1=0

This is another entry in a series of articles I wrote long ago for a school paper, which demonstrates why 'common sense' and mathematics do not always work well together. This article is reprinted here at the request of one of my former teachers who wanted to show it to his class, as an example of why proofs must be done carefully.

Consider the infinite sum, S = 1 - 1 + 1 - 1 + 1...

As everyone knows, brackets don't change anything, so let us write it as S = (1-1)+(1-1)+(1-1)+...

Each term in brackets is 0, and so the sum is clearly S = 0.

Now write it as S = 1 - (1-1)-(1-1) - (1-1)-...

Again each term in brackets is 0, so S = 1.

Therefore, S = 1 = 0 and so 1 = 0. And by mathematical induction, 0 = 1 = 2 = 3 = ...

Therefore we have proven that all integers equal zero.

It is amazing how many people cannot see the problem with this prove, and so figure it must be true. The secret is in the manipulation of infinite sums - it can't be done this way. The original sum is undefined since it never converges to a single number, and so this proof is utter nonsense!