====== One equals almost one ======

I came accross this one recently, and thought it was interesting. Feel free to discuss.

===== Statement =====

We wish to prove that <m>1 = 0.overline{9}</m>. (That is to say, 0.99999..., a repeating decimal.)

===== Initial Thoughts =====

Why would we want to prove something so seemingly rediculous? Well, we take for granted such things as approximation and rounding, and are more than happy to take limits as they approach infinity, so it would be interesting to prove that using one is just as good as using 0.99999..., if the situation allows.

===== The Proof =====

  * Let <m>x = 0.99999...</m>
  * And <m>10x = 9.99999...</m>
  * Then:

<m>tabular{0010}{000}{{10x=} {9.99999...} {-x=} {-0.99999...} {9x=} {9}}</m>

  * Thus, <m>x = 9/9 = 1</m>
  * But since <m>x = 0.overline{9}</m> as well, this goes to say <m>0.overline{9} = 1</m>! Which is what we wished to prove.

===== Final Thoughts? =====

Some interesting side notes... the logic in the proof plays out differently for other common repeating decimals. Take <m>1/3</m> for instance. Run through the same procedure and you will come out with <m>x = 3/9</m> which reduces to <m>1/3</m>. So there's no //trick// involved in the proof, but it's also worth noting that <m>0.overline{9}</m> is **not** representable as a rational number either.