Draft:Tau (mathematical constant)
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The number ๐ or tau (/taส/, /tษห/) is the mathematical constant equal to the ratio of a circle's circumference to its radius, approximately 6.28319. Equivalently, it is the number of radians in a turn, the circumference of the unit circle, and the period of the sine and cosine functions. ๐ is exactly two times the more well-known mathematical constant ฯ, the ratio of a circle's circumference to its diameter. However, some mathematicians have advocated for the use of a single letter to represent 2ฯ, stating that this value is more natural than ฯ.[citation needed] Like ฯ, ๐ is irrational, meaning it cannot be expressed as the quotient of two integers, and is transcendental, meaning it is not a solution to any nonzero polynomial with rational coefficients. However, its value can be expressed precisely using infinite series, integrals, or as the solution to equations involving trigonometric functions.[citation needed]
The value of ๐, to 50 decimal places, is:
Definition
[edit]๐ can be defined as the ratio of a circle's circumference C to its radius r. This ratio is constant, regardless of the size of the circle.[citation needed]
The circumference of a circle can be defined independently of geometry using limits, a concept in calculus. For example, one can directly compute the arc length of the unit circle using the following integral. (The factor of 2 is needed to calculate both halves of the unit circle, as the integral itself only calculates the length of the top half of the unit circle.)
๐ can also be defined using the sine and cosine functions, as follows:
- ๐ is the smallest positive real number such that cos(๐/4) = 0.
- ๐ is the smallest strictly positive real number such that sin(๐/2) = 0.
- ๐ is the period length of the sine and cosine functions, i.e. ๐ is the smallest strictly positive real number such that for any real or complex number x, sin(x) = sin(x+๐) and cos(x) = cos(x+๐).
Sine and cosine can be defined independently of geometry using Taylor series (see Sine_and_cosine#Series_definitions).
In addition, ๐ can be defined using the complex exponential function. Like sine and cosine, the exponential function can be defined as an infinite series. ๐ is the smallest strictly positive real number such that exp(i๐) = 1. The value exp(ix) = 1 is equal to 1 if and only if x is an integer multiple of ๐.