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I was reading in a book that ratio of circumference to diameter of a circle is $22:7$ or 3$55:113$.Why is it $355:113$?I can understand $22:7$ but what about $355:113$.Thanks for any help.

Soham
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1 Answers1

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Expressing $\pi$ as a ratio of two integers will always only give you an approximation. But the approximation improves if you have larger integers.

Try this. Using your calculator, divide $22$ by $7$. Write down the result. Now, divide $355$ by $113$. Write down the result. Finally, hit the key marked $\pi$ on your calculator. (This is also an approximation, but accurate to several decimal places.) Write that down.

Which ratio is closer to your calculator value of $\pi$? Can you find a closer approximation using a ratio of 4-digit integers?

MathAdam
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  • Sir,are the values calculated by trial and error method? – Soham Sep 30 '15 at 13:35
  • You could also choose the first few terms of a series whose sum approaches $\pi$. Or, you could take the ratio of 31415926535 : 10000000000, but that gets cumbersome. The point is to have a simple ratio that is close enough. – MathAdam Sep 30 '15 at 13:41
  • I can understand that but what interested me was that both the numerator and denominator in this case was mutually prime integers. – Soham Sep 30 '15 at 13:42
  • That ensures the ratio is fully simplified. – MathAdam Sep 30 '15 at 13:46
  • I am saying that how do they arrive at 355:113?The can't do so from 22:7 as 355 and 113 is not a multiple of 22 or 7. – Soham Sep 30 '15 at 13:48