Subject: 0.999999=1?

Name: Catherine

Who is asking: Student
Level: Secondary

Hi! My teacher told us that 0.9 repeating equals one. We discussed how this is true. But, I was wondering if there is a proof that this is true. If so what is this called? I was trying to find information, but, it's hard when you don't know the name.


Hi Catherine,

While there are various arguments that 'show' this equality is the 'right way to think about it', my favourite is the following.

IF the two numbers 0.99999... and 1 were NOT equal, you would have a number to represent the difference (the GAP or Distance between them) - and it should not be 0.

So what is the difference?

It is a number smaller than 0.0001, smaller than 0.000001, ....

So this really comes out to the question: is there a number smaller than 0.0001, 0.0000001, .... but bigger than 0?

When you check there is no such number. The gap between them really is 0. Of course it is a bit obscure since we are dealing with 'infinite processes' (sometimes called limits) whenever we work with decimal numbers which do no terminate. Hard to think about an infinite process with a finite mind and a finite amount of time. However, amazingly, we do manage to do it.

Walter Whiteley
York University

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