Exponentiation is a
mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
operation, written as , involving two numbers, the ''
base
Base or BASE may refer to:
Brands and enterprises
* Base (mobile telephony provider), a Belgian mobile telecommunications operator
*Base CRM
Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...
'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ".
When is a positive
integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
, 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 arithmetic, elementary Operation (mathematics), mathematical operations ...

of the base: that is, is the
product of multiplying bases:
:
The exponent is usually shown as a
superscript
Pro; the size of the subscript is about 62% of the original characters, dropped below the baseline by about 16%. The second typeface is Myriad
A myriad (from Ancient Greek
Ancient Greek includes the forms of the Greek language used in a ...

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".
One has , and, for any positive integers and , one has . To extend this property to non-positive integer exponents, is defined to be , and (with a positive integer and not zero) is defined as . In particular, is equal to , the ''
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another poly ...

'' of .
The definition of exponentiation can be extended to allow any real or
complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...
exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including
matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
.
Exponentiation is used extensively in many fields, including
economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant life and a bran ...

,
biology
Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

,
chemistry
Chemistry is the scientific
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is an occurrence in the real world. T ...

,
physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, and
computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...
, with applications such as
compound interest
Compound interest is the addition of interest
Interest, in finance and economics, is payment from a debtor, borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, ...

,
population growth
Population growth is the increase in the number of people in a population
Population typically refers the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size ...
,
chemical reaction kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry
Physical chemistry is the study of macroscopic
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with ...
,
wave
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

behavior, and
public-key cryptography
Public-key cryptography, or asymmetric cryptography, is a cryptographic system that uses pairs of keys
KEYS (1440 AM broadcasting, AM) is a radio station serving the Corpus Christi, Texas, Corpus Christi, Texas area with a talk radio, talk ...
.
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
Greek ( el, label=Modern Greek
Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...
δύναμις (''dúnamis'', here: "amplification"
) used by the
Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
mathematician
Euclid
Euclid (; grc-gre, Εὐκλείδης
Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

for the square of a line,
following
Hippocrates of Chios
Hippocrates of Kos (; grc-gre, Ἱπποκράτης ὁ Κῷος, Hippokrátēs ho Kôios; ), also known as Hippocrates II, was a Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), ...
. In ''
The Sand Reckoner
''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathemat ...
'',
Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

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 ( fa, محمد بن موسی خوارزمی, Moḥammad ben Musā Khwārazmi; ), or al-Khwarizmi and formerly Latinisation of names, Latinized as ''Algorithmi'', was a Persians, Persian polymath who produced ...
used the terms مَال (''māl'', "possessions", "property") for a
square
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...
—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 (''
'', "cube") for a
cube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
, which later
Islamic
Islam (; ar, اَلْإِسْلَامُ, al-’Islām, "submission o God
Oh God may refer to:
* An exclamation; similar to "oh no", "oh yes", "oh my", "aw goodness", "ah gosh", "ah gawd"; see interjection
An interjection is a word or expression that occurs as an utterance on its own and expresses a spontaneous feeling ...
) is an Abrahamic religions, Abrahamic monotheistic religion teaching that Muhammad is a Muhammad in Islam, messenger of God.Peters, F. E. 2009. "Allāh." In , ed ...
mathematicians represented in
mathematical notation
Mathematical notation is a system of symbol
A symbol is a mark, sign, or word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or prag ...
as the letters ''
mīm'' (m) and ''
kāf
Kaf (also spelled kaph) is the eleventh letter of the Semitic abjads, including Phoenician Kāp , Hebrew
Hebrew (, , or ) is a Northwest Semitic languages, Northwest Semitic language of the Afroasiatic languages, Afroasiatic language fam ...
'' (k), respectively, by the 15th century, as seen in the work of
Abū al-Hasan ibn Alī al-Qalasādī.
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
Kassel (; in Germany, spelled Cassel until 1926) is a city on the Fulda River in ...
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
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from ...
and
Michael Stifel
Michael Stifel or Styfel (1487 – April 19, 1567) was a German monk, Protestant reformer and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the ...
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 (c. 1512 – 1558) was a Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign
The plus and minus signs, and , are mathematical symbols used to represent the notions ...
used the terms square, cube, zenzizenzic (
fourth power
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...
), 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
Latinisation or Latinization can refer to:
* Latinisation of names, the practice of rendering a non-Latin name in a Latin style
* Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

in his text titled ''
La Géométrie
''La Géométrie'' was published
Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Traditionally, the term refers to the distribution of printed works ...
''; 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
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write
polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

introduced variable exponents, and, implicitly, non-integer exponents by writing:
"consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functionIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s, since in those the exponents must be constant."
Terminology
The expression is called "the
square
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...
of ''b''" or "''b'' squared", because the area of a square with side-length is .
Similarly, the expression is called "the
cube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
of ''b''" or "''b'' cubed", because the volume of a cube with side-length is .
When it is a
positive integer
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, 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 (from the Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...
s.
Positive exponents
The definition of the exponentiation as an iterated multiplication can be
formalized by using
induction
Induction may refer to:
Philosophy
* Inductive reasoning, in logic, inferences from particular cases to the general case
Biology and chemistry
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induction period, the t ...
, and this definition can be used as soon one has an
associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
with a multiplication that has an
identity
Identity may refer to:
Social sciences
* Identity (social science), personhood or group affiliation in psychology and sociology
Group expression and affiliation
* Cultural identity, a person's self-affiliation (or categorization by others ...
.
Intuitionally,
may be interpreted as the
empty product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 the branch of abstract algebra known as ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...
s in a multiplicative
monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
, that is, an
algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, with an associative multiplication and a
multiplicative identity
In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
denoted (for example, the
square matrices
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a given dimension). In particular, in such a structure, the inverse of an
invertible element
In the branch of abstract algebra known as ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...
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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. For example, . Also unlike addition and multiplication, exponentiation is not
associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. For example, , whereas . Without parentheses, the conventional
order of operations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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 exponentiation, powers of a binomial (polynomial), binomial. According to the theorem, it is possible to expand the polynomial into a summati ...
:
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
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
...
that is
commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. Otherwise, if and are, say,
square matrices
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the same size, this formula cannot be used. It follows that in
computer algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, many
algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

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
Mathematical software is software used to mathematical model, model, analyze or calculate numeric, symbolic or geometric data.
It is a type of applica ...

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
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
from a
set of elements to a set of elements (see
cardinal exponentiation). Such functions can be represented as -
tuple
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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
The decimal numeral system
A numeral system (or system of numeration) is a writing system
A writing system is a method of visually representing verbal communication
Communication (from Latin ''communicare'', meaning "to share") is t ...
) 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
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or coul ...
to denote large or small numbers. For instance, (the
speed of light
The speed of light in vacuum
A vacuum is a space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
in vacuum, in
metres per second
The metre per second is an SI derived unit
SI derived units are units of measurement derived from the
seven SI base unit, base units specified by the International System of Units (SI). They are either dimensionless quantity, dimensionless or ...
) 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 Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
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
The City of Colorado Springs is the List of cities and towns in Colorado#Home rule municipality, Home Rule Municipality that is the county seat and the List of ci ...
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 is the irreducible fraction resulting from dividing 1 (number), one by 2 (number), two or the fraction resulting from dividing any number by its double. Multiplication by one half is equivalent to division by two, or "halving"; con ...
'' and ''
''.
Powers of appear in
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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, i ...
, since a set with members has a
power set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, the set of all of its
subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s, which has members.
Integer powers of are important in
computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...
. The positive integer powers give the number of possible values for an -
bit
The bit is a basic unit of information in computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm
of an algorithm (Euclid's algo ...
integer
binary number
In mathematics and digital electronics
Digital electronics is a field of electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons
The electr ...
; for example, a
byte
The byte is a unit of digital information that most commonly consists of eight bit
The bit is a basic unit of information in computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It ...
may take different values. The
binary number system
In mathematics and digital electronics
Digital electronics is a field of electronics
Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter ...
expresses any number as a sum of powers of , and denotes it as a sequence of and , separated by a
binary pointIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s. For a similar discussion of powers of the complex number , see .
Large exponents
The
limit of a sequence
As the positive integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. ...
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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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 formIn calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
, are described in below.
Power functions

Real functions of the form
, where
, are sometimes called power functions. When
is an
integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
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 number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything ...
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 Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

are called
even functions.
When
is odd,
's
asymptotic
250px, A curve intersecting an asymptote infinitely many times.
In analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc ...
behavior reverses from positive
to negative
. For
,
will also tend towards positive
infinity
Infinity is that which is boundless, endless, or larger than any number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything ...
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
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, 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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
, 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
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

of the base and the
exponential function
The exponential function is a mathematical function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of ...

(, 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
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
Limits of rational exponents

Since any
irrational number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
can be expressed as the
limit of a sequence
As the positive integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. ...
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
Figure 3. A function that is not monotonic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
that have the same limit, denoted
This defines
for every positive and real as a
continuous function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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, and can be characterized in many ways. It is the base of a logarithm, base of the natural logarithm. It is the Limit of a sequence, limit of ...
. For avoiding
circular reasoning
Circular reasoning ( la, circulus in probando, "circle in proving"; also known as circular logic) is a logical fallacy in which the reasoner begins with what they are trying to end with. The components of a circular argument are often logically ...

, 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
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

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 satifies 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
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

function. Specifically, the fact that the
natural logarithm
The natural logarithm of a number is its logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
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 add ...
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
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
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 integer.
Euler's formula
Euler's formula, named after Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) incl ...

,
:
allows expressing the
polar form
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of
in terms of the
real and imaginary parts
In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
of , namely
:
where the
absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of the
trigonometric
Trigonometry (from Ancient Greek, Greek ''wikt:τρίγωνον, trigōnon'', "triangle" and ''wikt:μέτρον, metron'', "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The fiel ...

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
of the color is used to show the ''arg
Arg or ARG may refer to:
Places
*''Arg'' () means "citadel" in Persian, and may refer to:
**Arg, Iran, a village in Fars Province, Iran
**Arg (Kabul), presidential palace in Kabul, Afghanistan
**Arg, South ...
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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
as
:
where
is the
absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of , and
is its
argument
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
. The argument is defined
up to Two mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 in 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
in the whole complex plane, except for negative real values of the
radicand
In mathematics, an ''n''th 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 of degree 2 is called a ...
. 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
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic can also have the following meanings:
Natural sciences Chemistry
* ...

shows that the principal th root is the unique
complex differentiable
A rectangular grid (top) and its image under a conformal map ''f'' (bottom).
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
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
(they are multiplied by
). This shows that it is not possible to define a th root function that is not 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 Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...
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 ratio ...
).
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 number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of the
complex plane
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
, 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is defined, which is not continuous for the values of that are real and nonpositive, or
is defined as a
multivalued function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
In all cases, the
complex logarithm
of the color is used to show the ''arg
Arg or ARG may refer to:
Places
*''Arg'' () means "citadel" in Persian, and may refer to:
**Arg, Iran, a village in Fars Province, Iran
**Arg (Kabul), presidential palace in Kabul, Afghanistan
**Arg, South ...
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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

such that
:
for every in its
domain of definition.
Principal value
The
principal value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of the
complex logarithm
of the color is used to show the ''arg
Arg or ARG may refer to:
Places
*''Arg'' () means "citadel" in Persian, and may refer to:
**Arg, Iran, a village in Fars Province, Iran
**Arg (Kabul), presidential palace in Kabul, Afghanistan
**Arg, South ...
is the unique function, commonly denoted
such that, for every nonzero complex number ,
:
and the
imaginary part
In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
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 function (mathematics), functions are continuous. If a function is not continuous at a point in its domain of a function, domain, one says t ...
at negative real values of , and it is
holomorphic
Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom).
In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
(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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
, that is, there is an integer such that is an integer. This results from the
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 number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
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 discret ...

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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of ''. If
is the canonical form of ( and being real), then its polar form is
where
and
(see
atan2
The function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automati ...

for the definition of this function).
*''
Logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
of ''. The
principal value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of this logarithm is
where
denotes the
natural logarithm
The natural logarithm of a number is its logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
. 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 a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
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 any complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...
, and is a rational number, then is an algebraic number. This results from the theory of
algebraic extension
In abstract algebra, a field extension ''L''/''K'' is called algebraic if every element of ''L'' is algebraic over ''K'', i.e. if every element of ''L'' is a root
In vascular plants, the roots are the plant organ, organs of a plant that are ...
s. This remains true if is any algebraic number, in which case, all values of (as a
multivalued function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

) 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. Th ...
(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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
denoted as a multiplication.
[More generally, ]power associativityIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
is sufficient for the definition. The definition of
requires further the existence of a
multiplicative identity
In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
.
An
algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a
monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
. 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
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

. 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 people 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 identi ...
,
rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck
Ring may also refer to:
Sounds
* Ri ...
,
fields
File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe''
FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
,
square matrices
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(which form a ring). They apply also to
functions
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
from a
set to itself, which form a monoid under
function composition
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. This includes, as specific instances,
geometric transformation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 group ...
s of any
mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
.
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
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. ...
whose valued can be multiplied,
denotes the exponentiation with respect of multiplication, and
may denote exponentiation with respect of
function composition
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. That is,
:
and
:
Commonly,
is denoted
while
is denoted
In a group
A
multiplicative group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is a set with as
associative operation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
denoted as multiplication, that has an
identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, 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 (mathematics), 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 ...
. A group (or subgroup) that consists of all powers of a specific element is the
cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

generated by . If all the powers of are distinct, the group is
isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). 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 structure ...
of the integers. Otherwise, the cyclic group is
finite
Finite is the opposite of Infinity, 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 ...
(it has a finite number of elements), and its number of elements is the
order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
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 mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
. 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
Peter Ludwig Mejdell Sylow () (12 December 1832 – 7 September 1918) was a Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic ...
), and in the
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...
.
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, it may occur that some nonzero elements satisfy
for some integer . Such an element is said to be
nilpotent
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. In a
commutative ring
In ring theory
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...
, the nilpotent elements form an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics)
An ideal is a principle
A principle is a proposition or value that is a guide for behavior or evaluation. In law
Law is a system
A system is a group of Interaction, interacting ...
, called the
nilradical of the ring.
If the nilradical is reduced to the
zero ideal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(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
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, since the
coordinate ring
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
of an
affine algebraic set
Affine (pronounced /əˈfaɪn/) relates to 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 combination, a certai ...
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:
Arts and entertainment Music
*Radical (mixtape), ''Radical'' (mixtape), by Odd Future, 2010
*Radical (Smack album), ''Radical'' (Smack album), 1988
*"Radicals", a song by Tyler, The Creator from the 2011 album ''Goblin (album ...
of . The nilradical is the radical of the
zero ideal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. A
radical ideal is an ideal that equals its own radical. In a
polynomial ring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...