Definition:Pi

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.)

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

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