How did you know that?

To all those experienced students with decimals and fractions, we know that 0.33333... is 1/3 because only the 3 is repeating, so you simply put one 9 in the denominator to get 3/9 or 1/3. Or perhaps... 0.63 repeating can be expressed as 63/99 since two digits are being repeated to get 7/11. (if you didn't know this before hand, it's fine. Now you know :D). A harder question may ask 0.1256, where only 56 part is being repeated. This may be changed to 0.12+56/9900. Yes, there is a pattern, but I want to challenge you to derive this algebraically(use examples if it helps) BONUS: Turn 0.166666.. and 0.2118118118118.... into fractional form

#Algebra

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

These observations are a result of the following fact:

$0.\overline{d_1d_2 \cdots d_n}=\frac{d_1d_2 \cdots d_n}{10^n-1}$ where $d_1, d_2, \cdots, d_n$ denote digits.

The proof is quite simple.

A Former Brilliant Member - 5 years, 1 month ago

Thx i learned so much

Ian Chiu - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$