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: where ''a''0 is some Integer and all the other numbers ''a''''n'' are ''positive'' integers. Longer expressions are defined analogously. If the numerators are allowed to differ from unity, the resulting expression is a Generalized Continued Fraction . For clarity a ''non-generalized'' continued fraction is also called a simple continued fraction. MOTIVATION Continued fractions are motivated by a desire to have a "mathematically pure" representation for the Real Number s. Most people are familar with the Decimal Representation of real numbers: : where ''a''0, may be any integer, and each ''a''''i'' is an element of {0, 1, 2, ..., 9}. In this representation, the number π, for example, is represented by the sequence of integers {3, 1, 4, 1, 5, 9, 2, ...}. This representation has some problems, however. One problem is the appearance of the arbitrary constant 10 in the formula above. Why 10? This is because of a biological accident, not because of anything related to mathematics. 10 is used because it is the standard base of our number system; we may just as well use base 8 ( Octal ) or base 2 ( Binary ). Another problem is that many Rational Number s lack finite representations in this system. For example, the number 1/3 is represented by the infinite sequence {0, 3, 3, 3, 3, ....}. Continued fraction notation is a representation for the real numbers that evades both these problems. Let us consider how we might describe a number like 415/93, which is around 4.4624. This is approximately 4. Actually it is a little bit more than 4, about 4 + 1/2. But the 2 in the denominator is not correct; the correct denominator is a little bit ''more'' than 2, about 2 + 1/6, so 415/93 is approximately 4 + 1/(2 + 1/6). But the 6 in the denominator is not correct; the correct denominator is a little bit more than 6, actually 6+1/7. So 415/93 is actually 4+1/(2+1/(6+1/7)). This ''is'' exact. Dropping the trivial parts of the expression 4+1/(2+1/(6+1/7)) gives the abbreviated notation The continued fraction representation of real numbers can be defined in this way. It has several desirable properties:
This last property is extremely important, and is not true of the conventional decimal representation. Truncating the decimal representation of a number yields a rational approximation of that number, but not usually a very good approximation. For example, truncating 1/7 = 0.142857... at various places yields approximations such as 142/1000, 14/100, and 1/10. But clearly the best rational approximation is "1/7" itself. Truncating the decimal representation of π yields approximations such as 31415/10000 and 314/100. The continued fraction representation of π begins rational approximations 3, 22/7, 333/106, 355/113, 103993/33102, ... The denominators of 314/100 and 333/106 are almost the same, but the error in the approximation 314/100 is nineteen times as large as the error in 333/106. As an approximation to π, CALCULATING CONTINUED FRACTION REPRESENTATIONS Consider a real number ''r''. Let ''i'' be the integer part and ''f'' the fractional part of ''r''. Then the continued fraction representation of ''r'' is To calculate a continued fraction representation of a number ''r'', write down the integer part of ''r''. Subtract this integer part from ''r''. If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will halt if and only if ''r'' was rational.
SOME USEFUL THEOREMS If ''a''0, ''a''1, ''a''2, ... is an infinite sequence of positive integers, define the sequences and recursively: Theorem 1 For any positive : Theorem 2 The convergents of ''a''1, ''a''2, ... are given by : Theorem 3 If the ''n''th convergent to a continued fraction is , then : Corollary 1: Each convergent is in its lowest terms (for if and had a nontrivial common divisor it would divide , which is impossible). Corollary 2: The difference between successive convergents is a fraction whose numerator is unity: : Corollary 3: The continued fraction is equivalent to a series of alternating terms: : Corollary 4: The matrix :
Theorem 4 Each convergent is nearer to the ''n''th convergent than any of the preceding convergents. In symbols, if the ''n''th convergent is taken to be , then : Corollary 1: any convergent is nearer to the continued fraction than any other fraction whose denominator is less than that of the convergent Corollary 2: any convergent which immediately precedes a large quotient is a near approximation to the continued fraction. SEMICONVERGENTS If and are successive convergents, then any fraction of the form : where is a nonnegative integer and the numerators and denominators are between the and terms inclusive are called semiconvergents, secondary convergents, or intermediate fractions. Often the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent, rather than that a convergent is a kind of semiconvergent. The semiconvergents to the continued fraction expansion of a real number include all the rational approximations which are better than any approximation with a smaller denominator. Another useful property is that consecutive semiconvergents a/b and c/d are such that . BEST RATIONAL APPROXIMATIONS A ''best rational approximation'' to a real number ''x'' is a rational number ''n''⁄''d'', ''d'' > 0, that is closer to ''x'' than any approximation with a smaller denominator. The regular continued fraction for ''x'' generates ''all'' of the best rational approximations for ''x'' according to three rules: #Truncate the continued fraction, and possibly decrement its last term. #The decremented term cannot have less than half its original value. #If the final term is even, a special rule decides if half its value is admissible. (See below.) For example, 0.84375 has regular continued fraction {Link without Title} . Here are all of its best rational approximations. The strictly monotonic increase in the denominators as additional terms are included permits an algorithm to impose a limit, either on size of denominator or closeness of approximation. To incorporate a new term into a rational approximation, only the two previous convergents are necessary. If ''a'' is the new term, then the new numerator and denominator are n d The initial "convergents" (required for the first two terms) are 0⁄1 and 1⁄0. For example, here are the convergents for {Link without Title} . One formal description of the half rule is that the halved term ½ ''a''''k'' is admissible if and only if : ''a''''k''−1, …, ''a''1 > ''a''''k''+1, … . In practice, something like Euclid's GCD algorithm is often used to generate the terms sequentially, and the auxiliary values it provides allow a more convenient test. For example, here is the term generation for 0.84375 = 27⁄32. Using the ''f'' values so generated, the ½ ''a''''k'' admissibility test is ''d''''k''−2 / ''d''''k''−1 > ''f''''k'' / ''f''''k''−1. For ''a''3 of the example, ''d''1 / ''d''2 = 1⁄6 and ''f''3 / ''f''2 = 1⁄2, so ½ ''a''3 is not admissible; while for ''a''4, ''d''2 / ''d''3 = 6⁄13 and ''f''4 / ''f''3 = 0⁄1, so ½ ''a''4 is admissible. THE CONTINUED FRACTION EXPANSION OF Π To calculate the convergents of Pi we may set ''a''0 = ⌊π⌋ = 3 (where ⌊''x''⌋ denotes the Floor Function ), define ''u''1 = 1/(π − 3) ≈ 113/16 = 7.0625 and ''a''1 = ⌊''u''1⌋ = 7, ''u''2 = 1/(''u''1 − 7) ≈ 31993/2000 = 15.9965 and ''a''2 = ⌊''u''2⌋ = 15, ''u''3 = 1/(''u''2 − 15) ≈ 1003/1000 = 1.003. Continuing like this, one can determine the infinite continued fraction of π as 7, 15, 1, 292, 1, 1, ... . The third convergent of π is 7, 15, 1 = 355/113 = 3.14159292035..., which is fairly close to the true value of π. Let us suppose that the quotients found are, as above, 7, 15, 1 . The following is a rule by which we can write down at once the convergent fractions which result from these quotients without developing the continued fraction. The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, 3/1. Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, 22/7, which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small. Thus, the third quotient being 15, we have for our numerator (22 × 15 = 330) + 3 = 333, and for our denominator, (7 × 15 = 105) + 1 = 106. The third convergent, therefore, is 333/106. We proceed in the same manner for the fourth convergent. The fourth quotient being 1, we say 333 times 1 is 333, and this plus 22, the numerator of the fraction preceding, is 355; similarly, 106 times 1 is 106, and this plus 7 is 113. In this manner, by employing the four quotients 7, 15, 1 , we obtain the four fractions: : These convergents are alternately smaller and larger than the true value of π, and approach nearer and nearer to π. The difference between a given convergent and π is less than the reciprocal of the product of the denominators of that convergent and the next convergent. For example, the fraction 22/7 is greater than π, but 22/7 − π is less than 1/(7×106), that is 1/742 (in fact, 22/7 − π is just less than 1/790). The demonstration of the foregoing properties is deduced from the fact that if we seek the difference between one of the convergent fractions and the next adjacent to it we shall obtain a fraction of which the numerator is always unity and the denominator the product of the two denominators. Thus the difference between 22/7 and 3/1 is 1/7, in excess; between 333/106 and 22/7, 1/742, in deficit; between 355/113 and 333/106, 1/11978, in excess; and so on. The result being, that by employing this series of differences we can express in another and very simple manner the fractions with which we are here concerned, by means of a second series of fractions of which the numerators are all unity and the denominators successively be the product of every two adjacent denominators. Instead of the fractions written above, we have thus the series: : The first term, as we see, is the first fraction; the first and second together give the second fraction, 22/7; the first, the second and the third give the third fraction 333/106, and so on with the rest; the result being that the series entire is equivalent to the original value. OTHER CONTINUED FRACTION EXPANSIONS While one cannot discern any pattern in the infinite continued fraction expansion of π, this is not true for ''e'', the Base Of The Natural Logarithm : : : : : The numbers with periodic continued fraction expansion are precisely the solutions of Quadratic Equation s with integer coefficients. For example, the Golden Ratio φ = 1, 1, 1, 1, 1, ... and √ 2 = 2, 2, 2, 2, ... . However, most irrational numbers do not have any periodic or regular behavior in their continued fraction expansion. Nevertheless is a constant (known as Khinchin's Constant , ''K'' ≈ 2.6854520010...) independent of the value of ''x''. Paul Lévy showed that the ''n''th root of the denominator of the ''n''th convergent of the continued fraction expansion of almost all real numbers approaches an asymptotic limit, which is known as Lévy's Constant . PELL'S EQUATION Continued fractions play an essential role in the solution of Pell's Equation . For example, for positive integers and , if and only if is a convergent of . CONTINUED FRACTIONS AND CHAOS Continued fractions also play a role in the study of Chaos , where they tie together the Farey Fractions which are seen in the Mandelbrot Set with the Minkowski Question Mark Function and the Modular Group Gamma. The backwards of this map is called the Gauss-Kuzmin-Wirsing Operator . The distribution of the digits in continued fractions is given by the zero'th Eigenvector of this operator, and is called the Gauss-Kuzmin Distribution . SEE ALSO EXTERNAL LINKS
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