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Image by Sven Geier

One of my favourite proofs in the maths world is actually a very simple one. The reason that I love it so much is that it highlights the difficulty us humans face when trying to understand the concept of “infinity”.

Everything that we know of seems to have a beginning and an end. We die after a finite period of time. Pluto is extremely far away, but it is a finite distance away, meaning that it can be reached given enough time. At first, it is unnatural for most people to imagine something that never ends – and so to learn to treat an infinite object as naturally being infinite is often difficult.

The Proof

The statement that we want to prove is:

0.99999999… = 1

Note that the 3 dots after the 9s mean ‘recurring’, i.e. that there are an infinite number of nines after the decimal point.

Let x=0.9…

Then 10x = 9.9…

So 10x – x = 9x;
but also 10x – x = 9.9… – 0.9… = 9

(Many people get lost here. Even though we have multiplied x by 10, there are still an infinite number of 9s after the decimal point. So 0.9… and 9.9… both have infinite 9s after the decimal point. It is easy to feel that we must have shifted one of the 9s from after the decimal point to before it when we multiply by 10, but this is an illusion.)

Hence, 9x = 9.9… – 0.9… = 9

So 9x = 9. Dividing both sides of the equation by 9 gives us x=1.

But we already said that x=0.9… and so since x is also 1, we must have that 0.9… = 1.

Short but sweet. Understanding this opened up a whole new dimension of abstract thinking for me, and I hope that you can take something from it too.