Wednesday, May 01, 2013

2 = 1

All kinds of falsehoods can be "proven" true if subtle errors in reasoning are allowed to go unnoticed. I get a kick out of debunking these arguments, and thought I'd share one with you.

See if you can find what's wrong with the well-known "proof" that 2 = 1, outlined below:


a = b

Multiply both sides by a.
a² = ab

Add a² on both sides.
a² + a² = a² + ab

Simplify the left side.
2a² = a² + ab

Subtract 2ab from both sides.
2a² - 2ab = a² + ab - 2ab

Simplify right side.
2a² - 2ab = a² - ab

Factor 2 out of each term on left side.
2(a² - ab) = a² - ab

Divide both sides by a² - ab.
2(a² - ab)/(a² - ab) = (a² - ab)/(a² - ab)

Which "proves:"
2 = 1


Can you find the error?

Come on ... don't look ahead until you at least give it a try!

All right, here's the mistake:

Look at the eighth step. If a = b (given), then a² - ab = 0. Division by zero is impossible, and therefore not allowed. Breaking the "never divide by zero" rule breaks the proof.


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