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Published on Aug 26, 2015
This is the first part in a video series about continued fractions. In this video, we define a finite continued fraction and explore some basic examples. More to follow.
Continued fractions are believed to have first appeared around 300 BC, in Euclid's Elements. A continued fraction is a sequence of integers arranged in a "nested fraction" format: that is, it is an expression obtained by writing a number as the sum of a number plus the reciprocal of another number, then writing *that* number as a number plus the reciprocal of another number, and so on. In this way, one obtains a sequence of numbers, written [a_0, a_1, ..., a_n] in the case of a finite continued fraction (which converges to some rational number), or [a_0, a_1, ... ] in the case of an infinite continued fraction (which can converge to any real number).
The aim of this series is to apply continued fractions in a number of contexts, most notably, to the solution of Pell's equation, x^2 - dy^2 = 1 (where d is square-free), and as a corollary, to solve equations of the form x^2 + dy^2 = k, where d may be positive or negative.