Definition:Pi

Definition
The real number $$\pi$$ (pronounced pie) is an irrational number (see proof here) whose value is approximately $$3.14159\,26535\,89793\,23846\,2643 \ldots$$

Geometric
Take a circle whose circumference is $$C$$ and whose radius is $$r$$.

Then $$\pi$$ can be defined as $$\pi = \frac {C} {2r}$$.

(It can be argued that $$\pi = \frac {C} {d}$$, where $$d$$ is the circle's diameter, is a simpler and more straightforward definition. However, the radius is, in general, far more immediately "useful" than the diameter, hence the above more usual definition in terms of circumference and radius.)

Uniqueness of Pi
Note that $$\pi$$ is defined on a per-circle basis. For each circle with its own circumference $$C$$ and diameter $$d$$, $$\pi$$ is defined as the ratio between the two. It is conceivable, then, that $$\pi$$ has a difference value for each circle. It is also true, however, that all circles are similar and thus proportional in size. Thus, the value of $$\pi$$ is consistent between any two circles, and the constancy of $$\pi$$ is proven.

Algebraic
The real functions sine and cosine are shown to be periodic.

The period of both sine and cosine is $$2 \pi$$.