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Continued Fractions: Introduction and Gosper Arithmetic   

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Continued fractions are expressions of the form \[\lbrack a_0;a_1,a_2,\ldots\rbrack = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{\cdots}}}.\] Every real number has a unique expression as a continued fraction, so long as we exclude representations which end in 1, since \[a_0;a_1,\ldots,a_n,1] = [a_0;a_1,\ldots,a_n+1].\] Note that this is no worse (and probably quite a bit better) than the ambiguity in decimal expansions, for which \[1 = 0.999\cdots.\]