Is this really true?

[savio13]

The conjecture is that 0.9 recurring (i.e. 0.999....,
with the digits going on forever) is actually equal to 1

(For this exercise I will use the notation 0.999... as notation for 0.9 recurring,
the correct way would be to put a little dot above the 9, or a line on top like this: 0.9)

•

Let X = 0.999...
•

Then 10X = 9.999...

Subtract X from each side to give us:
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9X = 9.999... - X

but we know that X is 0.999..., so:
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9X = 9.999... - 0.999...
or: •

9X = 9


Divide both sides by 9:
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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:
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X = 0.999... = 1



•

Therefore 0.999... = 1