Mathematical constant

The circumference of a circle with diameter 1 is π.

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. However, it is debatable whether such appearances are fundamental in any sense. For example, the textbook nonrelativistic ground state wave function of the hydrogen atom is

${\displaystyle \psi (\mathbf {r} )={\frac {1}{(\pi a_{0}^{3})^{1/2}}}e^{-r/a_{0}},}$

where ${\displaystyle a_{0}}$ is the Bohr radius. This formula contains a π, but it is unclear if that is fundamental in a physical sense, or if it just reflects the π in the expression ${\displaystyle {4\pi r^{2}}}$ (for the surface area of a sphere with radius ${\displaystyle r}$).

Furthermore, this formula gives only an approximate description of physical reality, as it omits spin, relativity, and the quantal nature of the electromagnetic field itself. Likewise, the appearance of π in the formula for Coulomb's law in SI units is dependent on that choice of units, and a historical accident having to do with how the so-called permittivity of free space was introduced into the practice of electromagnetism by Giovanni Giorgi in 1901. It is true that once various constants are chosen in one relation, the appearance of π in other relationships is unavoidable, but that appearance is always for a mathematical reason as in the example of the hydrogen atom wave function above, and not a physical one.

The numeric value of π is approximately 3.1415926536 (sequence A000796 in the OEIS). Memorizing increasingly precise digits of π is a world record pursuit.

i in the complex or Cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis

The imaginary unit or unit imaginary number, denoted as i, is a mathematical concept which extends the real number system ℝ to the complex number system ℂ, which in turn provides at least one root for every polynomial P(x) (see algebraic closure and fundamental theorem of algebra). The imaginary unit's core property is that i² = −1. Here, the term "imaginary" is used because there is no real number having a negative square.

There are in fact two complex square roots of −1, namely i and −i, just as there are two complex square roots of every other real number (except zero, which has one double square root).

In contexts where i is ambiguous or problematic, j or the Greek ι (see alternative notations) is sometimes used. In the disciplines of electrical engineering and control systems engineering, the imaginary unit is often denoted by j instead of i, because i is commonly used to denote electric current in these disciplines.

Exponential growth (green) describes many physical phenomena.

Euler's number e, also known as the exponential growth constant, appears in many areas of mathematics, and one possible definition of it is the value of the following expression:

${\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}$

For example, the Swiss mathematician Jacob Bernoulli discovered that e arises in compound interest: An account that starts at $1, and yields interest at annual rate R with continuous compounding, will accumulate to e^R dollars at the end of one year.

The constant e also has applications to probability theory, where it arises in a way not obviously related to exponential growth. As an example, suppose that a slot machine with a one in n probability of winning is played n times, then for large n (e.g., one million), the probability that nothing will be won will tend to 1/e as n tends to infinity.

Another application of e, discovered in part by Jacob Bernoulli along with French mathematician Pierre Raymond de Montmort, is in the problem of derangements, also known as the hat check problem.[3] Here, n guests are invited to a party, and at the door each guest checks his hat with the butler, who then places them into labelled boxes. The butler does not know the name of the guests, and hence must put them into boxes selected at random. The problem of de Montmort is: what is the probability that none of the hats gets put into the right box. The answer is

${\displaystyle p_{n}=1-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+\cdots +(-1)^{n}{\frac {1}{n!}}}$

which, as n tends to infinity, approaches 1/e.

The numeric value of e is approximately 2.7182818284 (sequence A001113 in the OEIS).

The square root of 2 is equal to the length of the hypotenuse of a right-angled triangle with legs of length 1.

The square root of 2, often known as root 2, radical 2, or Pythagoras' constant, and written as √2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.

Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. Its numerical value truncated to 65 decimal places is:

1.41421356237309504880168872420969807856967187537694807317667973799... (sequence A002193 in the OEIS).

The square root of 2.

Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two is frequently used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10 ⁻⁵).

Bifurcation diagram of the logistic map.

Iterations of continuous maps serve as the simplest examples of models for dynamical systems.[4] Named after mathematical physicist Mitchell Feigenbaum, the two Feigenbaum constants appear in such iterative processes: they are mathematical invariants of logistic maps with quadratic maximum points[5] and their bifurcation diagrams.

The logistic map is a polynomial mapping, often cited as an archetypal example of how chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the Australian biologist Robert May,[6] in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst. The difference equation is intended to capture the two effects of reproduction and starvation.

The numeric value of α is approximately 2.5029. The numeric value of δ is approximately 4.6692.

Apery's constant is the sum of the series

$${\displaystyle \zeta (3)=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+\cdots }$$

Apéry's constant is an irrational number and its numeric value is approximately 1.2020569.

Despite being a special value of the Riemann zeta function, Apéry's constant arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.[7]

Golden rectangles in a regular icosahedron

${\displaystyle F_{n}={\frac {\varphi ^{n}-(1-\varphi )^{n}}{\sqrt {5}}}}$

An explicit formula for the nth Fibonacci number involving the golden ratio φ.

The number φ, also called the golden ratio, turns up frequently in geometry, particularly in figures with pentagonal symmetry. Indeed, the length of a regular pentagon's diagonal is φ times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. Also, it appears in the Fibonacci sequence, related to growth by recursion.[8] Kepler proved that it is the limit of the ratio of consecutive Fibonacci numbers.[9] The golden ratio has the slowest convergence of any irrational number.[10] It is, for that reason, one of the worst cases of Lagrange's approximation theorem and it is an extremal case of the Hurwitz inequality for Diophantine approximations. This may be why angles close to the golden ratio often show up in phyllotaxis (the growth of plants).[11] It is approximately equal to 1.6180339887498948482, or, more precisely 2⋅sin(54°) = ${\displaystyle \scriptstyle {\frac {1+{\sqrt {5}}}{2}}.}$

The area between the two curves (red) tends to a limit.

The Euler–Mascheroni constant is a recurring constant in number theory. The Belgian mathematician Charles Jean de la Vallée-Poussin proved in 1898 that when taking any positive integer n and dividing it by each positive integer m less than n, the average fraction by which the quotient n/m falls short of the next integer tends to ${\displaystyle \gamma }$ (rather than 0.5) as n tends to infinity. The Euler–Mascheroni constant also appears in Merten's third theorem and has relations to the gamma function, the zeta function and many different integrals and series. The definition of the Euler–Mascheroni constant exhibits a close link between the discrete and the continuous (see curves on the left).

The numeric value of ${\displaystyle \gamma }$ is approximately 0.57721.

${\displaystyle {\begin{matrix}1\\11\\21\\1211\\111221\\312211\\\vdots \end{matrix}}}$

Conway's look-and-say sequence

Conway's constant is the invariant growth rate of all derived strings similar to the look-and-say sequence (except for one trivial one).[12]

It is given by the unique positive real root of a polynomial of degree 71 with integer coefficients.[12]

The value of λ is approximately 1.30357.

If a real number r is written as a simple continued fraction:

${\displaystyle r=a_{0}+{\dfrac {1}{a_{1}+{\dfrac {1}{a_{2}+{\dfrac {1}{a_{3}+\cdots }}}}}},}$

where a_k are natural numbers for all k, then, as the Russian mathematician Aleksandr Khinchin proved in 1934, the limit as n tends to infinity of the geometric mean: (a₁a₂...a_n)^1/n exists and is a constant, Khinchin's constant, except for a set of measure 0.[13]

The numeric value of K is approximately 2.6854520010.

The Glaisher–Kinkelin constant is defined as the limit:

${\displaystyle A=\lim _{n\rightarrow \infty }{\frac {\prod _{k=1}^{n}k^{k}}{n^{n^{2}/2+n/2+1/12}e^{-n^{2}/4}}}}$

It is an important constant which appears in many expressions for the derivative of the Riemann zeta function. It has a numerical value of approximately 1.2824271291.

This Babylonian clay tablet gives an approximation of the square root of 2 in four sexagesimal figures: 1; 24, 51, 10, which is accurate to about six decimal figures.[14]

${\displaystyle c=\sum _{j=1}^{\infty }10^{-j!}=0.\underbrace {\overbrace {110001} ^{3!{\text{ digits}}}000000000000000001} _{4!{\text{ digits}}}000\dots }$

Liouville's constant is a simple example of a transcendental number.

Some constants, such as the square root of 2, Liouville's constant and Champernowne constant:

${\displaystyle C_{10}=0.{\color {blue}{1}}2{\color {blue}{3}}4{\color {blue}{5}}6{\color {blue}{7}}8{\color {blue}{9}}10{\color {blue}{11}}12{\color {blue}{13}}14{\color {blue}{15}}16\dots }$

are not important mathematical invariants but retain interest being simple representatives of special sets of numbers, the irrational numbers,[15] the transcendental numbers[16] and the normal numbers (in base 10)[17] respectively. The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum who proved, most likely geometrically, the irrationality of the square root of 2. As for Liouville's constant, named after French mathematician Joseph Liouville, it was the first number to be proven transcendental.[18]

In the computer science subfield of algorithmic information theory, Chaitin's constant is the real number representing the probability that a randomly chosen Turing machine will halt, formed from a construction due to Argentine-American mathematician and computer scientist Gregory Chaitin. Chaitin's constant, though not being computable, has been proven to be transcendental and normal. Chaitin's constant is not universal, depending heavily on the numerical encoding used for Turing machines; however, its interesting properties are independent of the encoding.

When unspecified, constants indicate classes of similar objects, commonly functions, all equal up to a constant—technically speaking, this may be viewed as 'similarity up to a constant'. Such constants appear frequently when dealing with integrals and differential equations. Though unspecified, they have a specific value, which often is not important.

Solutions with different constants of integration of ${\displaystyle y'(x)=-2y+e^{-x}}$.

Indefinite integrals are called indefinite because their solutions are only unique up to a constant. For example, when working over the field of real numbers

${\displaystyle \int \cos x\ dx=\sin x+C}$

where C, the constant of integration, is an arbitrary fixed real number.[19] In other words, whatever the value of C, differentiating sin x + C with respect to x always yields cos x.

In a similar fashion, constants appear in the solutions to differential equations where not enough initial values or boundary conditions are given. For example, the ordinary differential equation y^' = y(x) has solution Ce^x where C is an arbitrary constant.

When dealing with partial differential equations, the constants may be functions, constant with respect to some variables (but not necessarily all of them). For example, the PDE

${\displaystyle {\frac {\partial f(x,y)}{\partial x}}=0}$

has solutions f(x,y) = C(y), where C(y) is an arbitrary function in the variable y.

It is common to express the numerical value of a constant by giving its decimal representation (or just the first few digits of it). For two reasons this representation may cause problems. First, even though rational numbers all have a finite or ever-repeating decimal expansion, irrational numbers don't have such an expression making them impossible to completely describe in this manner. Also, the decimal expansion of a number is not necessarily unique. For example, the two representations 0.999... and 1 are equivalent[20][21] in the sense that they represent the same number.

Calculating digits of the decimal expansion of constants has been a common enterprise for many centuries. For example, German mathematician Ludolph van Ceulen of the 16th century spent a major part of his life calculating the first 35 digits of pi.[22] Using computers and supercomputers, some of the mathematical constants, including π, e, and the square root of 2, have been computed to more than one hundred billion digits. Fast algorithms have been developed, some of which — as for Apéry's constant — are unexpectedly fast.

${\displaystyle G=\left.{\begin{matrix}3\underbrace {\uparrow \ldots \uparrow } 3\\\underbrace {\vdots } \\3\uparrow \uparrow \uparrow \uparrow 3\end{matrix}}\right\}{\text{64 layers}}}$

Graham's number defined using Knuth's up-arrow notation.

Some constants differ so much from the usual kind that a new notation has been invented to represent them reasonably. Graham's number illustrates this as Knuth's up-arrow notation is used.[23][24]

It may be of interest to represent them using continued fractions to perform various studies, including statistical analysis. Many mathematical constants have an analytic form, that is they can be constructed using well-known operations that lend themselves readily to calculation. Not all constants have known analytic forms, though; Grossman's constant[25] and Foias' constant[26] are examples.

Symbolizing constants with letters is a frequent means of making the notation more concise. A standard convention, instigated by Leonhard Euler in the 18th century, is to use lower case letters from the beginning of the Latin alphabet ${\displaystyle a,b,c,\dots }$ or the Greek alphabet ${\displaystyle \alpha ,\beta ,\,\gamma ,\dots }$ when dealing with constants in general.

However, for more important constants, the symbols may be more complex and have an extra letter, an asterisk, a number, a lemniscate or use different alphabets such as Hebrew, Cyrillic or Gothic.[24]

Erdős–Borwein constant ${\displaystyle E_{B}}$
Embree–Trefethen constant ${\displaystyle \beta ^{*}}$
Brun's constant for twin prime ${\displaystyle B_{2}}$
Champernowne constants ${\displaystyle C_{b}}$
cardinal number aleph naught ${\displaystyle \aleph _{0}}$

Examples of different kinds of notation for constants.

Sometimes, the symbol representing a constant is a whole word. For example, American mathematician Edward Kasner's 9-year-old nephew coined the names googol and googolplex.[24][27]

${\displaystyle \mathrm {googol} =10^{100}\,\ ,\ \mathrm {googolplex} =10^{\mathrm {googol} }=10^{10^{100}}}$

The names are either related to the meaning of the constant (universal parabolic constant, twin prime constant, ...) or to a specific person (Sierpiński's constant, Josephson constant, and so on).

The universal parabolic constant is the ratio, for any parabola, of the arc length of the parabolic segment (red) formed by the latus rectum (blue) to the focal parameter (green).