| 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 |
Division by (x-1) is not allowed since x=1 and therefor x-1=0 and division by zero is an undefined operation.
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the square of -1 is 1 |
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divide both sides by -1 |
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identity operation; for all real (or complex) x, x = x/1 |
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take the square root of both sides (i=sqrt(-1)) |
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divide both sides by two |
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multiply both sides by i |
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substitute -1 for i2 and 1 for i/i |
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add 1 1/2 (3/2) to both sides |
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There are two square roots for -1, i and -i and for 1, 1 and -1.
| 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 = nxn-1 |
| 2x = x | x = 1+1+1+...+1 (x times) |
| 2 = 1 | divide both sides by x (x not equal to zero) |
Only continuous functions have derivatives. x2 as defined above is valid only for non-negative whole numbers and is therefor not a continuous function.