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$1 \textrm{mile} = 1.609344 \textrm{km}$

I know that using $1000000$ miles I can move the decimal point and get a full number of $1609344 $km.

But how can I find the smallest amount of full miles that is equal to full kilometers and do the same for other conversions?

Sven
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  • By "full number" do you mean a natural number? – Sujaan Kunalan Jun 03 '13 at 17:08
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    I think you mean "conversions" in your last question. That aside, just eliminate all common factors (we see both are divisible by $2$, for example, so eliminate the $2$ and continue on). This is assuming you don't have easy access to a calculator that lets you write the number as a fraction without any effort :) – Clayton Jun 03 '13 at 17:09
  • @SujaanKunalan: yes, a natural number. And yes, I meant conversions. – Sven Jun 03 '13 at 17:12

3 Answers3

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$$1.609344=\frac{25146}{15625}$$

So $15625$ miles is exactly $25146$ kilometers, and these are the smallest natural numbers because the above fraction is reduced.

Jared
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Keep eliminating the common factors from the numerator and denominator. It is obvious both are even by looking at their last digit, so lets start by dividing both numbers by 2.

If 1 mile$=$ 1.609344$\implies 1000000$miles$=1609344$.

Start with $\frac{1609344}{1000000}$. Divide both numbers by 2 and you get $\frac {804672} {500000}$. Divide both numbers by 2 again and you get $\frac{402336}{250000}$. Divide both numbers by 2 again, and repeat until the numbers can not be reduced by a common factor. Keep dividing like this and you will end up with$\frac{25146}{15625}$, which can not be reduced any further. (It should take you 6 divisions of both numbers by 2, which ultimately means that dividing both numbers by $2^{6}=64$ would have gotten your irreducible fraction in this case.)

Sujaan Kunalan
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Jared has a point. But actually, Wolfram alpha tells me that actually, 6336 kilometers is equal to 3937 miles, exactly.

Why? The original asker's value was not totally exact.

Anonymous Pi
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    I don't know where you saw that, but 1 mile is by definition exactly equal to 1609.344 metres. (1 mile = 1760 yards and 1 yard = 0.9144 metres exactly, by definition.) Your values give about 1.60934721869 km per mile, which corresponds to the US survey mile, a different unit. – ShreevatsaR Jun 03 '13 at 17:35
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    What a good idea it was to elaborate the metric system ! – ama Jun 03 '13 at 18:30
  • @ShreevatsaR Didn't know that! – Anonymous Pi Jun 03 '13 at 20:14