Pi

Pi, or π, is the ratio between the circumference and the diameter of a circle. The numerical value of π depends on context: which circle is under consideration, and how precise are one's measurements?

I think [TODO] that π should be thought of not as a number, but as a class of numbers, each number arising from the real circumference/diameter of a real circle.

The traditional concept

Standard mathematics says that π is the ratio between the circumference and the diameter of a perfect circle.

Standard mathematics "measures" this ratio using the methods of calculus.

Standard mathematics' notion of π is an irrational number---a number which cannot be written as a ratio of two whole numbers.

Thus in standard mathematics,

π = 3.141592653589793238462643383279502884197169399375105 8209749445923078164062862089986280348253421170679...

where the "..." signifies that the digits go on forever.

Examples

π shows up in many places, besides just circles.

a disk of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} has area Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi r^2}

a cone with height Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} and with base of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} , has area Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3}\pi r^2 h}

a sphere of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} has volume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4}{3} \pi r^3} , and surface area $4\pi r^{2}$

the area under a Gaussian distribution is proportional to some power of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{1/2}} .