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Useful Computational Methods

Square-roots via Continued Fractions

We saw elsewhere how to obtain the Continued Fraction representation of the square-root of a non-square number N as

Consider the continued fraction expansion of √2.

Let us look at the succession of fractions:

These successive partial sums of the continued fraction are called convergents. The convergents converge to the value of the number the continued fraction represents. So to find approximations to the square-root of a number, we compute successive convergents.

What we have seen above would apply also to finding approximations of other irrationals. The method is particularly suitable for approximating square-roots because of Louis Lagrange's Theorem that the conitinued fraction of every quadratic irrational is periodic after a while. The theory of convergents is beautiful. For instance, if we denote the n-th convergent by cn, the actual value C of the infinite continued fraction will lie between two strings, one string made up of the odd convergents, and the other of even convergents thus:
c1  <  c3  <  c5  ...  <  c2n+1  ...   C   ... <  c2n  ...  <  c6  <  c4  <  c2.

[1] G.H. Hardy and E.M. Wright, Introduction to the theory of numbers, Oxford University Press
[2] C.D. Olds, Continued Fractions, New Mathematics Library, Mathematical Association of America, 1963




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