# 0.999...

$0.999\ldots$ (or $0.\overline{9}$) is an equivalent representation of the real number $1$.

It is often mistaken that $0.999\ldots \neq 1$ for various reasons (that there can only be a finite number of $9$s, that there is a $\frac 1{\infty}$ term left over at the end, etc.).

## Proofs

### Fractions

Since $\frac 13 = 0.\overline{3} = 0.333\ldots$, multiplying both sides by $3$ yields $1 = 0.999\ldots$

Alternatively, $\frac 19 = 0.\overline{1} = 0.111\ldots$, and then multiply both sides by $9$.

### Manipulation

Let $x = 0.999\ldots$ Then

\begin{align*} 10x &= 9.999\ldots\\ x &= 0.999\ldots \end{align*} (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)

Subtracting,

\begin{align*} 9x &= 9\\ x &= 1 \end{align*} (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)

### Infinite series

$0.999\ldots = 0.9 + 0.09 + 0.009 + \ldots = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \ldots$

This is an infinite geometric series, so

$0.999\ldots = \frac{\frac{9}{10}}{1 - \frac{1}{10}} = 1$