The incorrect step in each proof is highlighted in red and an explanation is at the end of each proof.

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Errors in “Does 1 = 2?” Proofs

x = 1 |
set x equal to 1 |

x^{2} = x |
multiply both sides by x |

x^{2} - 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.`

(-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 |

i^{2}/2 = i/2i |
multiply both sides by i |

-1/2 = 1/2 |
substitute -1 for i^{2} and 1 for i/i |

-1/2 + 3/2 = 1/2 + 3/2 |
add 1 1/2 (3/2) to both sides |

1 = 2 |

`There are two square roots for -1, i and -i and for 1, 1 and -1.`

x^{2} = x+x+x+...+x (x times) |
definition of x^{2}; x not equal to zero |

2x = 1+1+1+...+1 (x times) |
take derivative of both sides;
derivative of x^{n} = nx^{n-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. x ^{2} as
defined above is valid only for non-negative whole numbers and is therefor
not a continuous function.`

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Copyright© 1998 Ross Nordeen. All rights reserved.