Date: 10/07/97 at 16:15:35 From: Sarah Subject: Continued fractions Dear Dr. Math, I was recently assigned a project in math class in which I must present to the class a lesson on continued fractions. Up until this point I have not been able to figure out exactly what "continued fractions" are. I have searched the Net and also a few math textbooks and I have not found even the simplest definition of the fractions. I would appreciate you sharing ANY knowlegde that you have on continued fractions. Thank you, Sarah
Date: 10/10/97 at 11:53:05 From: Doctor Luis Subject: Re: Continued fractions A continued fraction is an expression of the form, A = a_0 + 1/c_0 where c_0 = a_1 + 1/c_1 c_1 = a_2 + 1/c_2 . . (0 denominators are not allowed, of course) . If the process stopped at, say, c_n = a_(n+1) + 1/c_(n+1) then we would have a finite continued fraction. Now, the initial question is whether the process converges in the infinite case. If a_0 is an integer and the rest of the a's are positive integers, the process converges to a real number; and all real numbers can be represented in this way. Simple (or ``regular'', as some authors write) continued fractions are those with these requirements on a_0, a_1, a_2, .... It is standard to use [a_0,a_1,a_2,a_3,...] to represent a = a_0 + 1 -------------- a_1 + 1 --------- a_2 + 1 ---- a_3 + ... There's an interesting way you can find a continued fraction representation of 1 + the square root of 2. x = 1 + sqrt(2) x - 1 = sqrt(2) x^2 - 2x + 1 = 2 x^2 = 2x + 1 1 or x = 2 + --- x "Substituting" for x in the lefthand side 1 x = 2 + -------- 1 2 + --- x Repeating this process an infinite number of times you'll get the infinite continued fraction, 1 x = 2 + ------------------- 1 2 + ---------------- 1 2 + ----------- 1 2 + ------- 2 + ... Using the notation I introduced above, we have 1+sqrt(2) = [2,2,2,2,2,....] Note that this gives us the continued fraction representation of the square root of 2, namely sqrt(2) = [1,2,2,2,2,....] Which is interesting because, if you remember, the sqrt(2) has a nonterminating and nonrepeating (since it's irrational) decimal representation 1.414213562373...... BUT it has the repeating continued fraction representation [1,2,2,2,...] (Remember that rational numbers have repeating decimal representations, like: 1/3 = 0.3333333..... 2/7 = 0.285714285714285714....... 1/1 = 0.99999999999...... and so on ) Why don't you try to solve the following? Problem: Show that the continued fraction for sqrt(n(n+1)) is [n,2,2n,2,2n,2,2n...]. You should be able to find the theory of continued fractions in (almost) any elementary or standard number theory textbook. I certainly encourage you to find out more about this fascinating subject. Here are a couple of links I found that you might find interesting: Continued Fraction Representations involving e http://pauillac.inria.fr/algo/bsolve/constant/e/cntfrc.html Continued Fractions Involving Pi http://www.mathsoft.com/asolve/constant/pi/frc.html I hope this helped :) -Doctor Luis, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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