A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems.[1] Constants arise in many areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory, statistics, and calculus.
Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle . Other constants are notable more for historical reasons than for their mathematical properties. The more popular constants have been studied throughout the ages and computed to many decimal places.
All named 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.
See main article: Square root of 2.
The square root of 2, often known as root 2 or Pythagoras' constant, and written as, is the unique positive real 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 is an irrational number, possibly the first number to be known as such, and an algebraic number. Its numerical value truncated to 50 decimal places is:
.
Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two was frequently used before the common use of electronic calculators and computers. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10−5).
Its simple continued fraction is periodic and given by:
\sqrt2=1+
| 1 | |||||||||
|
See main article: Pi. The constant (pi) has a natural definition in Euclidean geometry as the ratio between the circumference and diameter of a circle. It may be found in many other places in mathematics: for example, the Gaussian integral, the complex roots of unity, 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 or its reciprocal factored out. For example, the ground state wave function of the hydrogen atom is
\psi(r)=
| 1 | |
| \sqrt{\pi{a0 |
3}}
| -r/a0 | |
| e |
,
where
a0
is an irrational number, transcendental number and an algebraic period.
The numeric value of is approximately:
.
Unusually good approximations are given by the fractions 22/7 and 355/113.
Memorizing as well as computing increasingly more digits of is a world record pursuit.
See main article: e (mathematical constant).
Euler's number, 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:
e=\limn\toinfty\left(1+
| 1 | |
| n |
\right)n
x\mapstoex
The Swiss mathematician Jacob Bernoulli discovered that arises in compound interest: If an account starts at $1, and yields interest at annual rate, then as the number of compounding periods per year tends to infinity (a situation known as continuous compounding), the amount of money at the end of the year will approach dollars.
The constant 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 probability of winning is played times, then for large (e.g., one million), the probability that nothing will be won will tend to as tends to infinity.
Another application of, 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.[2] Here, 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
pn=1-
| 1 | + | |
| 1! |
| 1 | - | |
| 2! |
| 1 | |
| 3! |
| ||||
| + … +(-1) |
which, as tends to infinity, approaches .
is an irrational number and a transcendental number.
The numeric value of is approximately:
.
See main article: Imaginary unit. The imaginary unit or unit imaginary number, denoted as, is a mathematical concept which extends the real number system
\R
\Complex.
There are in fact two complex square roots of −1, namely and, just as there are two complex square roots of every other real number (except zero, which has one double square root).
In contexts where the symbol is ambiguous or problematic, or the Greek iota is sometimes used. This is in particular the case in electrical engineering and control systems engineering, where the imaginary unit is often denoted by, because is commonly used to denote electric current.
These are constants which are encountered frequently in higher mathematics.
See main article: Golden ratio.
| F | ||||
|
An explicit formula for the th Fibonacci number involving the golden ratio .
The number, 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 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.[3] Kepler proved that it is the limit of the ratio of consecutive Fibonacci numbers.[4] The golden ratio has the slowest convergence of any irrational number.[5] 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).[6] It is approximately equal to:
.
or, more precisely
| 1+\sqrt{5 | |
See main article: Euler–Mascheroni constant.
Euler's constant or the Euler–Mascheroni constant is defined as the limiting difference between the harmonic series and the natural logarithm:
\begin{align} \gamma&=\limn\toinfty\left(-ln
| n | |
| n+\sum | |
| k=1 |
| 1{k} | |
| \right)\\[5px] \end{align} |
It appears frequently in mathematics, especially in number theoretical contexts such as Mertens' third theorem or the growth rate of the divisor function. It has relations to the gamma function and its derivatives as well as the zeta function and there exist many different integrals and series involving
\gamma
Despite the ubiquity of the Euler-Mascheroni constant, many of its properties remain unknown. That includes the major open questions of whether it is a rational or irrational number and whether it is algebraic or transcendental. In fact,
\gamma
\pi
e
The numeric value of
\gamma
.
See main article: Apéry's constant.
\zeta(s)
s=3
| \zeta(2)= | 16\pi |
| 2 |
\zeta(3)
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.
\zeta(3)
The numeric value of Apéry's constant is approximately:
.
See main article: Catalan's constant.
Catalan's constant is defined by the alternating sum of the reciprocals of the odd square numbers:
G=
| infty | |
| \sum | |
| n=0 |
| (-1)n | |
| (2n+1)2 |
=
| 1 | |
| 12 |
-
| 1 | |
| 32 |
+
| 1 | |
| 52 |
-
| 1 | |
| 72 |
+
| 1 | |
| 92 |
- …
\beta(s)
s=2
Questions about the arithmetic nature of this constant also remain unanswered,
G
Its is named after the French and Belgian mathematician Charles Eugène Catalan.
The numeric value of
G
.
See main article: Feigenbaum constants.
Iterations of continuous maps serve as the simplest examples of models for dynamical systems.[8] 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[9] and their bifurcation diagrams. Specifically, the constant α is the ratio between the width of a tine and the width of one of its two subtines, and the constant δ is the limiting ratio of each bifurcation interval to the next between every period-doubling bifurcation.
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,[10] 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 Feigenbaum constants in bifurcation theory are analogous to π in geometry and e in calculus. Neither of them is known to be irrational or even transcendental. However proofs of their universality exist.[11]
The respective approximate numeric values of δ and α are:
.
.
| infty | |
| c=\sum | |
| j=1 |
10-j!=0.\underbrace{\overbrace{110001}3!digits000000000000000001}4!digits000...
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:
C10=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...
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.
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[15] [16] 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.[17] 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.
G=\left.\begin{matrix}3\underbrace{\uparrow\ldots\uparrow}3\ \underbrace{\vdots}\ 3\uparrow\uparrow\uparrow\uparrow3\end{matrix}\right\}64layers
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.[18] [19]
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 and Foias' constant are examples.
a,b,c,...
\alpha,\beta,\gamma,...
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.[19]
EB
\beta*
B2
Cb
\aleph0
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.[19] [20]
googol=10100 , googolplex=10googol=10
| 10100 | |
Other 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).
See main article: List of mathematical constants.
| Symbol | Value | Name | Rational | Algebraic | Period | Field | Known digits | First described | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0.0000000000... | Zero | data-sort-value="0" style="background:#a6ffa7; | ✓ | data-sort-value="0" style="background:#a6ffa7; | ✓ | data-sort-value="0" style="background:#a6ffa7; | ✓ | Gen | all | data-sort-value="-500" align="right" | c. 500 BC | |
1 | 1.0000000000... | One | data-sort-value="0" style="background:#a6ffa7; | ✓ | data-sort-value="0" style="background:#a6ffa7; | ✓ | data-sort-value="0" style="background:#a6ffa7; | ✓ | Gen | all | data-sort-value="-5000" align="right" | Prehistory | |
i | 0 + 1i | Imaginary unit | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="0" style="background:#a6ffa7; | ✓ | data-sort-value="0" style="background:#a6ffa7; | ✓ | Gen, Ana | all | data-sort-value="1500" align="right" | 1500s | |
\pi | 3.1415926535... | Pi, Archimedes' constant | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="0" style="background:#a6ffa7; | ✓ | Gen, Ana | data-sort-value="200000" align="right" | 2.0 × 1014[21] | data-sort-value="-2600" align="right" | c. 2600 BC |
e | 2.7182818284... | e, Euler's number | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="1" style="background:#fcffa6; | ? | Gen, Ana | data-sort-value="35000" align="right" | 3.5 × 1013 | 1618 | |
\sqrt2 | 1.4142135623... | Square root of 2, Pythagoras' constant | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="0" style="background:#a6ffa7; | ✓ | data-sort-value="0" style="background:#a6ffa7; | ✓ | Gen | data-sort-value="20000" style="text-align:right;" | 2.0 × 1013 | data-sort-value="-800" align="right" | c. 800 BC |
\sqrt3 | 1.7320508075... | Square root of 3, Theodorus' constant | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="0" style="background:#a6ffa7; | ✓ | data-sort-value="0" style="background:#a6ffa7; | ✓ | Gen | data-sort-value="3100" style="text-align:right;" | 3.1 × 1012 | data-sort-value="-800" align="right" | c. 800 BC |
\varphi | 1.6180339887... | Golden ratio | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="0" style="background:#a6ffa7; | ✓ | data-sort-value="0" style="background:#a6ffa7; | ✓ | Gen | data-sort-value="20000" style="text-align:right;" | 2.0 × 1013 | data-sort-value="-200" align="right" | c. 200 BC |
\sqrt[3]{2} | 1.2599210498... | Cube root of two | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="0" style="background:#a6ffa7; | ✓ | data-sort-value="0" style="background:#a6ffa7; | ✓ | Gen | data-sort-value="1000" style="text-align:right;" | 1.0 × 1012 | data-sort-value="-380" align="right" | c. 380 BC |
ln(2) | 0.6931471805... | Natural logarithm of 2 | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="0" style="background:#a6ffa7; | ✓ | Gen, Ana | data-sort-value="3000" align="right" | 3.0 × 1012 | 1619 | |
\gamma | 0.5772156649... | Euler–Mascheroni constant | data-sort-value="1" style="background:#fcffa6; | ? | data-sort-value="1" style="background:#fcffa6; | ? | data-sort-value="1" style="background:#fcffa6; | ? | Gen, NuT | data-sort-value="1300" style="text-align:right;" | 1.3 × 1012 | 1735 | |
\zeta(3) | 1.2020569031... | Apéry's constant | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="1" style="background:#fcffa6; | ? | data-sort-value="0" style="background:#a6ffa7; | ✓ | Ana | data-sort-value="2000" align="right" | 2.0 × 1012 | 1780 | |
G | 0.9159655941... | Catalan's constant | data-sort-value="1" style="background:#fcffa6; | ? | data-sort-value="1" style="background:#fcffa6; | ? | data-sort-value="0" style="background:#a6ffa7; | ✓ | Com | data-sort-value="1200" style="text-align:right;" | 1.2 × 1012 | 1832 | |
\varpi | 2.6220575542... | Lemniscate constant | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="2" style="background:#ffa6a6; | ✗ | data-sort-value="0" style="background:#a6ffa7; | ✓ | Ana | data-sort-value="1200" align="right" | 1.2 × 1012 | data-sort-value="1700" align="right" | 1700s |
A | 1.2824271291... | Glaisher–Kinkelin constant | data-sort-value="1" style="background:#fcffa6; | ? | data-sort-value="1" style="background:#fcffa6; | ? | data-sort-value="1" style="background:#fcffa6; | ? | Ana | data-sort-value="500" align="right" | 5.0 × 105[22] | 1860 | |
K0 | 2.6854520010... | Khinchin's constant | data-sort-value="1" style="background:#fcffa6; | ? | data-sort-value="1" style="background:#fcffa6; | ? | data-sort-value="1" style="background:#fcffa6; | ? | NuT | data-sort-value="110" align="right" | 1.1 × 105[23] | 1934 | |
\delta | 4.6692016091... | Feigenbaum constants | data-sort-value="1" style="background:#fcffa6;" | ? | data-sort-value="1" style="background:#fcffa6;" | ? | data-sort-value="1" style="background:#fcffa6;" | ? | ChT | data-sort-value="10" style="text-align:right;" | 1,000+[24] | 1975 | |
\alpha | 2.5029078750... | data-sort-value="1" style="background:#fcffa6;" | ? | data-sort-value="1" style="background:#fcffa6;" | ? | data-sort-value="1" style="background:#fcffa6;" | ? | data-sort-value="10" style="text-align:right;" | 1,000+[25] | 1979 |
Gen – General, NuT – Number theory, ChT – Chaos theory, Com – Combinatorics, Ana – Mathematical analysis