Difference between revisions of "0.999..."
(create by collecting some random thoughts) |
I like pie (talk | contribs) m |
||
| Line 1: | Line 1: | ||
<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 == | ||
| Line 11: | Line 11: | ||
=== 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*} | ||
| Line 18: | Line 17: | ||
\end{align*} | \end{align*} | ||
</math></center> | </math></center> | ||
| − | |||
Subtracting, | Subtracting, | ||
| − | |||
<center><math> | <center><math> | ||
\begin{align*} | \begin{align*} | ||
| Line 38: | Line 35: | ||
== 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
