Fractions

There exists a interesting relationship between $27$ and $37$

$\frac{1}{27}=0.037037037...$ and $\frac{1}{37}=0.027027027...$ (That is, each number forms the other number's repeating decimal.)

This can be explained as follows :

Say $\frac{1}{X}=0.YYY....$ If $Y$ is n digits long,

then the fraction equals $Y$ times $10^{-n}+10^{-2n}+10^{-3n}...= \frac{Y}{(10^n-1)}$ .

Therefore, we need $XY=10^n-1$ . In the above case, $27*37=10^3-1$ ; other examples are $3*3=10^1-1$

So $\frac{1}{3}=0.333...$

or $11*9=10^2-1$ so $\frac{1}{11}=0.090909...$ and $\frac{1}{9}=0.11111...$ and so on.

#Fractions #Decimals #RecurringDecimals

Easy Math Editor

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`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`	$\sin \theta$
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Comments

Oh that's nice! Thanks for sharing this pattern that you noticed.

Calvin Lin Staff - 6 years, 4 months ago

Excellent

Kanthi Deep - 6 years, 4 months ago

nice

Akash Gupta - 6 years, 4 months ago

It is Fantastic! Thanks for sharing @Danish Ahmed

Sarthak Tanwani - 6 years, 4 months ago

!!!

Alfian Edgar - 6 years, 4 months 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}`	$\boxed{123}$