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Draft:Tau (mathematical constant)

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The number ๐œ or tau (/taสŠ/, /tษ”ห/) 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œ 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œ 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œ, to 50 decimal places, is:

6.28318530717958647692528676655900576839433879875021...

Definition

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๐œ 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œ can also be defined using the sine and cosine functions, as follows:

  • ๐œ is the smallest positive real number such that cos(๐œ/4) = 0.
  • ๐œ is the smallest strictly positive real number such that sin(๐œ/2) = 0.
  • ๐œ is the period length of the sine and cosine functions, i.e. ๐œ is the smallest strictly positive real number such that for any real or complex number x, sin(x) = sin(x+๐œ) and cos(x) = cos(x+๐œ).

Sine and cosine can be defined independently of geometry using Taylor series (see Sine_and_cosine#Series_definitions).

In addition, ๐œ can be defined using the complex exponential function. Like sine and cosine, the exponential function can be defined as an infinite series. ๐œ is the smallest strictly positive real number such that exp(i๐œ) = 1. The value exp(ix) = 1 is equal to 1 if and only if x is an integer multiple of ๐œ.