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

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

2 years, 4 months ago

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expect the unexpected ! · 1 year, 1 month ago

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. · 2 years, 4 months ago

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. · 2 years, 4 months ago

Do you have the proof of the property of recurring decimal expansions that they repeat for n-1 digits in a fraction 1/n? · 2 years, 4 months ago

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 $$n$$. · 2 years, 4 months ago

hmmm · 2 years, 4 months ago

thanx.But I recently learned LATEX and now I can't get enough of it. · 2 years, 4 months ago

I don't understand "I can't get enough of it", you can always learn it. It's really easy to learn. · 2 years, 4 months ago

"I can't get enough of it" means now I can't resist writing it.It is an idiom,I think. · 2 years, 4 months ago

BTW which class r u in? · 2 years, 4 months ago

I'm in class 10. · 2 years, 4 months ago

oh,ok. · 2 years, 4 months ago

@Krishna Ar · 2 years, 4 months ago

(y) · 2 years, 4 months ago

WOW @Krishna Ar, this is called popularity! · 2 years, 4 months ago

-_- · 2 years, 4 months ago