Is this really true?

(Here we write 0.999... as notation for 0.9 recurring, some people put a little dot above the 9, or a line on top like this: 0.9)

Does 0.999... = 1 ?

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