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The number š or tau (/taŹ/, /tÉĖ/) 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 length of the sine and cosine functions. Tau is exactly two times the more well-known mathematical constant Ļ, the ratio of a circle's circumference to its diameter. However, some mathematicians have advocated for the use of a single letter to represent 2Ļ, stating that this value is more natural than Ļ.[citation needed] Like Ļ, š 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 š, to 50 decimal places, is:
Definition
editTau 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.)
š can also be defined using the sine and cosine functions, as follows:
- š is the smallest positive real number such that cos(š/4) = 0.
- š is the smallest strictly positive real number such that sin(š/2) = 0.
- š is the period length of the sine and cosine functions, i.e. š is the smallest strictly positive real number such that for any real or complex number x, sin(x) = sin(x+š) and cos(x) = cos(x+š).
Sine and cosine can be defined independently of geometry using Taylor series (see Sine_and_cosine#Series_definitions).
In addition, š can be defined using the complex exponential function. Like sine and cosine, the exponential function can be defined as an infinite series. š is the smallest strictly positive real number such that exp(iš) = 1. The value exp(ix) = 1 is equal to 1 if and only if x is an integer multiple of š.