0.999...
This is an AoPSWiki Word of the Week for April 25-May 2 |
(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.).
Contents
[hide]Proofs
Fractions
Since , multiplying both sides by
yields
Alternatively, , and then multiply both sides by
.
Algebraic Manipulation
Let Then
10x &= 9.999\ldots\ x &= 0.999\ldots
\end{align*}$ (Error compiling LaTeX. Unknown error_msg)Subtracting,
9x &= 9\ x &= 1
\end{align*}$ (Error compiling LaTeX. Unknown error_msg)Infinite series
![$0.999\ldots = 0.9 + 0.09 + 0.009 + \ldots = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \ldots$](http://latex.artofproblemsolving.com/c/0/7/c0790728328bd92ec568c11fd893412ba7ee7aba.png)
This is an infinite geometric series, so
![$0.999\ldots = \frac{\frac{9}{10}}{1 - \frac{1}{10}} = 1$](http://latex.artofproblemsolving.com/a/e/2/ae2f54a6432e25455253212354eaa9db3238648c.png)
Limits
See Also
Related Threads
- <url>viewtopic.php?t=201302 ".9 rep = 1?"</url>
- <url>viewtopic.php?t=188041 "0.999999... =1?"</url>