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.).
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