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>. | ||
Revision as of 20:21, 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.).
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