Notes

Chapter 4: Systems Based on Numbers

Section 5: Mathematical Constants


[Computing] square roots

A standard way to compute n

\!\(\*SqrtBox[\(n\)]\) is Newton's method (actually used already in 2000 BC by the Babylonians), in which one takes an estimate of the value x and then successively applies the rule x 1/2 (x + n/x)
x  1/2 (x + n/x)
. After t steps, this method yields a result accurate to about t2
\!\(\*SuperscriptBox[\(t\),\(2\)]\)
digits.

Another approach to computing square roots is based on the fact that the ratio of successive terms in for example the sequence f[i] = 2 f[i - 1] + f[i - 2]

f[i] = 2 f[i - 1] + f[i - 2] with f[1] = f[2] = 1
f[1] = f[2] = 1
tends to 1 + 2
1+\!\(\*SqrtBox[\(2\)]\)
. This method yields about 2.5 t
2.5 t
base 2 digits after t steps.

The method of computing square roots shown in the main text is less efficient (it computes t digits in t steps), but illustrates more of the mechanisms involved. The basic idea is at every step t to maintain the relation s2 + 4 r 4t n

\!\(\*SuperscriptBox[\(s\),\(2\)]\)+4r==\!\(\*SuperscriptBox[\(4\),\(t\)]\)n, keeping r as small as possible so as to make s 2t n < s + 4
s<=\!\(\*SuperscriptBox[\(2\),\(t\)]\)\!\(\*SqrtBox[\(n\)]\)
. Note that the method works not only for integers, but for any rational number n for which 1 n < 4
1 ≤ n < 4
.



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From Stephen Wolfram: A New Kind of Science [citation]