Does 1 = 2?

I saw my first proof that one equals two when I was in seventh grade algebra. I've been collecting them ever since. There's an error in each proof, of course. If you know of any others, please email me. Click here to see the errors in these "proofs."

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Does 1 = 2?

1st Proof

x = 1

set x equal to 1

x² = x

multiply both sides by x

x² - 1 = x - 1

subtract 1 from both sides

(x - 1)(x + 1) = x - 1

separate left side into factors

x + 1 = 1

divide both sides by (x - 1)

1 + 1 = 1

substitute 1 for x

2 = 1

2nd Proof

`(-1)(-1) = 1`	`the square of -1 is 1`
`-1 = 1/-1`	`divide both sides by -1`
`-1/1 = 1/-1`	`identity operation; for all real (or complex) x, x = x/1`
`i/1 = 1/i`	`take the square root of both sides (i=sqrt(-1))`
`i/2 = 1/2i`	`divide both sides by two`
`i²/2 = i/2i`	`multiply both sides by i`
`-1/2 = 1/2`	`substitute -1 for i² and 1 for i/i`
`-1/2 + 3/2 = 1/2 + 3/2`	`add 1 1/2 (3/2) to both sides`
`1 = 2`

3rd Proof

`x² = x+x+x+...+x (x times)`	`definition of x²; x not equal to zero`
`2x = 1+1+1+...+1 (x times)`	`take derivative of both sides;` `derivative of xⁿ = nx^n-1`
`2x = x`	`x = 1+1+1+...+1 (x times)`
`2 = 1`	`divide both sides by x (x not equal to zero)`

This page was last updated on November 3, 2009.
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Copyright© 1998 Ross Nordeen. All rights reserved.