# Irrationality of $\pi$ and circumference to diameter ratio. [duplicate]

How is $\pi$ actually defined? If it is defined as the ratio of the circumference of a circle to its diameter then from this definition itself either of the circumference and diameter has to be irrational. The circumference and diameter both are finite but $\pi$ is irrational. An irrational number is a number with never ending numbers after decimal, till now no recurrence has been discovered.

Yet there are methods to represent them in a number line which means they have finite length. So, if we take a finite length and make a circle around it as diameter, does it mean circumference is irrational but finite? In our everyday life, do we always take rounded off values of circumference or diameter?

• Yes, if you take a diameter which is an integer, say, the circumference must be irrational, and vice versa. You can also notice this with triangles: if you take a right angle triangle with the two orthogonal sides having length $1$, the length of the hypotenuse will be $\sqrt{2}$, which is also irrational. Apr 17, 2014 at 6:32
• Note that $\frac13$ also has "never ending numbers after decimal". This doesn't contradict the fact that $\frac13$ is finite. Apr 17, 2014 at 6:41
• Watch this. Apr 17, 2014 at 11:23
• "Finite" and "irrational" are not mutually exclusive. Having an infinite decimal expansion and being infinite are not the same thing. $\pi$ has infinitely many digits after the decimal point, but $\pi$ is still a finite number, just like the circumference and the diameter are finite numbers. What do you mean by "recurrence" when you say "no recurrence has been discovered"?
– user307169
May 24, 2017 at 16:15
• Ah, didn't realize how insanely old this question is...
– user307169
May 24, 2017 at 16:16

• A better way to write that sum would be $\sum_1^{\infty} \frac {4(-1)^r }{2r-1}$ Apr 22, 2015 at 20:29