Pi, usu­ally writ­ten \(π\), is a num­ber equal to the ra­tio of a cir­cle’s cir­cum­fer­ence to its di­ame­ter. The value of \(π\) is ap­prox­i­mately \(3.141593\).

If the length of a curve seems like an ill-defined con­cept to you (maybe you only un­der­stand how lines could have lengths), con­sider big­ger and big­ger reg­u­lar poly­gons that make bet­ter and bet­ter ap­prox­i­ma­tions of the cir­cle. As the num­ber of sides \(N\) of the poly­gon goes to \(∞\), the per­ime­ter will ap­proach a length of \(π\) times the di­ame­ter.

One could also define \(π\) to be the area of a cir­cle di­vided the area of a square, whose edge is the ra­dius of the cir­cle.

What Kind of Num­ber It Is

It’s not an In­te­ger.

If the di­ame­ter here is 1, then the per­ime­ter of the hexagon is 3, the per­ime­ter of the square is 4, and the cir­cum­fer­ence of the cir­cle is in be­tween. There are no in­te­gers be­tween 3 and 4.

It’s not ra­tio­nal, nor is it alge­braic. It’s tran­scen­den­tal.


  • Pi is irrational

    The num­ber pi is fa­mously not ra­tio­nal, in spite of jok­ing at­tempts at leg­is­la­tion to fix its value at 3 or 227.


  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.