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 == | ||
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=== 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> | ||
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Subtracting, | Subtracting, | ||
− | |||
<center><math> | <center><math> | ||
\begin{align*} | \begin{align*} | ||
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== 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. Unknown error_msg)Subtracting,
9x &= 9\ x &= 1
\end{align*}$ (Error compiling LaTeX. Unknown error_msg)Infinite series
This is an infinite geometric series, so