0.999...

This is an AoPSWiki Word of the Week for April 25-May 2

$0.999\ldots$ (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 $9$ .

Algebraic Manipulation

Let $x = 0.999\ldots$ Then

$\begin{align*}

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

\end{align*}$ (Error compiling LaTeX. Unknown error_msg)

Subtracting,

$\begin{align*}

9x &= 9\\ x &= 1

\end{align*}$ (Error compiling LaTeX. Unknown error_msg)

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

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

Limits

$0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1$

Related Threads

<url>viewtopic.php?t=201302 ".9 rep = 1?"</url>
<url>viewtopic.php?t=188041 "0.999999... =1?"</url>

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