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:

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

In binary:

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