Exponentiation is a
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...
, written as , involving two numbers, the ''
base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ".
When is a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
, exponentiation corresponds to repeated
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
of the base: that is, is the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Pr ...
of multiplying bases:
The exponent is usually shown as a
superscript
A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...
to the right of the base. In that case, is called "''b'' raised to the ''n''th power", "''b'' (raised) to the power of ''n''", "the ''n''th power of ''b''", "''b'' to the ''n''th power", or most briefly as "''b'' to the ''n''th".
Starting from the basic fact stated above that, for any positive integer
,
is
occurrences of
all multiplied by each other, several other properties of exponentiation directly follow. In particular:
In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that
must be equal to 1, as follows. For any
,
. Dividing both sides by
gives
.
The fact that
can similarly be derived from the same rule. For example,
. Taking the cube root of both sides gives
.
The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what
should mean. In order to respect the "exponents add" rule, it must be the case that
. Dividing both sides by
gives
, which can be more simply written as
, using the result from above that
. By a similar argument,
.
The properties of fractional exponents also follow from the same rule. For example, suppose we consider
and ask if there is some suitable exponent, which we may call
, such that
. From the definition of the square root, we have that
. Therefore, the exponent
must be such that
. Using the fact that multiplying makes exponents add gives
. The
on the right-hand side can also be written as
, giving
. Equating the exponents on both sides, we have
. Therefore,
, so
.
The definition of exponentiation can be extended to allow any real or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
.
Exponentiation is used extensively in many fields, including
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
,
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary in ...
,
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties, ...
,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
, and
computer science, with applications such as
compound interest,
population growth
Population growth is the increase in the number of people in a population or dispersed group. Actual global human population growth amounts to around 83 million annually, or 1.1% per year. The global population has grown from 1 billion in 1800 to ...
,
chemical reaction kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...
,
wave behavior, and
public-key cryptography
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic a ...
.
History of the notation
The term ''power'' ( la, potentia, potestas, dignitas) is a mistranslation
of the
ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
δύναμις (''dúnamis'', here: "amplification"
) used by the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
mathematician
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
for the square of a line,
following
Hippocrates of Chios
Hippocrates of Chios ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 BC) was an ancient Greek mathematician, geometer, and astronomer.
He was born on the isle of Chios, where he was originally a merchant. After some misadve ...
. In ''
The Sand Reckoner
''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes, an Ancient Greek mathematician of the 3rd century BC, in which he set out to determine an upper bound for the number of grains of sand that fit into the unive ...
'',
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientist ...
discovered and proved the law of exponents, , necessary to manipulate powers of . In the 9th century, the Persian mathematician
Muhammad ibn Mūsā al-Khwārizmī used the terms مَال (''māl'', "possessions", "property") for a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-lengt ...
—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"
—and كَعْبَة (''
kaʿbah'', "cube") for a
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
, which later
Islamic
Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God (or ''Allah'') as it was revealed to Muhammad, the ma ...
mathematicians represented in
mathematical notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathema ...
as the letters ''
mīm'' (m) and ''
kāf
Kaph (also spelled kaf) is the eleventh letter of the Semitic abjads, including Phoenician kāp , Hebrew kāf , Aramaic kāp , Syriac kāp̄ , and Arabic kāf (in abjadi order).
The Phoenician letter gave rise to the Greek kappa (Κ), Lat ...
'' (k), respectively, by the 15th century, as seen in the work of
Abū al-Hasan ibn Alī al-Qalasādī
Abū'l-Ḥasan ibn ʿAlī ibn Muḥammad ibn ʿAlī al-Qurashī al-Qalaṣādī ( ar, أبو الحسن علي بن محمد بن علي القرشي البسطي; 1412–1486) was a Muslim Arab mathematician from Al-Andalus specializing in Is ...
.
In the late 16th century,
Jost Bürgi used Roman numerals for exponents.
Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by
Henricus Grammateus
Henricus Grammateus (also known as Henricus Scriptor, Heinrich Schreyber or Heinrich Schreiber; 1495 – 1525 or 1526) was a German mathematician. He was born in Erfurt. In 1507 he started to study at the University of Vienna, where he subsequen ...
and
Michael Stifel
Michael Stifel or Styfel (1487 – April 19, 1567) was a German monk, Protestant reformer and mathematician. He was an Augustinian who became an early supporter of Martin Luther. He was later appointed professor of mathematics at Jena Univer ...
in the 16th century. The word ''exponent'' was coined in 1544 by Michael Stifel.
Samuel Jeake introduced the term ''indices'' in 1696.
In the 16th century,
Robert Recorde
Robert Recorde () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557.
Biography
Born around 1512, Robert Recorde was the second and last ...
used the terms square, cube, zenzizenzic (
fourth power
In arithmetic and algebra, the fourth power of a number ''n'' is the result of multiplying four instances of ''n'' together. So:
:''n''4 = ''n'' × ''n'' × ''n'' × ''n''
Fourth powers are also formed by multiplying a number by its cube. Furthe ...
), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and
zenzizenzizenzic (eighth).
''Biquadrate'' has been used to refer to the fourth power as well.
Early in the 17th century, the first form of our modern exponential notation was introduced by
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathe ...
in his text titled ''
La Géométrie''; there, the notation is introduced in Book I.
Some mathematicians (such as
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the ...
) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s, for example, as .
Another historical synonym, involution, is now rare and should not be confused with
its more common meaning.
In 1748,
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
introduced variable exponents, and, implicitly, non-integer exponents by writing:
Terminology
The expression is called "the
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-lengt ...
of ''b''" or "''b'' squared", because the area of a square with side-length is .
Similarly, the expression is called "the
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
of ''b''" or "''b'' cubed", because the volume of a cube with side-length is .
When it is a
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
, the exponent indicates how many copies of the base are multiplied together. For example, . The base appears times in the multiplication, because the exponent is . Here, is the ''5th power of 3'', or ''3 raised to the 5th power''.
The word "raised" is usually omitted, and sometimes "power" as well, so can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiation can be expressed as "''b'' to the power of ''n''", "''b'' to the ''n''th power", "''b'' to the ''n''th", or most briefly as "''b'' to the ''n''".
A formula with nested exponentiation, such as (which means and not ), is called a tower of powers, or simply a tower.
Integer exponents
The exponentiation operation with integer exponents may be defined directly from elementary
arithmetic operations.
Positive exponents
The definition of the exponentiation as an iterated multiplication can be
formalized by using
induction, and this definition can be used as soon one has an
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
multiplication:
The base case is
:
and the
recurrence is
:
The associativity of multiplication implies that for any positive integers and ,
:
and
:
Zero exponent
By definition, any nonzero number raised to the power is :
:
This definition is the only possible that allows extending the formula
:
to zero exponents. It may be used in every
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
with a multiplication that has an
identity.
Intuitionally,
may be interpreted as the
empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in questio ...
of copies of . So, the equality
is a special case of the general convention for the empty product.
The case of is more complicated. In contexts where only integer powers are considered, the value is generally assigned to
but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context.
Negative exponents
Exponentiation with negative exponents is defined by the following identity, which holds for any integer and nonzero :
:
.
Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (
).
This definition of exponentiation with negative exponents is the only one that allows extending the identity
to negative exponents (consider the case
).
The same definition applies to
invertible elements in a multiplicative
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
, that is, an
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
, with an associative multiplication and a
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
denoted (for example, the
square matrices of a given dimension). In particular, in such a structure, the inverse of an
invertible element is standardly denoted
Identities and properties
The following
identities, often called , hold for all integer exponents, provided that the base is non-zero:
:
Unlike addition and multiplication, exponentiation is not
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
. For example, . Also unlike addition and multiplication, exponentiation is not
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
. For example, , whereas . Without parentheses, the conventional
order of operations
In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.
For examp ...
for
serial exponentiation in superscript notation is top-down (or ''right''-associative), not bottom-up
(or ''left''-associative). That is,
:
which, in general, is different from
:
Powers of a sum
The powers of a sum can normally be computed from the powers of the summands by the
binomial formula
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
:
However, this formula is true only if the summands commute (i.e. that ), which is implied if they belong to a
structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
that is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
. Otherwise, if and are, say,
square matrices of the same size, this formula cannot be used. It follows that in
computer algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
, many
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose
computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. Th ...
s use a different notation (sometimes instead of ) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.
Combinatorial interpretation
For nonnegative integers and , the value of is the number of
functions from a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of elements to a set of elements (see
cardinal exponentiation). Such functions can be represented as -
tuples from an -element set (or as -letter words from an -letter alphabet). Some examples for particular values of and are given in the following table:
:
Particular bases
Powers of ten
In the base ten (
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic num ...
) number system, integer powers of are written as the digit followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, and .
Exponentiation with base is used in
scientific notation
Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, ...
to denote large or small numbers. For instance, (the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit for ...
in vacuum, in
metres per second
The metre per second is the unit of both speed (a scalar quantity) and velocity (a vector quantity, which has direction and magnitude) in the International System of Units (SI), equal to the speed of a body covering a distance of one metre in ...
) can be written as and then
approximated as .
SI prefix
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
es based on powers of are also used to describe small or large quantities. For example, the prefix
kilo
KILO (94.3 FM, 94.3 KILO) is a radio station broadcasting in Colorado Springs and Pueblo, Colorado. It also streams online.
History
KLST and KPIK-FM
The 94.3 signal signed on the air on August 22, 1962, as KLST, owned by Little London Br ...
means , so a kilometre is .
Powers of two
The first negative powers of are commonly used, and have special names, e.g.: ''
half
One half ( : halves) is the irreducible fraction resulting from dividing one by two or the fraction resulting from dividing any number by its double. Multiplication by one half is equivalent to division by two, or "halving"; conversely, ...
'' and ''
quarter''.
Powers of appear in
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, since a set with members has a
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
, the set of all of its
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s, which has members.
Integer powers of are important in
computer science. The positive integer powers give the number of possible values for an -
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented ...
integer
binary number
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one).
The base-2 numeral system is a positional notatio ...
; for example, a
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable un ...
may take different values. The
binary number system expresses any number as a sum of powers of , and denotes it as a sequence of and , separated by a
binary point, where indicates a power of that appears in the sum; the exponent is determined by the place of this : the nonnegative exponents are the rank of the on the left of the point (starting from ), and the negative exponents are determined by the rank on the right of the point.
Powers of one
The powers of one are all one: .
The first power of a number is the number itself:
Powers of zero
If the exponent is positive (), the th power of zero is zero: .
If the exponent is negative (), the th power of zero is undefined, because it must equal
with , and this would be
according to above.
The expression
is either defined as 1, or it is left undefined.
Powers of negative one
If is an even integer, then .
If is an odd integer, then .
Because of this, powers of are useful for expressing alternating
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s. For a similar discussion of powers of the complex number , see .
Large exponents
The
limit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limit ...
of powers of a number greater than one diverges; in other words, the sequence grows without bound:
: as when
This can be read as "''b'' to the power of ''n'' tends to
+∞ as ''n'' tends to infinity when ''b'' is greater than one".
Powers of a number with
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
less than one tend to zero:
: as when
Any power of one is always one:
: for all if
Powers of alternate between and as alternates between even and odd, and thus do not tend to any limit as grows.
If , alternates between larger and larger positive and negative numbers as alternates between even and odd, and thus does not tend to any limit as grows.
If the exponentiated number varies while tending to as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is
: as
See ' below.
Other limits, in particular those of expressions that take on an
indeterminate form
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...
, are described in below.
Power functions

Real functions of the form
, where
, are sometimes called power functions. When
is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
and
, two primary families exist: for
even, and for
odd. In general for
, when
is even
will tend towards positive
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions a ...
with increasing
, and also towards positive infinity with decreasing
. All graphs from the family of even power functions have the general shape of
, flattening more in the middle as
increases.
Functions with this kind of
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
are called
even functions.
When
is odd,
's
asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
behavior reverses from positive
to negative
. For
,
will also tend towards positive
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions a ...
with increasing
, but towards negative infinity with decreasing
. All graphs from the family of odd power functions have the general shape of
, flattening more in the middle as
increases and losing all flatness there in the straight line for
. Functions with this kind of symmetry are called
odd functions.
For
, the opposite asymptotic behavior is true in each case.
Table of powers of decimal digits
Rational exponents
If is a nonnegative
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, and is a positive integer,
or
denotes the unique positive real
th root of , that is, the unique positive real number such that
If is a positive real number, and
is a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
, with and integers, then
is defined as
:
The equality on the right may be derived by setting
and writing
If is a positive rational number,
by definition.
All these definitions are required for extending the identity
to rational exponents.
On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real th root, which is negative, if is
odd, and no real root if is even. In the latter case, whichever complex th root one chooses for
the identity
cannot be satisfied. For example,
:
See and for details on the way these problems may be handled.
Real exponents
For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (, below), or in terms of the
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
of the base and the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
(, below). The result is always a positive real number, and the
identities and properties shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
exponents.
On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values (see ). One may choose one of these values, called the
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
, but there is no choice of the principal value for which the identity
:
is true; see . Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a
multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
.
Limits of rational exponents

Since any
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
can be expressed as the
limit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limit ...
of rational numbers, exponentiation of a positive real number with an arbitrary real exponent can be defined by
continuity with the rule
:
where the limit is taken over rational values of only. This limit exists for every positive and every real .
For example, if , the
non-terminating decimal representation and the
monotonicity
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain
:
So, the upper bounds and the lower bounds of the intervals form two
sequences
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
that have the same limit, denoted
This defines
for every positive and real as a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
of and . See also
Well-defined expression.
The exponential function
The ''exponential function'' is often defined as
where
is
Euler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an express ...
. For avoiding
circular reasoning
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet)
A flyer (or flier) is a form of paper
Paper is a thin sheet material pro ...
, this definition cannot be used here. So, a definition of the exponential function, denoted
and of Euler's number are given, which rely only on exponentiation with positive integer exponents. Then a proof is sketched that, if one uses the definition of exponentiation given in preceding sections, one has
:
There are
many equivalent ways to define the exponential function, one of them being
:
One has
and the ''exponential identity''
holds as well, since
:
and the second-order term
does not affect the limit, yielding
.
Euler's number can be defined as
. It follows from the preceding equations that
when is an integer (this results from the repeated-multiplication definition of the exponentiation). If is real,
results from the definitions given in preceding sections, by using the exponential identity if is rational, and the continuity of the exponential function otherwise.
The limit that defines the exponential function converges for every
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
value of , and therefore it can be used to extend the definition of
, and thus
from the real numbers to any complex argument . This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.
Powers via logarithms
The definition of as the exponential function allows defining for every positive real numbers , in terms of exponential and
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
function. Specifically, the fact that the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, ...
is the
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when ...
of the exponential function means that one has
:
for every . For preserving the identity
one must have
:
So,
can be used as an alternative definition of for any positive real . This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.
Complex exponents with a positive real base
If is a positive real number, exponentiation with base and
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
exponent is defined by means of the exponential function with complex argument (see the end of , above) as
:
where
denotes the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, ...
of .
This satisfies the identity
:
In general,
is not defined, since is not a real number. If a meaning is given to the exponentiation of a complex number (see , below), one has, in general,
:
unless is real or is an integer.
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...
,
:
allows expressing the
polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of
in terms of the
real and imaginary parts of , namely
:
where the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
of the
trigonometric
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
factor is one. This results from
:
Non-integer powers of complex numbers
In the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only. For other bases, difficulties appear already with the apparently simple case of th roots, that is, of exponents
where is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to th roots, this case deserves to be considered first, since it does not need to use
complex logarithm
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to be ...
s, and is therefore easier to understand.
th roots of a complex number
Every nonzero complex number may be written in
polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
as
:
where
is the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
of , and
is its
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
. The argument is defined
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R' ...
an integer multiple of ; this means that, if
is the argument of a complex number, then
is also an argument of the same complex number.
The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an th root of a complex number can be obtained by taking the th root of the absolute value and dividing its argument by :
:
If
is added to
, the complex number is not changed, but this adds
to the argument of the th root, and provides a new th root. This can be done times, and provides the th roots of the complex number.
It is usual to choose one of the th root as the
principal root. The common choice is to choose the th root for which
that is, the th root that has the largest real part, and, if they are two, the one with positive imaginary part. This makes the principal th root a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
in the whole complex plane, except for negative real values of the
radicand. This function equals the usual th root for positive real radicands. For negative real radicands, and odd exponents, the principal th root is not real, although the usual th root is real.
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
shows that the principal th root is the unique
complex differentiable function that extends the usual th root to the complex plane without the nonpositive real numbers.
If the complex number is moved around zero by increasing its argument, after an increment of
the complex number comes back to its initial position, and its th roots are
permuted circularly (they are multiplied by
). This shows that it is not possible to define a th root function that is continuous in the whole complex plane.
Roots of unity

The th roots of unity are the complex numbers such that , where is a positive integer. They arise in various areas of mathematics, such as in
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a compl ...
or algebraic solutions of algebraic equations (
Lagrange resolvent).
The th roots of unity are the first powers of
, that is
The th roots of unity that have this generating property are called ''primitive th roots of unity''; they have the form
with
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
with . The unique primitive square root of unity is
the primitive fourth roots of unity are
and
The th roots of unity allow expressing all th roots of a complex number as the products of a given th roots of with a th root of unity.
Geometrically, the th roots of unity lie on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
at the vertices of a
regular -gon with one vertex on the real number 1.
As the number
is the primitive th root of unity with the smallest positive
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
, it is called the ''principal primitive th root of unity'', sometimes shortened as ''principal th root of unity'', although this terminology can be confused with the
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
of
which is 1.
Complex exponentiation
Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for
. So, either a
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
is defined, which is not continuous for the values of that are real and nonpositive, or
is defined as a
multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
.
In all cases, the
complex logarithm
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to be ...
is used to define complex exponentiation as
:
where
is the variant of the complex logarithm that is used, which is, a function or a
multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
such that
:
for every in its
domain of definition.
Principal value
The
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
of the
complex logarithm
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to be ...
is the unique function, commonly denoted
such that, for every nonzero complex number ,
:
and the
imaginary part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of satisfies
:
The principal value of the complex logarithm is not defined for
it is
discontinuous at negative real values of , and it is
holomorphic (that is, complex differentiable) elsewhere. If is real and positive, the principal value of the complex logarithm is the natural logarithm:
The principal value of
is defined as
where
is the principal value of the logarithm.
The function
is holomorphic except in the neighbourhood of the points where is real and nonpositive.
If is real and positive, the principal value of
equals its usual value defined above. If
where is an integer, this principal value is the same as the one defined above.
Multivalued function
In some contexts, there is a problem with the discontinuity of the principal values of
and
at the negative real values of . In this case, it is useful to consider these functions as
multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
s.
If
denotes one of the values of the multivalued logarithm (typically its principal value), the other values are
where is any integer. Similarly, if
is one value of the exponentiation, then the other values are given by
:
where is any integer.
Different values of give different values of
unless is a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
, that is, there is an integer such that is an integer. This results from the
periodicity
Periodicity or periodic may refer to:
Mathematics
* Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups
* Periodic function, a function whose output contains values tha ...
of the exponential function, more specifically, that
if and only if
is an integer multiple of
If
is a rational number with and
coprime integers
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
with
then
has exactly values. In the case
these values are the same as those described in
§ th roots of a complex number. If is an integer, there is only one value that agrees with that of .
The multivalued exponentiation is holomorphic for
in the sense that its
graph
Graph may refer to:
Mathematics
* Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
* Graph (topology), a topological space resembling a graph in the sense of disc ...
consists of several sheets that define each a holomorphic function in the neighborhood of every point. If varies continuously along a circle around , then, after a turn, the value of
has changed of sheet.
Computation
The ''canonical form''
of
can be computed from the canonical form of and . Although this can be described by a single formula, it is clearer to split the computation in several steps.
*''
Polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of ''. If
is the canonical form of ( and being real), then its polar form is
where
and
(see
atan2
In computing and mathematics, the function atan2 is the 2- argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive for the definition of this function).
*''
Logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
of ''. The
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
of this logarithm is
where
denotes the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, ...
. The other values of the logarithm are obtained by adding
for any integer .
*''Canonical form of
'' If
with and real, the values of
are
the principal value corresponding to
*''Final result.'' Using the identities
and
one gets
with
for the principal value.
=Examples
=
*
The polar form of is
and the values of
are thus
It follows that
So, all values of
are real, the principal one being
*
Similarly, the polar form of is
So, the above described method gives the values
In this case, all the values have the same argument
and different absolute values.
In both examples, all values of
have the same argument. More generally, this is true if and only if the
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of is an integer.
Failure of power and logarithm identities
Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined ''as single-valued functions''. For example:
Irrationality and transcendence
If is a positive real
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the ...
, and is a rational number, then is an algebraic number. This results from the theory of
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ex ...
s. This remains true if is any algebraic number, in which case, all values of (as a
multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
) are algebraic. If is
irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
(that is, ''not rational''), and both and are algebraic, Gelfond–Schneider theorem asserts that all values of are
transcendental (that is, not algebraic), except if equals or .
In other words, if is irrational and
then at least one of , and is transcendental.
Integer powers in algebra
The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any
associative operation denoted as a multiplication.
[More generally, power associativity is sufficient for the definition.] The definition of
requires further the existence of a
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
.
An
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
. In such a monoid, exponentiation of an element is defined inductively by
*
*
for every nonnegative integer .
If is a negative integer,
is defined only if has a
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
. In this case, the inverse of is denoted
and
is defined as
Exponentiation with integer exponents obeys the following laws, for and in the algebraic structure, and and integers:
:
These definitions are widely used in many areas of mathematics, notably for
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
,
rings,
fields,
square matrices (which form a ring). They apply also to
functions from a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
to itself, which form a monoid under
function composition. This includes, as specific instances,
geometric transformation
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...
s, and
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
s of any
mathematical structure
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...
.
When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if is a
real function
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
whose valued can be multiplied,
denotes the exponentiation with respect of multiplication, and
may denote exponentiation with respect of
function composition. That is,
:
and
:
Commonly,
is denoted
while
is denoted
In a group
A
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred ...
is a set with as
associative operation denoted as multiplication, that has an
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
, and such that every element has an inverse.
So, if is a group,
is defined for every
and every integer .
The set of all powers of an element of a group form a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgro ...
. A group (or subgroup) that consists of all powers of a specific element is the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative ...
generated by . If all the powers of are distinct, the group is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the
additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structur ...
of the integers. Otherwise, the cyclic group is
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb
Traditionally, a finite verb (from la, fīnītus, past partici ...
(it has a finite number of elements), and its number of elements is the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of . If the order of is , then
and the cyclic group generated by consists of the first powers of (starting indifferently from the exponent or ).
Order of elements play a fundamental role in
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen a ...
. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the ''order'' of the group). The possible orders of group elements are important in the study of the structure of a group (see
Sylow theorems
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fix ...
), and in the
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or el ...
.
Superscript notation is also used for
conjugation
Conjugation or conjugate may refer to:
Linguistics
*Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the change ...
; that is, , where ''g'' and ''h'' are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely
and
In a ring
In a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, it may occur that some nonzero elements satisfy
for some integer . Such an element is said to be
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
. In a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, the nilpotent elements form an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
, called the
nilradical of the ring.
If the nilradical is reduced to the
zero ideal (that is, if
implies
for every positive integer ), the commutative ring is said
reduced. Reduced rings important in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometric ...
, since the
coordinate ring of an
affine algebraic set is always a reduced ring.
More generally, given an ideal in a commutative ring , the set of the elements of that have a power in is an ideal, called the
radical
Radical may refer to:
Politics and ideology Politics
*Radical politics, the political intent of fundamental societal change
*Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
of . The nilradical is the radical of the
zero ideal. A
radical ideal is an ideal that equals its own radical. In a
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variabl ...