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2 ≠ 1 February 19, 2007

Posted by Brian L. Belen in Academically Speaking.
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It’s funny how some things learned back in the day sort of stick with you. I’ve never quite forgotten the algebraic sequence below, which seems to demonstrate that math isn’t quite as exact as it is often made out to be.

With the exception of the difference of squares from the polynomial in the third line onwards, the algebra is pretty straightforward. If we let a = b:

a2 = b2

a2 = ab

a2−b2= a2−ab

(a+b)(a−b)= a(a−b)

[(a+b)(a−b)]/(a−b)= [a(a−b)]/(a−b)

a + b = a

a + a = a

2a = a

2 = 1

This cannot be the case, of course. The reason is simple but easy to miss, and it’s quite amusing how people can pore over this and be mystified (if the reactions of my friends and former students are anything to go by). If you know the explanation behind the “anomaly”, try not to spoil it! But do leave a comment either way.

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