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 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
* Produ ...
of multiplying bases:
The exponent is usually shown as a
superscript 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 i ...
,
chemistry,
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 r ...
, and
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
, with applications such as
compound interest,
population growth,
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
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
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 alg ...
.
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 p ...
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 discovered and proved the law of exponents, , necessary to manipulate powers of . In the 9th century, the Persian mathematician
Muhammad ibn Mūsā al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
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-length a ...
—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, which later
Islamic 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 mathem ...
as the letters ''
mīm
Mem (also spelled Meem, Meme, or Mim) is the thirteenth letter of the Semitic abjads, including Hebrew mēm , Aramaic Mem , Syriac mīm ܡ, Arabic mīm and Phoenician mēm . Its sound value is .
The Phoenician letter gave rise to the Greek m ...
'' (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), Lati ...
'' (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
Jost Bürgi (also ''Joost, Jobst''; Latinized surname ''Burgius'' or ''Byrgius''; 28 February 1552 – 31 January 1632), active primarily at the courts in Kassel and Prague, was a Swiss clockmaker, a maker of astronomical instruments and a ma ...
used Roman numerals for exponents.
Nicolas Chuquet
Nicolas Chuquet (; born ; died ) was a French mathematician. He invented his own notation for algebraic concepts and exponentiation. He may have been the first mathematician to recognize zero and negative numbers as exponents.
In 1475, Jehan A ...
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 Universi ...
in the 16th century. The word ''exponent'' was coined in 1544 by Michael Stifel.
Samuel Jeake
Samuel Jeake (1623–1690), dubbed the Elder to distinguish him from his son, was an English merchant, nonconformist, antiquary and astrologer from Rye, East Sussex, England.
Life
Born at Rye in Sussex, on 9 October 1623, he may have belong ...
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 las ...
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. Further ...
), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and
zenzizenzizenzic
Zenzizenzizenzic is an obsolete form of mathematical notation representing the eighth power of a number (that is, the zenzizenzizenzic of ''x'' is ''x''8), dating from a time when powers were written out in words rather than as superscript numbers. ...
(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. Ma ...
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 grea ...
) 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 example ...
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 ma ...
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-length a ...
of ''b''" or "''b'' squared", because the area of a square with side-length is .
Similarly, the expression is called "the
cube of ''b''" or "''b'' cubed", because the volume of a cube with side-length is .
When it is a
positive integer, 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 operation
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th cen ...
s.
Positive exponents
The definition of the exponentiation as an iterated multiplication can be
formalized by using
induction
Induction, Inducible or Inductive may refer to:
Biology and medicine
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
, and this definition can be used as soon one has an
associative multiplication:
The base case is
:
and the
recurrence
Recurrence and recurrent may refer to:
*''Disease recurrence'', also called relapse
*''Eternal recurrence'', or eternal return, the concept that the universe has been recurring, and will continue to recur, in a self-similar form an infinite number ...
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 with a multiplication that has an
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
.
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 question ...
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 element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
s 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.
Monoid ...
, that is, an
algebraic structure, with an associative multiplication and a
multiplicative identity denoted (for example, the
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
of a given dimension). In particular, in such a structure, the inverse of an
invertible element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
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. For example, , whereas . Without parentheses, the conventional
order of operations for
serial exponentiation
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 ...
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 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
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
of the same size, this formula cannot be used. It follows that in
computer algebra, 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. The d ...
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
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
). Such functions can be represented as -
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s 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) 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, o ...
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 ...
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 a ...
) can be written as and then
approximated as .
SI prefixes based on powers of are also used to describe small or large quantities. For example, the prefix
kilo
KILO (94.3 FM broadcasting, FM, 94.3 KILO) is a radio station broadcasting in Colorado Springs, Colorado, Colorado Springs and Pueblo, Colorado, Pueblo, Colorado. It also streams online.
History
KLST and KPIK-FM
The 94.3 signal signed on th ...
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 A quarter is one-fourth, , 25% or 0.25.
Quarter or quarters may refer to:
Places
* Quarter (urban subdivision), a section or area, usually of a town
Placenames
* Quarter, South Lanarkshire, a settlement in Scotland
* Le Quartier, a settlement ...
''.
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 conce ...
, 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 post ...
, the set of all of its
subsets, which has members.
Integer powers of are important in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
. 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 represente ...
integer
binary number; 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 uni ...
may take different values. The
binary number system
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 ...
expresses any number as a sum of powers of , and denotes it as a sequence of and , separated by a
binary point
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
, 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 calle ...
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 limi ...
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 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 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 are called
even functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power s ...
.
When
is odd,
's
asymptotic behavior reverses from positive
to negative
. For
,
will also tend towards positive
infinity 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
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power ser ...
.
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 rat ...
, 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
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
, 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, ...
(, 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, 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 a ...
.
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 limi ...
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
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, is ...
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 t ...
that have the same limit, denoted
This defines
for every positive and real as a
continuous function of and . See also
Well-defined expression
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
.
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 expressi ...
. For avoiding
circular reasoning, 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 is the
inverse 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 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 fo ...
,
:
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 a ...
of
in terms of the
real and imaginary parts
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 a ...
of , namely
:
where the
absolute value 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 logarithms, 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 a ...
as
:
where
is the
absolute value of , and
is its
argument. The argument is defined
up to 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
In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'':
:r^n = x,
where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A roo ...
. 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 the whole complex plane, except for negative real values of the
radicand
In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'':
:r^n = x,
where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A root ...
. 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 n ...
shows that the principal th root is the unique
complex differentiable
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
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 comple ...
or algebraic solutions of algebraic equations (
Lagrange resolvent
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a rati ...
).
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 Eucli ...
of the
complex plane 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, 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 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 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 a ...
.
In all cases, the
complex logarithm 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 a ...
such that
:
for every in its
domain of definition
In mathematics, a partial function from a Set (mathematics), set to a set is a function from a subset of (possibly itself) to . The subset , that is, the Domain of a function, domain of viewed as a function, is called the domain of defini ...
.
Principal value
The
principal value of the
complex logarithm 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
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of a ...
at negative real values of , and it is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
(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 a ...
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 rat ...
, 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 discre ...
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 a ...
of ''. If
is the canonical form of ( and being real), then its polar form is
where
and
(see
atan2 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 of this logarithm is
where
denotes the
natural logarithm. 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, 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 ext ...
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 a ...
) 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
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 f ...
denoted as a multiplication.
[More generally, ]power associativity In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity.
Definition
An algebra (or more generally a magma) is said to be power-associative if the subalgebra ge ...
is sufficient for the definition. The definition of
requires further the existence of a
multiplicative identity.
An
algebraic structure 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.
Monoid ...
. 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''/ ...
. 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
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 ...
,
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
,
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
(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.
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 interv ...
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 referre ...
is a set with as
associative operation
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 f ...
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 su ...
, 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 subgroup ...
. 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 bina ...
generated by . If all the powers of are distinct, the group is
isomorphic 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 structures ...
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, a verb form that has a subject, usually being inflected or marke ...
(it has a finite number of elements), and its number of elements is the
order 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 group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. 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 fixe ...
), 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 else i ...
.
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 chang ...
; 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, 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
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive identi ...
(that is, if
implies
for every positive integer ), the commutative ring is said
reduced. Reduced rings important in
algebraic geometry, since the
coordinate ring
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal ...
of an
affine algebraic set
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine com ...
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 of . The nilradical is the radical of the
zero ideal
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive identi ...
. A
radical ideal
In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is called ' ...
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 variables ...