# 0.999...

Revision as of 18:59, 24 April 2008 by I like pie (talk | contribs)

(or ) is an equivalent representation of the real number .

It is often mistaken that for various reasons (that there can only be a finite number of s, that there is a term left over at the end, etc.).

## Proofs

### Fractions

Since , multiplying both sides by yields

Alternatively, , and then multiply both sides by .

### Manipulation

Let Then

10x &= 9.999\ldots\\ x &= 0.999\ldots

\end{align*}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)Subtracting,

9x &= 9\\ x &= 1

\end{align*}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)### Infinite series

This is an infinite geometric series, so