Method for [Mentally] Calculating Squares

Proposal

Let a and b be real numbers for which a2 is readily known and b2 has yet to be calculated. Also let h be "step size", and be equal to b - a. Then:

b2 = a2 + 2(h)(a) + h2

Example

Find: 17.12

Nearby 17.1 is 20, and we know that 202 is 400. 'h' then is 17.1-20, which is -2.9, and (-2.9)2 is 8.41, so now we can mentally solve for 17.12.

17.12 = 400 + 2(-2.9)(20) + 8.41 = 400 - 116 + 8.41 = 292.41

Proof of Method

b2 = a2 + 2(h)(a) + h2
b2 = a2 + (2)(b - a)(a) + (b - a)2
b2 = a2 + (2ab - 2a2) + (b2 - 2ab + a2)
b2 = a2 + 2ab - 2a2 + b2 - 2ab + a2
b2 = (a2 - 2a2 + a2) + (2ab - 2ab) + b2

b2 = b2

About

I noticed this pattern in an effort to gain an edge in a "Can you guess the square root of... " game that I like to play with people in math classes. I'm sure that hundreds of people have noticed this relation before me, however I do not know who found it first. Please contact me if you know!

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© 2006 Nic Reveles
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