# Difference between revisions of "0.999..."

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<math>0.999\ldots</math> (or <math>0.\overline{9}</math>) is an equivalent representation of the [[real number]] <math>1</math>. | <math>0.999\ldots</math> (or <math>0.\overline{9}</math>) is an equivalent representation of the [[real number]] <math>1</math>. | ||

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

== Proofs == | == Proofs == | ||

Line 11: | Line 11: | ||

=== Manipulation === | === Manipulation === | ||

Let <math>x = 0.999\ldots</math> Then | Let <math>x = 0.999\ldots</math> Then | ||

− | |||

<center><math> | <center><math> | ||

\begin{align*} | \begin{align*} | ||

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\end{align*} | \end{align*} | ||

</math></center> | </math></center> | ||

− | |||

Subtracting, | Subtracting, | ||

− | |||

<center><math> | <center><math> | ||

\begin{align*} | \begin{align*} | ||

Line 38: | Line 35: | ||

== Related threads == | == Related threads == | ||

− | *[http://www.artofproblemsolving.com/Forum/viewtopic.php? | + | *[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=201302] |

## Revision as of 18:59, 24 April 2008

(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