Calculate square root with pen and paper only (2)
2007-01-01This is a continuation of previous article; here I’ll explain the reasoning behind the method. It’s basically related to the expansion of square of polynomial:
Or arranged alternatively:
These terms may look strikingly similar to the ‘division’ operation: in the first round, the last digit a is doubled and shifted left, and we need to find the digit b so that 
is the largest possible number deductible.
Now let’s start. Denote n as the number of which we want to find the square root, as well as representing its square root as a sum of series of digits (the series can be finite or infinite). Expressed in equation form:
with 
= 0-9 for any i. Think about replacing the a, b, c, d… with 
and so on, as well as replacing 
with n:
When the terms in right hand side are reduced from n in succession (i.e. after each iteration), the remainder 
becomes the following infinite series:
……
The remaining task is to show that this series (the remainder) converges to zero, so we can get arbitrary precision with enough iteration. Try to express as sum of remaining terms instead:
which can be arbitrarily small when i is large enough. The conclusion is, after enough iteration, the difference between the real square root and the number we get approaches to zero (since the square of this difference, , approaches to zero as well).
2007-03-03 update: Edited formula so that it works with LatexRender.
Actually I just want to showcase the wordpress FigureRender plugin (superseded by LatexRender plugin and my ScoreRender plugin) 
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