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

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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?

## marked as duplicate by J. M. is a poor mathematician, Misha Lavrov, tilper, Lord Shark the Unknown, JonMark PerryMay 25 '17 at 0:09

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• 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. – user139388 Apr 17 '14 at 6:32
• Note that $\frac13$ also has "never ending numbers after decimal". This doesn't contradict the fact that $\frac13$ is finite. – Hagen von Eitzen Apr 17 '14 at 6:41
• Watch this. – Shaun Apr 17 '14 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"? – tilper May 24 '17 at 16:15
• Ah, didn't realize how insanely old this question is... – tilper May 24 '17 at 16:16

## 1 Answer

Pi is defined as a circle's circumference divided by its diameter in a Euclidean plane. But it is computable in many ways, most of them totally independent of a Euclidean circle. The simplest to remember that I know of is: 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ...

Basically, the numerator is always 4, the denominator is successive odd numbers, and you just keep alternating subtract and add operations.

Given the many different ways to compute pi and given the flexibility of English, you could say that each of them is a definition of pi. In other words, you could say "pi is defined as the alternating difference/sum of fractions where 4 is the numerator and the successive odd number is the denominator." And you wouldn't be inaccurate, but you would be imprecise. As far as I can determine, the consensus definition used by mathematicians of the word "pi" remains the same regardless of the method used to compute it.

As for the second part of your question... you make a mistake when you say "The circumference and diameter both are finite but pi is irrational". Being irrational does NOT make pi infinite. Pi is a finite quantity. It is an exact point on the number line. We just fail to express it when we write it out as digits. When we "write it out" by drawing a perfect circle, then we express it exactly.

• A better way to write that sum would be $\sum_1^{\infty} \frac {4(-1)^r }{2r-1}$ – imulsion Apr 22 '15 at 20:29
• I disagree. To be perfectly frank, a whole lot more people can read and understand the notation I used than the sigma notation. It would take me hours to work out that your notation is equivalent to mine. This is a high school level question. It shouldn't use a college level answer. – SRM Apr 29 '15 at 14:35