Theorem needed to legitimize definition of $\pi$

Question

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!

Answer 1

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.

Answer 2

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.

Answer 3

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?

Answer 4

That definition is technically not precise. When you use any specific circle to calculate $\pi$, what you're really doing is calculating the value of $\pi$ defined using that circle. It's the same number but a different definition for each circle. What you're really doing is blindly applying the rule that since you get that number for one circle, you will get the same number for all circles.