Quandaries and Queries


Who is asking: Other
Level: All

Q1. What is the simplest way of finding a square root of any number using just a pen and paper? (I am asking this question because I browsed a few sites a didn't find any method that is simpler than the one I use. so I am just curious.)

Q2. Is it possible that you take an equation and turn it into an inequality by performing same mathematical operations on both sides?

I shall appreciate your opinion on this.




Hi Waheed,


I think that using Newton's method on the polynomial f(x) = x2 - B is
the simplest way to find good approximations of the square root of B:
set A0 to a guess. If you don't have a guess just let A0 = B and

An+1(An2 + B)/2An until you get a good enough approximation.

For example if B is 236 then I know that 152 is 225 so I would let x0 = 15. This gives

A1(152 + 236)/2(15)461/30 = 15.3667

15.36672 = 236.1355 so that may be close enough. If not try A2

A2(15.36672 + 236)/2(15.3667)472.1355/2(15.3667) = 15.3623

15.36232 = 236.0002


I can, for instance take

(x+2)2 = x2 + 4x + 4

and subtract 4 on both sides to get

(x+2)2 - 4 <= x2 + 4x.

However I could not get a strict inequality. A "mathematical operation"
outputs unambiguous results that only depend on the input. The inputs
from both sides of an equation are the same, so the outputs are also the

Claude and Penny



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