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Continued Fractions A Presentation by Jeffrey Sachs, Bill Gottesman, and Tim Mossey Introduction Continued Fractions provide insight into many mathematical problems, particularly into the nature of numbers. Consider the quadratic: =0 By rearranging this, and dividing by “x” we can produce: This can be represented as: We can substitute for x: Our best approximation of this is 3 Our best approximation of this is 3 and 1/3 = 3.333333 Our best approximation of this is 3 and 3/10 = 3.3 Our best approximation of this is 3 and 10/33 = 3.30303 Using the quadratic formula, x = = 3.30278… Our approximations become more and more precise with each expansion of the continued fraction. This one will continue on forever. These types of continued fractions are called infinite continued fractions. Creating Continued Fractions Consider the fraction: We want to turn this into a continued fraction, but how? Using the Euclidean algorithm we can start to form our continued fraction: 67 – 29(2) = 9 The The 2 is known as the partial quotient. Thus, our continued fraction begins as: can be also represented as: ⇒ The goes into the Euclidean Algorithm as such: This process continues, now with the 29 – 9(3) = 2: 9 – 2(4) = 1: 2 – 1(2) = 0 = will be our final faction because it will have no remainder in the Euclidean Algorithm. And so, our finite continued fraction of is: Unwinding a Continued Fraction Consider the continued fraction from before: To find the initial rational number we must start from the bottom. Begin with: This equals: We now have: We now have: Thus, the continued fraction represents the rational number . Continued Fraction Notation A continued fraction is an expression of the form: This is actually known as a “simple” continued fraction. is usually a negative or positive integer, and all subsequent are positive integers. A finite continued fraction is called a “terminating” continued fraction. Convenient ways to write continued fractions include: This is the set of partial quotients of the simple continued fraction. Infinite Continued Fractions Infinite continued fractions never terminate, but they “converge.” A good example is pi. pi = 3.141592653589… What is the first good rational approximation of pi? What is the second good rational approximation of pi? It’s not because that isn’t the best approximation of pi. 3.1 whereas is 3.14. It is a simpler rational number. Is only This all can be found with continued fractions with what is called “convergents.” Convergents are successive rational representations of the continued fraction. Each successive convergent is always a better approximation of the original number than the one before it. Convergents of pi Using the Euclidean algorithm we can begin to construct the infinite continued fraction of pi. From this, the construction of the continued fraction looks like this: The third convergent: The fourth convergent: = = = 3.141509… = 3.1415929… Thank you to number theorist Professor John Voight for helping us with this topic. Questions? Bibliography: Khinchin, Aleksandr. Continued Fractions 1964, Chicago University Press. English translation edited by Herbert Eagle. Olds, Carl Douglas. Continued fractions. 1963, Random house. Euler, Leonard. Introduction to analysis of the infinite, Book 1. 1742; English translation by J. D. Blanton 1988, Springer. Pianos and Continued Fractions. Edward G. Dunne, www.research.att.com/~njas/sequences/DUNNE/TEMPERAMENT.HTML (Applause) To be continued… Homework Consider the improper fraction: Construct its “terminating” continued fraction.