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Pi is defined as circumference/diameter, but it is an irrational number. And by definition an irrational number can't be defined by a fraction. So how is it that pi is circumference/diameter and on a side note: how are important irrational numbers like e normally found?

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marked as duplicate by MJD, Eric Wofsey, ncmathsadist, user99914, BLAZE Nov 11 '15 at 3:58

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    $\begingroup$ Irrational numbers cannot be expressed as a fraction of integers. Thus, a circle cannot have a diameter and circumference which are both integers. $\endgroup$ – Alex S Nov 10 '15 at 17:29
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    $\begingroup$ In practice this is no problem because we can use arbitary good rational approximations of $\pi$, a very good one is $\frac{355}{113}$ $\endgroup$ – Peter Nov 10 '15 at 17:30
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    $\begingroup$ Archimedes was the first to calculate $\pi$ upto some digits. $\endgroup$ – Peter Nov 10 '15 at 17:31
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    $\begingroup$ He approximated the circumfence of a circle with regular polygons inside ans outside the circle and took the middle value of the results. $\endgroup$ – Peter Nov 10 '15 at 17:33
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    $\begingroup$ The definition is usually : The first positive root of $cos(x)$ is $\frac{\pi}{2}$ $\endgroup$ – Peter Nov 10 '15 at 17:35