Constants: the 25 most written about

Pi

Number, approximately 3.14

A mathematical constant, written π: the ratio between the circumference of any circle and its diameter. Its value is about 3.141592653589793238462643383279…

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Golden ratio

Number, approximately 1.618

Two quantities are in the golden ratio when dividing their sum by the larger of the two gives the same ratio as the two quantities themselves.

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Euler's number

Base of natural logarithms

A mathematical constant, written e, that serves as the base of the natural logarithm and of the exponential function.

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Imaginary unit

Principal square root of minus 1

A mathematical constant, usually written i, that solves the quadratic equation x² = −1, which no real number solves. Multiplying it by any real number gives what is called an imaginary number.

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Euler's constant

Difference between logarithm and harmonic series

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Square root of 2

Unique positive real number which when multiplied by itself gives 2

The positive real number whose square, the number times itself, is 2.

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Square root of 3

Unique positive real number which when multiplied by itself gives 3

The positive real number that, multiplied by itself, makes 3.

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Apéry's constant

Sum of the inverses of the positive cubes

A mathematical constant found by adding up, without end, one divided by each positive integer cubed.

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Catalan's constant

Number, approximately 0.916

A constant, written G, found by alternately adding and subtracting one over each odd square number.

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Plastic ratio

Number, approximately 1.3247

A geometrical proportion, defined as the only real number x for which x³ = x + 1. Written as a decimal, it starts 1.324717957244746…

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Square root of 5

Positive real number which when multiplied by itself gives 5

Written √5, the positive real number that gives the natural number 5 when squared. Together with its conjugate −√5 it solves x² − 5 = 0, which makes it a quadratic integer, one kind of algebraic number.

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Champernowne constant

Transcendental number(s) with all positive integers in order

A transcendental real constant, written C₁₀, whose decimal expansion has important properties. It takes its name from D. G. Champernowne, an economist and mathematician who published it in 1933 while still an undergraduate.

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Silver ratio

Number, approximately 2.41421

A geometrical proportion whose exact value is 1 + √2, the positive solution of x² = 2x + 1.

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Omega constant

Solution to x * e^x = 1

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Erdős–Borwein constant

Sum of the reciprocal of the Mersenne numbers

Named after Paul Erdős and Peter Borwein, the constant you get by adding together one over each Mersenne number.

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Gelfond's constant

Constant e raised to the power of pi

The number e to the power π, written e^π and known as Gelfond's constant; it is a real number.

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Landau–Ramanujan constant

Constant value used in mathematics

A positive real number b from number theory. It appears in a theorem Edmund Landau proved in 1908: for large x, the count of positive integers below x that are sums of two squares grows asymptotically like bx/√(log(x)).

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Legendre's constant

Constant of proportionality of prime number density

A constant in a formula Adrien-Marie Legendre devised to approximate how the prime-counting function π(x) behaves. The value that matches that behaviour exactly in the limit is now known to be 1.

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Copeland–Erdős constant

Irrational number based on primes

The number formed by writing "0." followed by the prime numbers, in order, in base 10.

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Tau

Constant equal to twice pi

A mathematical constant, written τ: how many times a circle's radius fits into its circumference. That is exactly 2π, approximately 6.283185.

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Chaitin's constant

Halting probability of a random computer program

A real number from algorithmic information theory, a branch of computer science, also called a halting probability. Informally, it is the chance that a program built at random will halt; these numbers come from a construction by Gregory Chaitin.

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Meissel–Mertens constant

Mathematical constant

A constant of number theory named after Ernst Meissel and Franz Mertens. It is also known as the prime reciprocal constant, as Kronecker's constant, as the Mertens constant, and as the Hadamard–de la Vallée-Poussin constant.

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Mills' constant

Prime-generating mathematical constant

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Khinchin's constant

Mathematical constant in number theory

A constant of number theory connected with the expansions of many real numbers as simple continued fractions. Aleksandr Yakovlevich Khinchin proved that for almost every real number x, when x is written as a continued fraction its coefficients aᵢ have a finite geometric mean, one that does not depend on x.

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Reciprocal Fibonacci constant

Mathematical constant

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