_1,a_2,a_3,\dots ,a_n+1]$$

For example:

$$\begin{gathered} 2.25=\frac{9}{4}=[2;3,1]=[2;4] \\ -4.2=-\frac{21}{5}=[-5;1,3,1]=[-5;1,4] \end{gathered}$$

6. Reciprocal of continued fractions

The continued fraction representation of a rational number is the same as its reciprocal except that it is shifted left or right by one bit depending on whether the number is less than or greater than 1, respectively. In other words, $[a_0;a_1,a_2,a_3,\dots ,a_n]$ 和 $[0;a_0,a_1,a_2,\dots ,a_n]$ are reciprocals of each other. This is because if $a$ is an integer, then if$x<1$ ，则 $x=0+\frac{1}{a+\frac{1}{b}}$ 且 $\frac{1}{x}=a+\frac{1}{b}$, and if $x>1$ ,$x=a+\frac{1}{b}$ 且 $\frac{1}{x}=0+\frac{1}{a+\frac{1}{b}}$Generate pair xx with last number$x$ and its reciprocal are remainders of the same continued fraction.

For example:

$$\begin{gathered} 2.25=\frac{9}{4}=[2;4] \\ \frac{1}{2.25}=\frac{4}{9}=[0;2,4] \end{gathered}$$

7. Infinitely continued fractions

All infinitely continued fractions are irrational numbers, and all irrational numbers can be represented in an exact way as infinitely continued fractions.

The infinite continued fraction representation of irrational numbers is very useful because its initial segment provides an excellent rational approximation to this number. These rational numbers can be called the convergence of this continued fraction (convergent, also translated as "asymptotic"). All even numbers converge less than the original number, and odd numbers converge greater than it.

For continued fractions $[a_0;a_1,a_2,\dots ]$ , the first four convergent (number$0$ to$3$ is

$$\frac{a_0}{1},\frac{a_1{a}_0+1}{a_1},\frac{a_2(a_1{a}_0+1)+a_0}{a_2{a}_1+1},\frac{a_3\left[a_2\right(a_1{a}_0+1)+a_0]+(a_1{a}_0+1)}{a_3(a_2{a}_1+1)+a_1}$$

In plain language, number $The\; 3$ convergent molecules are obtained by the 3rd$3$ quotients ($a_2$) times the $2$ convergent numerators, plus the 1st$1$ convergent molecule. The formation of the denominator is similar.

If a continuous convergence is found, with molecules $h_1,h_2,\dots$ and denominators$k_1,k_2,\dots$ , then the relevant recurrence relation is:

$$h_n=a_n{h}_{n-1}+h_{n-2},k_n=a_n{k}_{n-1}+k_{n-2}$$

Successive convergence is given by

$$\frac{h_n}{k_n}=\frac{a_n{h}_{n-1}+h_{n-2}}{a_n{k}_{n-1}+k_{n-2}}$$

8. Some useful theorems

如果 $a_0,a_1,a_2,...$ is an infinite sequence of positive integers, recursively defined sequence$h_n$and $k_n$：

$$\begin{gathered} h_n=a_n{h}_{n-1}+h_{n-2}{h}_{-1}=1h_{-2}=0 \\ {k}_n=a_n{k}_{n-1}+k_{n-2}{k}_{-1}=0k_{-2}=1 \end{gathered}$$

Theorem 1

For any positive number $x\in \mathbb{R}$

$$\left[a_0;a_1,\dots ,a_{n-1},x\right]=\frac{x{h}_{n-1}+h_{n-2}}{x{k}_{n-1}+k_{n-2}}$$

Theorem 2

$[a_0,a_1,a_2,\dots ]$ converge to

$$\left[a_0;a_1,\dots ,a_n\right]=\frac{h_n}{k_n}$$

give.

Theorem 3

If the $n$ convergence is$h_n/k_n$,but

$$k_n{h}_{n-1}-k_{n-1}{h}_n=(-1{)}^n$$

• Corollary 1: Every convergence is in its lowest terms (if $h_n$and $k_n$has unusual common divisor, then it divides $k_n{h}_{n-1}-k_{n-1}{h}_n$, which is of course impossible).

• Corollary 2: The difference between successive convergences is a unit fraction:

$$\left|\frac{h_n}{k_n}-\frac{h_{n-1}}{k_{n-1}}\right|=\left|\frac{h_n{k}_{n-1}-k_n{h}_{n-1}}{k_n{k}_{n-1}}\right|=\frac{1}{k_n{k}_{n-1}}$$

• Corollary 3: Continued fractions are equivalent to series of alternating terms:

$$a_0+\sum _{n=0}^{\infty }\frac{(-1{)}^n}{k_{n+1}{k}_n}$$

• Corollary 4: Matrix

$$\left[\begin{array}{cc}{h}_n& h_{n-1}\\ k_n& k_{n-1}\end{array}\right]$$

The determinant value of is positive $1$ or minus$1$ , thus belonging to$2\times 2$ unimodular matrix$S^{\ast }L(2,Z)$ group.

Theorem 4

Each (th $s$ ) are greater than any preceding ($r$ ) converges closer to the subsequent ($n$ ) converge. In notational terms, if the$n$ convergence is$[a_0;a_1,a_2,\dots a_n]=x_n$,but

$$\left|x_r-x_n\right|>|x_s-x_n|$$

For all $r<s<n$。

• Corollary 1: Odd convergence (at $n$ before) keeps increasing and is always less than$x_n$。

• Corollary 2: Convergence in even numbers (at $n$ before) keeps decreasing and is always greater than$x_n$。

Theorem 5

$$\frac{1}{k_n(k_{n+1}+k_n)}<\left|x-\frac{h_n}{k_n}\right|<\frac{1}{k_n{k}_{n+1}}$$

• Corollary 1: Any convergent fraction is closer to this continued fraction than any other fraction whose denominator is smaller than the convergent denominator.

• Corollary 2: Any convergence immediately preceding a large quotient is a close approximation to this continued fraction.

9. Semi-convergent

If $\frac{h_{n-1}}{k_{n-1}}$and $\frac{h_n}{k_n}$is continuous convergence, then any fraction of the form

$$\frac{h_{n-1}+a{h}_n}{k_{n-1}+to{k}_n}$$

aa here$a$ is a non-negative integer, and the numerator and denominator are in$n$ and$n+1$ term (inclusive), called "semi-convergent", sub-convergent or intermediate fractions. The term often means that ruling out the possibility is convergence, not that convergence is a kind of semi-convergence.

For the real number $The\; semiconvergence\; of\; the\; continued\; fraction\; expansion\; of\; x$ includes all rational approximations that are better than any approximation with a smaller denominator. Another useful property is that the continuous semiconvergence$\frac{a}{b}$and $\frac{c}{d}$With $ad-bc=\pm 1$。

10. Best Rational Number Approximation

The theory of continued fractions plays a fundamental role in the field of Diophantine approximation, and can solve the problem of optimal approximation of real numbers. For details, please refer to the corresponding main page. In fact, the initial motivation for developing the theory of continued fractions was precisely to solve the problem of the best approximation of real numbers.

11. Continued fraction history

• 300 BC - Euclid, "Elements" - The algorithm for the greatest common divisor produces a continued fraction as a by-product.
• Aryabhatiya 499 contains the solution of uncertain equations using continued fractions.
• 1572 - Rafael Bombelli, L'Algebra Opera - Method for extracting square roots related to continued fractions.
• 1613 - Pietro Cataldi, "Trattato del modo brevissimo di trovar la radice quadra delli numeri" - The first notation for continued fractions
  Cataldi expresses continued fractions as $a_0.\&\frac{n_1}{d_1}。\&\frac{n_2}{d_2}。\&\frac{n_3}{d_3}$with dots indicating where subsequent continued fractions go.
• 1695 - John Wallis, Opera Mathematica - Introduces the term "continued fractions".
• 1737 Leonhard Euler, De fractionibus continuis dissertatio - provides the first comprehensive account of the properties of continued fractions and is the first to demonstrate the number $e$ is an irrational number.
• 1748 Leonhard Euler, "Introductio in analysis in infinitorum". Vol. I, Chapter 18 - Proves the equivalence of continued fractions of a particular form and generalized infinite series. Proved that every rational number can be written as a finite continued fraction, and proved that the continued fraction of irrational numbers is infinite.
• 1768 Joseph-Louis Lagrange - Provides a general solution to Pell's equations using continued fractions similar to Bombelli.
• 1761 Johann Lambert - using $The\; continued\; fraction\; of\; tan\left(x\right)$ proved for the first time thatπ \piThe irrationality of$\pi \; .$
• 1770 Lagrange - proved that quadratic irrational numbers extend to periodic continued fractions.
• 1813 - Carl Friedrich Gauss, Werke, Book III, pp. 134-138 - Deduces very general complex-valued continued fractions by a clever identity involving hypergeometric series.
• In 1892 Henri Padé defined the Padé approximation
• 1972 Bill Gosper - First exact algorithm for continued fraction arithmetic.

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• references

(Former Soviet Union) A. Ya. Khinchin (A. Ya. Khinchin), translated by Liu Shijun and Liu Shaoyue. Continued Fractions. Shanghai: Shanghai Science and Technology Press. 1965.

Wikipedia: Continued fractions

wiki: Continued fraction