They also provide a simple way to construct a variety of transcendental numbers. Here's the definition. We shall only be concerned with the latter variety. The key step of the algorithm is exactly this: "Keep dividing the smaller number into the larger one. The above explains why the terms a i are usually called quotients.
Also, it shows that every rational number can be associated with a finite continued fraction.
Numbers and Functions as Continued Fractions - Numericana
On the other hand, given a finite fraction [a 0 ;a 1 ,a 2 ,a 3 , Which will ultimately lead to a rational number r. Since, obviously, [a 0 ;a 1 ,a 2 ,a 3 , With this convention, the correspondence between rational numbers and finite continued fractions becomes Irrational numbers can also be uniquely associated with simple continued fractions. Quite naturally the terms r i are called remainders. As we see, the association between real numbers and continued fractions is defined recursively.
It will come then as no surprise that many of the features of the continued fractions are expressed in recursive terms. Then we have the following fundamental theorem that can be proved by induction. First subtracting 2 multiplied by p k-1 from 1 multiplied by q k-1 and then subtracting 2 multiplied by p k-2 from 1 multiplied by q k-2 we further get. Note that this is the same inequality we once proved using the Pigeonhole principle. The new proof supplies a constructive way to approximate an irrational number. All of the above fractions are periodic in the sense we apply to decimal fractions.
Le comte J. Lagrange proved that this is a characteristic property of roots of quadratic polynomials.
On the evolution of continued fractions in a fixed quadratic field
The next two examples are due to S. Ramanujan , one of the greatest mathematical geniuses. Being a poor uneducated clerk, in Ramanujan sent a letter to G. Hardy who was by this time a world famous mathematician. He was accustomed to receiving letters from cranks.
Advanced Studies in Pure Mathematics
So he was bored more than anything else. These rational numbers are called the convergents of the continued fraction. Other numbers like e have only small terms early in their continued fraction, which makes them more difficult to approximate rationally. In this sense, therefore, it is the "most irrational" of all irrational numbers.
Even-numbered convergents are smaller than the original number, while odd-numbered ones are larger.
For a continued fraction [ a 0 ; a 1 , a 2 , The numerator of the third convergent is formed by multiplying the numerator of the second convergent by the third quotient, and adding the numerator of the first convergent. The denominators are formed similarly. Therefore, each convergent can be expressed explicitly in terms of the continued fraction as the ratio of certain multivariate polynomials called continuants. If successive convergents are found, with numerators h 1 , h 2 , Thus to incorporate a new term into a rational approximation, only the two previous convergents are necessary.
For example, here are the convergents for [0;1,5,2,2]. When using the Babylonian method to generate successive approximations to the square root of an integer, if one starts with the lowest integer as first approximant, the rationals generated all appear in the list of convergents for the continued fraction. Comparing the convergents with the approximants derived from the Babylonian method:. Baire space is a topological space on infinite sequences of natural numbers.
Expansion of Functions
The infinite continued fraction provides a homeomorphism from Baire space to the space of irrational real numbers with the subspace topology inherited from the usual topology on the reals. The infinite continued fraction also provides a map between the quadratic irrationals and the dyadic rationals , and from other irrationals to the set of infinite strings of binary numbers i. Theorem 1. Theorem 2. Theorem 3. Corollary 2: The difference between successive convergents is a fraction whose numerator is unity:.
Theorem 4. Corollary 1: A convergent is nearer to the limit of the continued fraction than any fraction whose denominator is less than that of the convergent. Corollary 2: A convergent obtained by terminating the continued fraction just before a large term is a close approximation to the limit of the continued fraction. Sometimes the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent i.
The simple continued fraction for x can be used to generate all of the best rational approximations for x by applying these three rules:. For example, 0. 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. The convergents to x are "best approximations" in a much stronger sense than the one defined above. When both x and y are irrational and.
For example, the decimal representation 3. The continued fraction representations of 3. The numbers x and y are formed by incrementing the last coefficient in the two representations for z.
Let us suppose that the quotients found are, as above, [3;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. 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.
We proceed in the same manner for the fourth convergent.
The fourth quotient being 1, we say times 1 is , and this plus 22, the numerator of the fraction preceding, is ; similarly, times 1 is , and this plus 7 is 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. 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:. To illustrate the use of generalized continued fractions, consider the following example. The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients; rational solutions have finite continued fraction expansions as previously stated. Another, more complex pattern appears in this continued fraction expansion for positive odd n :. Then for all nonnegative rationals, we have. Many of the formulas can be proved using Gauss's continued fraction.
Most irrational numbers do not have any periodic or regular behavior in their continued fraction expansion. Lochs' theorem states that n th convergent of the continued fraction expansion of almost all real numbers determines the number to an average accuracy of just over n decimal places. Generalized continued fractions are used in a method for computing square roots. Continued fractions play an essential role in the solution of Pell's equation. Continued fractions also play a role in the study of dynamical systems , where they tie together the Farey fractions which are seen in the Mandelbrot set with Minkowski's question mark function and the modular group Gamma.
The transfer operator of this map is called the Gauss—Kuzmin—Wirsing operator.
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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. The Lanczos algorithm uses a continued fraction expansion to iteratively approximate the eigenvalues and eigenvectors of a large sparse matrix. Continued fractions have also been used in modelling optimization problems for wireless network virtualization to find a route between a source and a destination. From Wikipedia, the free encyclopedia. It is not to be confused with Repeating decimal. Main article: Generalized continued fraction.
Main article: Periodic continued fraction.