Statement | Comment: |

-20 = -20 | Hard to argue with! |

16 - 36 = 25 - 45 | Rewriting |

16 - 36 + 81/4 = 25 - 45 + 81/4 | Add 81/4 to each side |

(4 - 9/2)^{2} = (5 - 9/2)^{2}
| Rewriting |

4 - 9/2 = 5 - 9/2 | Sqrt of each side |

4 = 5 | Add 9/2 to each side |

1 = 2 | Subtract 3 from each side |

Statement | Comment: |

16 = 16 | Reflexive property of "=" |

4 + 12 = 16 | From definition of "+" |

4 - 12 = 16 - 24 | Subtract 24 from both sides |

4 - 12 + 9 = 16 - 24 + 9 | Add 9 to both sides, completing squares |

(2 - 3)^{2} = (4 - 3)^{2}
| Factor, rewriting as squares |

2 - 3 = 4 - 3 | Sqrt of each side |

2 = 4 | Add 3 to each side |

1 = 2 | Divide by 2 |

Exposing the error: one side squares a negative number and the other squares its additive inverse (I think that's correct term, it's been 30 years). That's equivalent to saying that 3 = -3 because 3^{2} = (-3)^{2}.

Here's a different one, with a different flaw:

x = y

x^{2} = xy

x^{2} - y^{2} = xy - y^{2}

(x - y)(x + y) = y(x - y)

(x - y)(x + y)/(x - y) = y(x - y)/(x - y)

x + y = y

y + y = y

2y = y

2 = 1

The flaw is this:

x = y

x - y = 0

(x - y)(x + y)/(x - y) is division by zero.