NonStop Chatter

I spotted this old trick algebraic proof over at wikiHow. The challenge is to figure out how this works, or rather, why it isn’t a valid proof. It took me a minute, but it’s been a number of years since I took any math class. Not to mention dealing with a rather nasty virus right now. The kind that gives you a cold, not the computer type, thank goodness.

Assume a=b. The do the following mathematical operations:

3a = 3b	(multiplying both sides by the real number)
11a = 11b	(multiplying both sides by the “real” number)
3a2 = 3ab	(multiplying by a on both sides)
11ab = 11b2	(multiplying by b on both sides)
3a2 - 11ab = 3ab - 11b2	(subtracting the above two equations to make one)
3a2 - 3ab = 11ab - 11b2	(subtracting 3ab and adding 11ab to both sides)
3a2 - 3ab + ab - b2 = 12ab - 12b2	(adding ab and subtracting b2 from both sides)
3a(a-b) + b(a-b) = 12b(a-b)	(factoring out common terms )
3a + b = 12b	(removing common terms)
3a = 11b	(subtracting b from both sides)
3b = 11b	(substituting a for b, remembering that they are equal)
3 = 11	(removing common terms)

Think about it. Where’s the mistake?

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If a=b, then dividing by a-b means you divided by zero, which is against the rules of arithmetic.

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