Tutoring math, you often hear the comment that math involves itself in silly pursuits.  The tutor gets the point, and apologizes.

Sometimes, non-math people will point out that math harbours absurdities. The infinite string of numbers with a finite sum (see my post here) is one example. Irrational numbers might also qualify (see my post here), if due to their name alone.

I’ve heard people joking that there’s a proof for the fact that 1 is not equal to 0. Who would need to prove something so obvious? Surely, those math people are just amusing themselves, while everyone else has to put up with the output.

Those sentiments are definitely understandable; I’ve even heard my own math professors say so. However, there are two mollifying considerations:

1. The ideas that seem absurd are often taken out of context.
2. Often, the most obvious ideas are the hardest to prove.

The fact that 1 is not equal to 0 is obviously true in our common number system. We all count on its truth. Yet, to discover further ideas about our numbers, we often have to prove what we already know. That’s why math people so often embark on apparently ridiculous errands.

For those who still want to see proof that 1 is not equal to 0, here’s my version:

I’ll use an indirect proof (see my post here about that): I’ll first suppose the opposite of what I want to prove.

Suppose 1 is equal to 0.

In our number system, a number is itself times 1. For example, 10=10(1).

In general terms, the number n is equal to n(1).

Now, if 1=0, then n(1)=n(0). Therefore, n=0.

It follows that if 1=0, then every number is equal to 0. However, we know that not every number is equal to 0. Therefore, 1 cannot equal 0.

No doubt, there are other ways to prove 1 is not equal to 0. Is it worth proving? In some circles, yes – but probably not for most people:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.