Fractions

There exists a interesting relationship between 2727 and 3737

127=0.037037037...\frac{1}{27}=0.037037037... and 137=0.027027027...\frac{1}{37}=0.027027027... (That is, each number forms the other number's repeating decimal.)

This can be explained as follows :

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

then the fraction equals YY times 10n+102n+103n...=Y(10n1)10^{-n}+10^{-2n}+10^{-3n}...= \frac{Y}{(10^n-1)}.

Therefore, we need XY=10n1XY=10^n-1. In the above case, 2737=103127*37=10^3-1; other examples are 33=10113*3=10^1-1

So 13=0.333...\frac{1}{3}=0.333...

or 119=102111*9=10^2-1 so 111=0.090909...\frac{1}{11}=0.090909... and 19=0.11111...\frac{1}{9}=0.11111... and so on.

Note by Danish Ahmed
4 years, 8 months ago

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!!!

Alfian Edgar - 4 years, 7 months ago

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It is Fantastic! Thanks for sharing @Danish Ahmed

Sarthak Tanwani - 4 years, 8 months ago

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nice

Akash Gupta - 4 years, 8 months ago

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Excellent

Kanthi Deep - 4 years, 8 months ago

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

Calvin Lin Staff - 4 years, 8 months ago

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