Draft:Tau (mathematical constant)

The number 𝜏 or tau (/taʊ/, /tɔː/) 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. 𝜏 is exactly two times the more well-known mathematical constant $π$ , the ratio of a circle's circumference to its diameter. However, some mathematicians have advocated for the use of a single letter to represent 2 $π$ , stating that this value is more natural than $π$ .^{[citation needed]} Like $π$ , 𝜏 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 𝜏, to 50 decimal places, is:

6.28318530717958647692528676655900576839433879875021...

Definition

𝜏 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]} $\tau ={\frac {C}{r}}$

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.) $\tau =2\int _{-1}^{1}{\frac {1}{\sqrt {1-x^{2}}}}\,dx$

𝜏 can also be defined using the sine and cosine functions, as follows:

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

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

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