# Definition:Pi/Definition 1

## Definition

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

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

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

(It can be argued that $\pi = \dfrac C d$, 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 different 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.

## Decimal Expansion

The decimal expansion of $\pi$ starts:

$\pi \approx 3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$

### Binary Expansion

The binary expansion of $\pi$ starts:

$\pi \approx 11 \cdotp 00100 \, 10000 \, 11111 \, 1011 \ldots$