# Theorem needed to legitimize definition of $\pi$

Reading through Euclidean and Non-Euclidean Geometry I stumbled upon this problem. If the number $\pi$ is defined as the ratio of the circumference of any circle to its diameter, what theorem must first be proved to legitimize this definition ?. The problem also mentions that the required theorem is proved in section 21.2 of Moise. Why would you need to prove a theorem before defining an arbitrary term, couldn't you just state that that is the definition of $\pi$. I am not a math student so I don't have much practice but I do want to get better at proofs and the more formal side of mathematics. Thanks a lot in advance!

• If $\pi$ is defined as a single value, I believe you would first need to show that the diameter and circumference of any circle is unique and exists for any arbitrary circle. Then you would need to show that the ratio between the two is always the same, regardless of the circle. – Dair Aug 9 '17 at 2:38
• Well, I can define kipple to be the ratio of a person height to the person weight. So can I now use kipple as a well defined mathematical constant? – fleablood Aug 9 '17 at 2:42
• Some related questions: google.com/… – Hans Lundmark Aug 9 '17 at 7:32

## 3 Answers

One would need first to show that, for any circle of diameter $D$ and circumference $C,$ the value of $C/D$ is the same, then $C/D$ does not depend on the size of the circle. That is, that $C/D$ stays constant no matter how "large" or "small" the circle may be.

• In addition to needing that $C/D$ is independent of the circle's size, one also needs that it's independent of the circle's location. – Andreas Blass Aug 9 '17 at 2:47
• @AndreasBlass Technically right, but then are we considering circles as certain subsets of $\mathbb{R}^2,$ or only "synthetic geometry" constructs? – coffeemath Aug 9 '17 at 2:59
• Or are we perhaps even considering circles in $\mathbb R^3$? This definition of $\pi$ can keep a repair crew busy for quite a while. – Andreas Blass Aug 9 '17 at 3:07

Usually when someone says that they are defining $\pi$ to be the ratio of the circumference of any circle to its diameter, it is implied that the value of $\pi$ does not change for any circle. You would need to show that the ratio is always the same for any circle.

Furthermore, if you want to get really, really pedantic, you need to prove the existence of all the objects. You would need to illustrate that circles exist, and that every circle has a exactly one circumference and diameter associated with it.

If I defined a mathematical constant of kipple $= \frac {\text {persons height}}{\text {circumference of person's waist}}$ is that a valid constant?

Why not?

So if I define $\pi = \frac {\text{circle's circumference}}{\text{circle's diameter}}$? Is that a valid constant?

Why?

What makes one okay and the other not?

Is there some theorem that applies to circles, their diameters and circumferences that does not apply to people, their height and their waists?