A "proof" that 1 + 1 = 1
| 1) | Let x | = | 1. | |
| Multiply both sides by x: | ||||
| 2) | x² | = | x | |
| Subtract 1 from both sides: | ||||
| 3) | x² − 1 | = | x − 1 | |
| Factor: | ||||
| 4) | (x − 1)(x + 1) | = | x − 1 | |
| Divide both sides by x − 1: | ||||
| 5) | x + 1 | = | 1 | |
| But x = 1 -- Line 1. Therefore | ||||
| 6) | 1 + 1 | = | 1. | |
Where is the error?
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Consider the problem yourself first!
The error is in going from Line 4) to Line 5). Since x = 1, then x −1 = 0. We have divided by 0, an operation that is not permitted. See Lesson 6.
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