
Useful Computational Methods 

Squareroots via Continued Fractions


We saw elsewhere how to obtain the Continued Fraction representation of the squareroot of a nonsquare number N as
Convergents
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 squareroot 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 squareroots 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 nth convergent by c_{n},
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:
c_{1} < c_{3} < c_{5} ... < c_{2n+1} ... C ... < c_{2n} ... < c_{6} < c_{4} < c_{2}.
REFERENCES
[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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