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Why is PI considered irrational if it is defined as ratio of circumference to diameter? And how do we close the value of circumference and diameter to a certain number without giving up accuracy at some point, be it after trillion digits? I know it is approximated via other rigorous mathematical techniques but how come we do not ever involve losing any accuracy at any point. How are we so certain of the irrationality of PI?

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    $\begingroup$ Because circumference and diameter are never simultaneously integers. $\endgroup$
    – user239203
    Dec 31, 2019 at 11:54
  • $\begingroup$ Well, we don't approximate the circumference but rather take the ideal circumference of an ideal circle in mathematics. Historically, it took quite a while until the irrationality of $\pi$ was proven. $\endgroup$
    – Qi Zhu
    Dec 31, 2019 at 11:57
  • $\begingroup$ @JoséCarlosSantos This is indeed a duplicate but I think the match I found has a better answer than the accepted one to your match. $\endgroup$ Dec 31, 2019 at 12:01
  • $\begingroup$ @EthanBolker Indeed. $\endgroup$ Dec 31, 2019 at 12:04

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