Tau (mathematics)

The number τ (; spelled out as tau) is a mathematical constant, equal to 2π, that has been proposed as an alternative to π by various proponents, who argue that mathematical notation would be simpler, clearer or pedagogically preferable if formulas and quantities (including angles) were described in terms of τ instead of π. A constant named tau is implemented in some programming languages and calculator apps as an alias for 2π. τ is approximately equal to

It is the ratio of a circle's circumference to its radius, whereas π instead corresponds to the ratio of a circle's circumference to its diameter. Like π, τ can be defined in several other equivalent ways: notably, τ is equal to the period of the complex exponential function. Like π, τ is irrational and transcendental.

While π is used almost exclusively in mainstream mathematical education and practice, it has been proposed in recent decades, most notably by physicist Michael Hartl in his 2010 article The Tau Manifesto, that τ should be used instead. Hartl and other proponents argue that τ is the more natural circle constant and its use leads to conceptually simpler and more intuitive mathematical notation; in particular, τ is the angle measure of a whole turn in radians, whereas π corresponds only to a half-turn, so proponents argue that angles in radians are more intuitive when expressed as fractions of τ compared to π.

Critics have responded that the benefits of using τ over π are trivial and that given the ubiquity and historical significance of π a change is unlikely to occur.

The proposal did not initially gain widespread acceptance in the mathematical community, but awareness of τ has become more widespread, including having been added to several major programming languages and calculators.