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regular continued fraction (continued fraction result)


Continued fraction definition:

A continued fraction xi is said to be regular if it has the form\nxi = b_0 + 1\/(b_1 + 1\/(b_2 + 1\/(descending ellipsis))), \nwhere b_k  element Z for all k = 0, 1, 2, ... and where b_k>0 for k>=1. The regular fraction xi above can also be written xi = [b_0 ;b_1, b_2, ...] or, using Gauss notation,\nxi = b_0 + (continued fraction k)_(m=1)^infinity 1\/b_m.\nThe terms b_k are said to be both the partial quotients and the partial denominators of xi, as the partial numerators of xi are all identically 1.\nIt is not uncommon in literature for the unmodified term "continued fraction" to mean "regular continued fraction," and despite an apparent loss of generality in doing so, no such loss exists. Indeed, a well-known result in the study of continued fractions is the existence of an equivalence transformation r = {r_m} between any generalized continued fraction xi and an associated regular continued fraction xi_reg, whereby it follows that any theory for generalized continued fractions holds for regular fractions and vice versa. Regular continued fractions are especially useful when representing irrationals, for example, because the convergents of regular continued fractions are the so-called best rational approximations thereof.

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