0.999...
(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).
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