# Does 0.999... = 1 ?

The idea is that 0.9 recurring
(0.999... with the digits going on forever)
is actually equal to 1

### Is this really true?

You decide! But here we give some nice arguments as to why it does.

## Three Thirds

Times 3:33 = 3 × 0.333...
So:1 = 0.999...

## Using Algebra

Let us start by having x = 0.999...

x = 0.999...

10x = 9.999...

Subtract x from each side to give us:

9x = 9.999... − x

but we know that x is 0.999..., so:

9x = 9.999... − 0.999...

9x = 9

Divide both sides by 9:

x = 1

But hang on a moment I thought we said x was equal to 0.999... ?

Yes, it does, but from our calculations x is also equal to 1, so:

x = 0.999... = 1

And so:

0.999... = 1

## How Many Nines?

If 0.999... and 1 are the same number, then their difference will be zero.

1 nine:1 − 0.9 = 0.1
2 nines:1 − 0.99 = 0.01
3 nines:1 − 0.999 = 0.001
0.001 = 1103:1 − 0.999 = 1103
n nines:1 − 0.(n 9s) = 110n

As n goes to infinity 110n goes to zero.

So the difference between 1 and 0.999... is zero

0.999... = 1

## Infinite Geometric Series

We can think of 0.999... as being equal to:

=0.9 + 0.09 + 0.009 + 0.0009 + ...
=0.9×0.10 + 0.9×0.11 + 0.9×0.12 + ...

This is an Infinite Geometric Series where a = 0.9 and r = 0.1 with the series being convergent because r is between −1 and +1. The formula for the sum is:

a1 − r

So our sum is equal to:

0.91 − 0.1 = 0.90.9 = 1

0.999... = 1

Footnote: we use 0.999... as notation for 0.9 recurring, some people put a line, or little dot, above the 9 like this: 0.9