Mathematical fact about representing numbers (Jonathan)

[James Crook]

Jonathan wrote:

here's another fact that's interesting, and much simpler:

.999999...=1

Proof:

Let x=.999...
10x=9.999...
10x-x=9.999...-.999...
9x=9
x=1

So, .999...=1!

That's pretty cool, I think.



Yes. One way to look at it is that '...' (or rather the notation with
dots above the start and end of the repeating part) is an operator
that operates on a number. Then it's no longer that different from
being able to represent the same number either as 0.5 or 1/2. Without
the '...' notation each decimal number can only be represented one way
as a decimal number (let's outlaw minus zero ). That's a nice
property to have. Unfortunately you can't represent 1/3 without using
an operator nor can you represent a lot of other rational numbers.
But is that so very bad? You can't represent sqrt(2) without using an
operator either.