Difference between revisions of "0.999..."
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Alternatively, <math>\frac 19 = 0.\overline{1} = 0.111\ldots</math>, and then multiply both sides by <math>9</math>. | Alternatively, <math>\frac 19 = 0.\overline{1} = 0.111\ldots</math>, and then multiply both sides by <math>9</math>. | ||
− | === Manipulation === | + | === Algebraic Manipulation === |
Let <math>x = 0.999\ldots</math> Then | Let <math>x = 0.999\ldots</math> Then | ||
<center><math> | <center><math> | ||
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0.999\ldots = \frac{\frac{9}{10}}{1 - \frac{1}{10}} = 1 | 0.999\ldots = \frac{\frac{9}{10}}{1 - \frac{1}{10}} = 1 | ||
</math></center> | </math></center> | ||
+ | |||
+ | ===Limits=== | ||
+ | <math>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</math> | ||
+ | |||
+ | ==See Also== | ||
+ | *[[Geometric sequence]] | ||
== Related threads == | == Related threads == | ||
*[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=201302] | *[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=201302] |
Revision as of 20:28, 24 April 2008
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
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. ! 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
Limits