Draft:Tau (mathematical constant)

The number 𝜏 or tau 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 $\pi$, the ratio of a circle's circumference to its diameter. However, some mathematicians have advocated for the use of a single letter to represent 2π, stating that this value is more natural than π. 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.

The value of 𝜏, to 50 decimal places, is:

Definition
Tau 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. $$ \tau = \frac{C}{r}$$

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: Sine and cosine can be defined independently of geometry using Taylor series (see Sine_and_cosine).
 * 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 $cos(x) = cos(x+𝜏)$.

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 $exp(i𝜏) = 1$. The value $exp(ix) = 1$ is equal to 1 if and only if x is an integer multiple of 𝜏.