Mathematical constant

A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, and calculus.

What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste, and some mathematical constants are notable more for historical reasons than for their intrinsic mathematical interest. The more popular constants have been studied throughout the ages and computed to many decimal places.

All mathematical constants are definable numbers and usually are also computable numbers (Chaitin's constant being a significant exception).

These are constants which one is likely to encounter during pre-college education in many countries.

The constant π (pi) has a natural definition in Euclidean geometry (the ratio between the circumference and diameter of a circle), but may be found in many places in mathematics: for example, the Gaussian integral in complex analysis, the roots of unity in number theory, and Cauchy distributions in probability. However, its ubiquity is not limited to pure mathematics. It appears in many formulas in physics, and several physical constants are most naturally defined with π or its reciprocal factored out. It is debatable, however, if such appearances are fundamental in any sense. For example, the textbook nonrelativistic ground state wave function of the hydrogen atom is

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