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Me and one of my friends had an argument and he said that using $22/7$ as value of $\pi$ is sufficient for any calculation. Can we always take it $22/7$, or is there some example of some calculation in which if we use $\pi=22/7$, some noticeable error which makes quite a significant difference, appears. Kind of calculative contradiction.

Thanks!

Bhaskar Vashishth
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    Fourier analysis, or any analysis which depends on cancelling trigonometric sums, would almost certainly go wrong- unless the value were corrected. – Mark Bennet Dec 23 '14 at 16:37
  • Relevant : http://math.stackexchange.com/questions/1043192/how-come-pi-is-usually-approximated-as-3-14-or-22-7 – Hippalectryon Dec 23 '14 at 16:37
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    In using such an approximation you are wrong by an amount of about $0.04$%. I think there is no difference with regard to any other approximation you can do in calculations. – Vincenzo Tibullo Dec 23 '14 at 16:37
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    Any function/algorithm with repeated need of the $\pi$ number or any function with chaotic behavior (Lorentz attractor) would be significantly affected by this approximation, most likely. – Matthew Levy Dec 23 '14 at 16:57
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    Circles would be regular $96$-gons if this were so. – David Mitra Dec 23 '14 at 17:17
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    $$\frac1{7\pi-22}$$ –  Dec 23 '14 at 17:18
  • Related: http://math.stackexchange.com/questions/838467/when-the-approximation-pi-simeq-3-14-is-not-sufficent – njuffa Dec 23 '14 at 19:13

4 Answers4

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$$\int_0^1 dx \frac{x^4 (1-x)^4}{1+x^2} = \frac{22}{7}-\pi$$

That integral is clearly not zero.

Ron Gordon
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Consider calculating the position of a satellite in a circular orbit about the earth. After a few hundred rotations, the satellite will be noticeably in the wrong place.

Simon S
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  • Interesting, I am just wondering if you could explain why this would take place? I mean why after several hundred rotations? – Quality Dec 23 '14 at 16:42
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    $22/7$ overestimate $\pi$ by around 0.04 percent. With one rotation, the error will be small and possibly not material. After say 100 rotations using $22/7$, one might have calculated the satellite to have returned to its original position, however it has over rotated by approx 4% of 360 degrees, or approx 14 degrees. – Simon S Dec 23 '14 at 16:45
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Compute the third decimal digit of $\pi$.

Perhaps even better, because it is as much wrong as it can be:

Compute the ninth binary digit (in the fractional part) of $\pi$

egreg
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What quadrant is $1\ 000\ 000$ radians in? The answer you get to that question is wrong if you use $22/7$ as an approximation to $\pi$.

Michael Hardy
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