Pi: Difference between revisions
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'''Pi''', or '''π''', is the [[ratio]] between the circumference and the diameter of a [[circle]]. | '''Pi''', or '''π''', is the [[ratio]] between the circumference and the diameter of a [[circle]]. π is not a number, because the numerical value of π depends on context: which circle is under consideration, and how precise are one's measurements? π should be thought of as more like a number-valued variable, which stands only for a very specific, limited, class of things. | ||
== The traditional concept == | == The traditional concept == | ||
Standard mathematics says that π is the ratio between the circumference and the diameter of a [[Circle#The traditional concept|''perfect'' circle]]. | Standard mathematics says that π is the ratio between the circumference and the diameter of a [[Circle#The traditional concept|''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 <math>r</math> has area <math>\pi r^2</math> | |||
a cone with height <math>h</math> and with base of radius <math>r</math>, has area <math>\frac{1}{3}\pi r^2 h</math> | |||
a sphere of radius <math>r</math> has volume <math>\frac{4}{3} \pi r^3</math>, and surface area <math>4 \pi r^2</math> | |||
the area under a Gaussian distribution is proportional to some power of <math>\pi^{1/2}</math>. | |||
Latest revision as of 01:44, 2 July 2024
Pi, or π, is the ratio between the circumference and the diameter of a circle. π is not a number, because the numerical value of π depends on context: which circle is under consideration, and how precise are one's measurements? π should be thought of as more like a number-valued variable, which stands only for a very specific, limited, class of things.
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 has area
a cone with height and with base of radius , has area
a sphere of radius has volume , and surface area
the area under a Gaussian distribution is proportional to some power of .
