# 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

## Using Algebra

Let us start by having **x** = 0.999...

**x** = 0.999...

10**x** = 9.999...

*Subtract x from each side to give us:*

9**x** = 9.999... − **x**

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

9**x** = 9.999... − 0.999...

9**x** = 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.

**n**nines:1 − 0.(

**n**9s) = \frac{1}{10^{n}}

As **n** goes to infinity \frac{1}{10^{n}} 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}+ 0.9×0.1

^{1 }+ 0.9×0.1

^{2}+ ...

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:

\frac{a}{1 − r}

So our sum is equal to:

\frac{0.9}{1 − 0.1} = \frac{0.9}{0.9} = 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*