Not Just Physics > Maths > Another proof that -1=1

# Another proof that -1=1

Here is a little maths teaser for you. Since I was a student, I always loved those “proofs” that zero equals one. Of course, most of the time, this was achieved by sneakily dividing by zero somewhere along the way.

But yesterday I came across a proof that used a different, slightly more subtle trick and uses complex numbers. I apologise to any reader not familiar with complex numbers. Anyone interested can find a quick introduction here.

Enough introduction, here is the “proof”:

Looks OK, but it can’t be right of course. So where is the error in this equation? Can you find out?

Update:
You can find the answer here.

1. frosty840 | |

It’s been a few years since A-level maths, but I think there are some brackets implied in the third stage, which then suffer some terrible fate in between the third and fourth stages…

2. Robert Lewis | |

This is an old one. The problem is that sqrt is not a function: every real and complex number has two square roots.

3. holger | |

That depends on your definition of sqrt. If I define sqrt to be the principal square root then it is a function. But the problem above persists.

4. Bobby Thomas | |

The subtle error in the proof is the assumption that the property “sqrt(x * y) = sqrt(x) * sqrt(y)” holds for all real numbers. It is only valid for all x, y >= 0 where x, y are real numbers. The above “proof” can even be modified to yield an actual proof demonstrating that the property does not hold for ANY negative real numbers.