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\(A\ very\ simple,yet\ elegant\ NT\ proof(corrected)\)

\[It\ is\ a\ well\ known\ fact\ that\ if\ in\ a\ fraction,the\ denominator\ is\ of\ the\ \\ form\ 2^{m}*5^{n}\ where\ m\ and\ n\ are\ non-negative\ integers\ the\ fraction\ \\ has\ a\ \\ terminating\ decimal.\] \(Here\ is\ the\ proof:\) \(Let\ us\ say\ that\ there\ is\ a\ fraction\ in\ which\ the\ numerator\ is\ N\ and\ \\ the\ denominator\ be\ 2^{m}*5^{n}\ where\ m+n=x\) \(The\ fraction\ can\ be\ written\ as:\\ \dfrac{N}{2^{x-n}*5^{x-m}}\) \(Multiplying\ by\ \dfrac{2^{n}}{2^{n}}\ and\ then\ by\ \dfrac{5^{m}}{5^{m}}\\ we\ have\ a\ fraction\ which\ has\ a\ new\ numerator,let\ us\ call\ it\ A.\)\ \(The\ new\ denominator\ is\ 10^{x}\\ Thus,the\ new\ fraction\ is\ \dfrac{A}{10^{x}}.\)\ \(Which,obviously,has\ a\ terminating\ decimal\ expansion.\)

Note by Adarsh Kumar
3 years ago

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expect the unexpected !

Raven Herd - 1 year, 10 months ago

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It is a very simple proof which was in my RD SHARMA but I thought it too elegant not to share it with you guys.

Adarsh Kumar - 3 years ago

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Nice note, Adarsh. You may also like to add the number of decimal digits after a number of such a form terminates. It's closely related to your proof. Also, you don't need to latex the whole thing. If you are using a latex editor, then there's no need. If you want to learn it, you can easily do so here. Thanks.

Satvik Golechha - 3 years ago

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Do you have the proof of the property of recurring decimal expansions that they repeat for n-1 digits in a fraction 1/n?

Adarsh Kumar - 3 years ago

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That's not necessary. It is only true in case of cyclic numbers. I was referring to the number of after-decimal digits after \(\frac{x}{2^m5^n}\) terminates. It's the larger of \(m\) and \(n\).

Satvik Golechha - 3 years ago

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@Satvik Golechha hmmm

Adarsh Kumar - 3 years ago

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thanx.But I recently learned LATEX and now I can't get enough of it.

Adarsh Kumar - 3 years ago

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I don't understand "I can't get enough of it", you can always learn it. It's really easy to learn.

Satvik Golechha - 3 years ago

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@Satvik Golechha "I can't get enough of it" means now I can't resist writing it.It is an idiom,I think.

Adarsh Kumar - 3 years ago

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BTW which class r u in?

Adarsh Kumar - 3 years ago

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@Adarsh Kumar I'm in class 10.

Satvik Golechha - 3 years ago

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@Satvik Golechha oh,ok.

Adarsh Kumar - 3 years ago

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@Krishna Ar

Adarsh Kumar - 3 years ago

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(y)

Krishna Ar - 3 years ago

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WOW @Krishna Ar, this is called popularity!

Satvik Golechha - 3 years ago

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-_-

Krishna Ar - 3 years ago

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hehe

Adarsh Kumar - 3 years ago

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