# Dictionary Definition

constant adj

1 persistent in occurrence and unvarying in
nature; "maintained a constant temperature"; "a constant beat";
"principles of unvarying validity"; "a steady breeze" [syn:
changeless, invariant, steady, unvarying]

2 continually recurring or continuing without
interruption; "constant repetition of the exercise"; "constant
chatter of monkeys"

3 steadfast in purpose or devotion or affection;
"a man constant in adherence to his ideals"; "a constant lover";
"constant as the northern star" [ant: inconstant]

4 uninterrupted in time and indefinitely long
continuing; "the ceaseless thunder of surf"; "in constant pain";
"night and day we live with the incessant noise of the city"; "the
never-ending search for happiness"; "the perpetual struggle to
maintain standards in a democracy"; "man's unceasing warfare with
drought and isolation"; "unremitting demands of hunger" [syn:
ceaseless, incessant, never-ending,
perpetual, unceasing, unremitting]

### Noun

1 a quantity that does not vary [syn: constant
quantity]

2 a number representing a quantity assumed to
have a fixed value in a specified mathematical context; "the
velocity of light is a constant"

# User Contributed Dictionary

## English

### Etymology

From constantem.### Adjective

#### Translations

unchanged through time

- Finnish: pysyvä, muuttumaton, vakaa, vakio-
- French: constant , constante
- German: beständig, konstant
- Hungarian: állandó, változatlan
- Italian: costante
- Swedish: konstant

consistently recurring over time

steady

- Finnish: vakaa, muuttumaton
- French: constant , constante
- German: konstant, fest
- Hungarian: állhatatos, kitartó
- Italian: costante
- Swedish: konstant

### Noun

- That which is permanent or invariable.
- A quantity that remains at a fixed value throughout a given discussion.
- Any property of an experiment, determined numerically, that does not change under given circumstances.
- An identifier that is bound to an invariant value.

#### Translations

that which is permanent or invariable

- Finnish: vakio
- German: Konstante, Fixwert banking
- Hungarian: állandó, konstans
- Italian: costante
- Swedish: konstant

algebra: quantity that remains fixed

- Finnish: vakio
- German: Konstante
- Hungarian: állandó, konstans
- Italian: costante
- Swedish: konstant

science: property that does not change

- Finnish: vakio
- German: Konstante
- Hungarian: állandó, konstans
- Italian: costante
- Swedish: konstant

identifier that is bound to an invariant value

- Croatian: konstanta
- Czech: konstanta
- Finnish: vakio
- German: Konstante
- Hungarian: állandó, konstans
- Italian: costante
- Swedish: konstant

### Related terms

- constantly adverb
- constancy noun

### See also

## French

### Pronunciation

- lang=fr|/kɔ̃s.tɑ̃/
- SAMPA: /kO~s.tA~/

### Adjective

constant#### Related terms

# Extensive Definition

Constants are real numbers
or numerical values which are significantly interesting in some
way. The term "constant" is used both for mathematical
constants and for physical
constants, but with quite different meanings.

One always talks about definable,
and almost always also computable,
mathematical constants — Chaitin's
constant being a notable exception. However for some computable
mathematical constants only very rough numerical estimates are
known.

When dealing with physical dimensionful
constants, a set of units must be chosen. Sometimes, one unit is
defined in terms of other units. For example, the metre is defined as 1/(299\ 792\
458) of a light-second.
This definition implies that, in metric units,
the speed of light in vacuum is exactly 299\ 792\ 458 metres
per second. No increase in the precision
of the measurement of the speed of light could alter this numerical
value expressed in metres per second.

## Mathematical constants

Ubiquitous in many different fields of science,
such recurring constants include \pi, e
and the Feigenbaum
constants which are linked to the mathematical
models used to describe physical phenomena, Euclidean
geometry, analysis
and logistic
maps respectively. However, mathematical constants such as
Apéry's
constant and the Golden ratio
occur unexpectedly outside of mathematics.

### Archimedes' constant π

Pi, though having a
natural definition in
Euclidean
geometry (the circumference of a
circle of diameter 1), may be found in
many different places in mathematics. Key examples include the
Gaussian
integral in complex
analysis, nth roots of
unity in number
theory and Cauchy
distributions in probability. However, its
universality is not limited to mathematics. Indeed, various
formulas in physics, such as
Heisenberg's uncertainty principle, and constants such as the
cosmological
constant bear the constant pi. The presence of pi in physical
principles, laws
and formulas can have
very simple explanations. For example, Coulomb's
law, describing the inverse square proportionality of the
magnitude
of the electrostatic
force between two electric
charges and their distance, states that, in
SI units, F = \frac\frac.

### The exponential growth – or Napier's – constant e

The exponential
growth constant appears in many parts of applied mathematics.
For example, as the Swiss
mathematician Jacob
Bernoulli discovered, e\, arises in compound
interest. Indeed, an account that starts at $1, and yields
1+R\, dollars at simple interest, will yield e^R\, dollars with
continuous compounding. e\, also has applications to probability
theory, where it arises in a way not obviously related to
exponential growth. Suppose that a gambler plays a slot machine
with a one in n probability and plays it n times. Then, for large n
(such as a million) the probability that the gambler
will win nothing at all is (approximately) 1/e\,. Another
application of e\,, also 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. 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. But the butler does not know the name of
the guests, and so 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 p_n =
1-\frac+\frac-\frac+\cdots+(-1)^n\frac and as n\, tends to
infinity, p_n\, approaches 1/e\,.

### The Feigenbaum constants α and δ

Iterations of continuous maps serve as the
simplest examples of models for dynamical
systems. 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 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
English
biologist
Robert May, 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 or reproduction and starvation.

### Apéry's constant ζ(3)

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. Also, Pascal
Wallisch noted that \sqrt\approxeq\frac, where m_n,m_e,\varphi
are the neutron mass,
the electron mass and
the Golden ratio
respectively.

### The golden ratio φ

F\left(n\right)=\frac An explicit formula for the
nth Fibonacci
number involving the golden
ratio.

The number \varphi turns up frequently in
geometry, particularly
in figures with pentagonal symmetry. Indeed, the length of
a regular pentagon's
diagonal is \varphi
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.

Adolf Zeising, whose main interests were
mathematics and philosophy, found the golden ratio expressed in the
arrangement of branches along the stems of
plants and of veins in leaves.
He extended his research to the skeletons of animals and the
branchings of their veins and nerves, to the proportions of
chemical compounds and the geometry of crystals, even to the use of
proportion in artistic endeavours. In these phenomena he saw the
golden ratio operating as a universal law. Zeising wrote in
1854:

[The Golden Ratio is a universal law] in which is
contained the ground-principle of all formative striving for beauty
and completeness in the realms of both nature and art, and which
permeates, as a paramount spiritual ideal, all structures, forms and
proportions, whether cosmic or individual, organic
or inorganic, acoustic or
optical; which finds its
fullest realization, however, in the human form.

### The Euler-Mascheroni constant γ

The
Euler–Mascheroni constant is a recurring constant in number
theory. The French
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 \gamma as n
tends to infinity.
Surprisingly, this average doesn't tend to one half. 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 right).

### Conway's constant λ

\begin 1 \\ 11 \\ 21 \\ 1211 \\ 111221 \\ 312211
\\ \vdots \end 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 two trivial ones). It is given by the unique
positive real root of a polynomial of degree 71 with
integer coefficients.

### Khinchin's constant K

If a real number r\, is written using simple
continued fraction

- r=a_0+\dfrac,

then, as Russian
mathematician Aleksandr
Khinchin proved in 1934, the limit
as n\, tends to infinity of the geometric
mean (a_1a_2\cdots a_n)^ exists, and, except for a set of
measure
0, this limit is a constant, Khinchin's
constant.

## Physical constants

In physics, universal constants appear in the
basic theoretical equations upon which the entire science rests or
are the properties of the fundamental particles of physics of which
all matter is constituted (the electron
charge e, the electron
mass m_e and the fine-structure
constant \alpha).

### The speed of light c and Planck's constant h

The speed of
light and the Planck
constant are examples of quantities that occur naturally
in the mathematical formulation of certain fundamental physical
theories, the former in James
Clerk Maxwell's theory of electric
and magnetic
fields and Albert
Einstein's theories of relativity, and the latter in quantum
theory. For example, in special
relativity, mass and energy are equivalent:
E = mc2 where c^2\, is the constant of
proportionality. In quantum
mechanics, the energy and frequency of a photon are
related by E=h\nu\,.

The speed of light is also used to express other
fundamental constants such as the electric
constant \epsilon_0=(4\pi 10^ c^2)^\,, Coulomb's
constant k=10^ c^2\, and the
characteristic impedance of vacuum Z_0=4\pi10^c\,.

### The electron charge e and the electron mass m_e

The electron
charge and the electron
mass are examples of constants that characterize the basic, or
elementary, particles
that constitute matter, such as the electron, alpha
particle, proton,
neutron, muon, and pion. Many constants can be
expressed using the fundamental constants h,\,c,\,e. For example.
it is a property of a supercurrent
(superconducting electrical current) that the magnetic
flux passing through any area bounded by such a current is
quantized. The magnetic
flux quantum \Phi_0=hc/(2e)\, is a physical constant, as it is
independent of the underlying material as long as it is a superconductor. Also, the
fundamental fine-structure
constant \alpha=\mu_0ce^2/(2h)\, where the
permeability of free space \mu_0 is just a numerical constant
equal to .

## Mathematical curiosities, specific physical facts and unspecified constants

### Simple representatives of sets of numbers

c=\sum_^\infty 10^=0.\underbrace_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|Champernowne constant C_ =
\color0.\color1\color2\color3\color4\color5\color6\color7\color8\color9\color10\color11\color12\color13\color14\color15\color16\dots]]
are not important mathematical invariants but retain interest being
simple representatives of special sets of numbers, the irrational
numbers, the transcendental
numbers and the normal
numbers (in base 10) respectively. The discovery of the
irrational
numbers is usually attributed to the Pythagorean
Hippasus
of Metapontum who proved, most likely geometrically, the
irrationality of \sqrt. As for Liouville's constant, named after
French
mathematician Joseph
Liouville, it was the first transcendental number ever
constructed.

### Chaitin's constant Ω

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. Amusingly, Chaitin's
constant, though not being computable,
has been proven transcendental
and normal.

\text 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.

Commonly, constants in the physical sciences are
represented using the scientific
notation, with, when appropriate, the inaccuracy - or measurement
error - attached. When writing the Planck
constant h=6.626\ 068\ 96(33) \times 10^\ \mbox\cdot\mbox it is
meant that h=(6.626\ 068\ 96 \plusmn 0.000\ 000\ 003\ 3)\times 10^\
\mbox\cdot\mbox\,. Only the significant
figures are shown and a greater precision
would be superfluous, extra figures coming from experimental
inaccuracies. When writing Isaac
Newton's gravitational
constant G = \left(6.67428 \plusmn 0.00067 \right) \times 10^ \
\mbox^3 \ \mbox^ \ \mbox^ \, only 6 significant figures are
given.

For mathematical constants, 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 constructed using well-known operations
that lend themselves readily to calculation. However, Grossman's
constant has no known analytic form.

### Symbolizing and naming of constants

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 a,b,c,\dots\, or the Greek
alphabet \alpha,\beta,\,\gamma,\dots\, when dealing with
constants in general.

Erdős–Borwein constant E_B\,Embree-Trefethen
constant \beta*\,Brun's
constant for twin prime
B_2\,Rydberg
constant R_\inftycardinal
number aleph naught
\aleph_0 Different kinds of notation.

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,...).

### Lumping constants

A common practice in physics is to lump constants
to simplify the equations and algebraic manipulations. For example,
Coulomb's
constant \kappa =(4\pi\epsilon_0)^\, is just \epsilon_0\,,
\pi\, and 4\, lumped together. Also, combining old constants does
not necessarily make the new one less fundamental. For example, the
-notably- dimensionless fine-structure
constant \alpha=\mu_0ce^2/(2h)\, is a fundamental constant of
quantum electrodynamics and in the quantum theory of the
interaction among electrons, muons and photons.

### A notation simplifier : the Avogadro constant N_a

The Avogadro
constant is the number of entities in one mole,
commonly used in chemistry, where the entities
are often atoms or molecules. Its unit is inverse
mole. However, the mole being a counting unit, we can consider the
Avogadro
constant dimensionless, and, contrary to the speed of light,
the Avogadro constant doesn't convert units, but acts as a scaling
factor for dealing practically with large
numbers.

## Mystery and aesthetics behind constants

e^+1=0\, Euler's
identity relating five of the most important mathematical
constants.

For some authors, constants, either mathematical
or physical may be mysterious, beautiful or fascinating. For
example, English mathematician
Glaisher (1915) writes

During the 1920s until his death, British
astrophysicist
Eddington
increasingly concentrated on what he called "fundamental
theory" which was intended to be a unification of quantum
theory, relativity
and gravitation. At
first he progressed along "traditional" lines, but turned
increasingly to an almost numerological analysis of the
dimensionless ratios of fundamental constants. In a similar
fashion, British
theoretical
physicist Paul Dirac
studied ratios of fundamental physical constant to build his
large numbers hypothesis.

## See also

Mathematical
constantPhysical
constantAstronomical
constant ScalarCoefficientNumber
Constant
functionConstant
of integrationCosmological
constant

## References

## Further reading

## External links

constant in Min Nan: Tiāⁿ-sò͘

constant in Bulgarian: Константа

constant in Danish: Konstant

constant in German: Konstante

constant in Estonian: Konstant

constant in Spanish: Constante

constant in Esperanto: Konstanto

constant in French: Constante

constant in Scottish Gaelic: Cunbhal

constant in Korean: 상수

constant in Indonesian: Konstanta

constant in Italian: Costante

constant in Lithuanian: Konstanta

constant in Dutch: Constant (eigenschap)

constant in Japanese: 定数

constant in Norwegian: Konstant

constant in Portuguese: Constante

constant in Russian: Константа

constant in Simple English: Constant

constant in Serbian: Константа

constant in Finnish: Vakio

constant in Swedish: Konstant

constant in Thai: ค่าคงตัว

constant in Ukrainian: Константа

constant in Chinese: 常数

# Synonyms, Antonyms and Related Words

abiding, accordant, active, age-long, aged, ageless, alike, ancient, antique, ardent, articulated, assiduous, atom, atomic mass, atomic number,
atomic weight, automatic, balanced, beaten, catenated, ceaseless, changeless, chattering, chronic, clinging, close, coeternal, colorfast, committed, compliant, concatenated, confirmed, conforming, connected, conscientious, consistent, consonant, continual, continued, continuing, continuous, correspondent, cyclical, dateless, dedicated, deep-dyed, delicate, dependable, determined, devoted, devout, diligent, direct, diuturnal, dogged, double-dyed, durable, duteous, dutiful, dyed-in-the-wool,
endless, enduring, equable, equal, eternal, eterne, even, ever-being, ever-durable,
ever-during, evergreen, everlasting, everliving, exact, express, fadeless, faithful, fast, featureless, fine, firm, fixed, flat, flinty, frequent, frozen, gapless, habitual, hackneyed, hardy, homogeneous, immediate, immemorial, immobile, immovable, immutable, inalterable, incessant, incommutable, inconvertible, indefatigable, indefeasible, indelible, indestructible, indomitable, industrious, inerrable, inerrant, inert, infallible, infinite, inflexible, ingrain, ingrained, insistent, insusceptible of
change, intact, interminable, intransient, intransmutable, invariable, inveterate, invincible, inviolate, irretrievable, irreversible, irrevocable, joined, jointless, lasting, level, liege, linked, long-lasting, long-lived,
long-standing, long-term, longeval, longevous, loyal, machine gun, macrobiotic,
marble-constant, mathematical, measured, mechanical, methodic, methodical, meticulous, micrometrically
precise, microscopic, mindful, monolithic, monotonous, never-ceasing,
never-ending, never-tiring, nice, noble, nonreturnable, nonreversible, nonstop, nonterminating, nonterminous, observant, obstinate, of a piece, of long
duration, of long standing, olamic, ordered, orderly, oscillating, patient, patient as Job,
perdurable, perduring, perennial, periodic, permanent, perpetual, perseverant, persevering, persistent, persisting, pertinacious, pinpoint, plodding, plugging, practicing, precise, preoccupied, pulsating, punctilious, punctual, quantum, quiescent, rapid, rapt, recurrent, recurring, refined, regardful, regular, regular as clockwork,
relentless, religious, religiously exact,
remaining, repeated, repetitive, resolute, reverseless, rigid, rigorous, robotlike, round-the-clock,
routine, running, scientific, scientifically
exact, scrupulous,
seamless, sedulous, sempervirent, sempiternal, serried, set, settled, severe, single-minded, sleepless, slogging, smooth, solid, sot, square, stabile, stable, staccato, static, stationary, staunch, staying, steadfast, steady, steely, stereotyped, straight, strict, stubborn, stuttering, subtle, sustained, systematic, tenacious, tested, timeless, tireless, torpid, tough, tried, tried and true, trite, true, true-blue, trusty, twenty-four-hour,
unabating, unalterable, unalterative, unaltered, unbending, unbroken, unceasing, unchangeable, unchanged, unchanging, unchecked, unconquerable, undaunted, undeflectable, undestroyed, undeviating, undifferentiated,
undiscouraged,
undiversified,
undrooping, unending, unerring, unfading, unfailing, unfaltering, unflagging, unflappable, unflinching, uniform, unintermitted, unintermittent, unintermitting, uninterrupted, unmodifiable, unmovable, unnodding, unrelaxing, unrelenting, unrelieved, unremitting, unrestorable, unreturnable, unruffled, unshakable, unshakeable, unshaken, unshifting, unsleeping, unstopped, unsusceptible, unswerving, untiring, unvariable, unvaried, unvarying, unwavering, unwearied, unwearying, unwinking, unyielding, utterly
attentive, valence,
vibrating, vital, weariless, well-trodden,
well-worn, without end