Exponentiation is a
mathematical 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, exponentiation corresponds to repeated
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
of the base: that is, is the
product of multiplying bases:
The exponent is usually shown as a
superscript
A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...
to the right of the base. In that case, is called "''b'' raised to the ''n''th power", "''b'' (raised) to the power of ''n''", "the ''n''th power of ''b''", "''b'' to the ''n''th power", or most briefly as "''b'' to the ''n''th".
Starting from the basic fact stated above that, for any positive integer
,
is
occurrences of
all multiplied by each other, several other properties of exponentiation directly follow. In particular:
In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that
must be equal to 1, as follows. For any
,
. Dividing both sides by
gives
.
The fact that
can similarly be derived from the same rule. For example,
. Taking the cube root of both sides gives
.
The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what
should mean. In order to respect the "exponents add" rule, it must be the case that
. Dividing both sides by
gives
, which can be more simply written as
, using the result from above that
. By a similar argument,
.
The properties of fractional exponents also follow from the same rule. For example, suppose we consider
and ask if there is some suitable exponent, which we may call
, such that
. From the definition of the square root, we have that
. Therefore, the exponent
must be such that
. Using the fact that multiplying makes exponents add gives
. The
on the right-hand side can also be written as
, giving
. Equating the exponents on both sides, we have
. Therefore,
, so
.
The definition of exponentiation can be extended to allow any real or
complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including
matrices.
Exponentiation is used extensively in many fields, including
economics,
biology,
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
,
physics, and
computer science, with applications such as
compound interest,
population growth
Population growth is the increase in the number of people in a population or dispersed group. Actual global human population growth amounts to around 83 million annually, or 1.1% per year. The global population has grown from 1 billion in 1800 to ...
,
chemical reaction kinetics,
wave behavior, and
public-key cryptography.
History of the notation
The term ''power'' ( la, potentia, potestas, dignitas) is a mistranslation
of the
ancient Greek δύναμις (''dúnamis'', here: "amplification"
) used by the
Greek mathematician
Euclid for the square of a line,
following
Hippocrates of Chios. In ''
The Sand Reckoner'',
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
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 Persians, Persian polymath from Khwarazm, who produced vastly influential works in Mathematics ...
used the terms مَال (''māl'', "possessions", "property") for a
square—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
The Kaaba (, ), also spelled Ka'bah or Kabah, sometimes referred to as al-Kaʿbah al-Musharrafah ( ar, ٱلْكَعْبَة ٱلْمُشَرَّفَة, lit=Honored Ka'bah, links=no, translit=al-Kaʿbah al-Musharrafah), is a building at the c ...
'', "cube") for a
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
, which later
Islamic
Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God (or '' Allah'') as it was revealed to Muhammad, the mai ...
mathematicians represented in
mathematical notation as the letters ''
mīm'' (m) and ''
kāf
Kaph (also spelled kaf) is the eleventh letter of the Semitic abjads, including Phoenician kāp , Hebrew kāf , Aramaic kāp , Syriac kāp̄ , and Arabic kāf (in abjadi order).
The Phoenician letter gave rise to the Greek kappa (Κ), Latin ...
'' (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''; Latinisation of names, 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 astronomica ...
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 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 Universit ...
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 used the terms square, cube, zenzizenzic (
fourth power), 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 in his text titled ''
La Géométrie
''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''La Géométrie ...
''; there, the notation is introduced in Book I.
Some mathematicians (such as
Isaac Newton) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write
polynomials, for example, as .
Another historical synonym, involution, is now rare and should not be confused with
its more common meaning.
In 1748,
Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing:
Terminology
The expression is called "the
square of ''b''" or "''b'' squared", because the area of a square with side-length is .
Similarly, the expression is called "the
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
of ''b''" or "''b'' cubed", because the volume of a cube with side-length is .
When it is a
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
, the exponent indicates how many copies of the base are multiplied together. For example, . The base appears times in the multiplication, because the exponent is . Here, is the ''5th power of 3'', or ''3 raised to the 5th power''.
The word "raised" is usually omitted, and sometimes "power" as well, so can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiation can be expressed as "''b'' to the power of ''n''", "''b'' to the ''n''th power", "''b'' to the ''n''th", or most briefly as "''b'' to the ''n''".
A formula with nested exponentiation, such as (which means and not ), is called a tower of powers, or simply a tower.
Integer exponents
The exponentiation operation with integer exponents may be defined directly from elementary
arithmetic operations.
Positive exponents
The definition of the exponentiation as an iterated multiplication can be
formalized by using
induction
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
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 ...
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
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
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 of copies of . So, the equality
is a special case of the general convention for the empty product.
The case of is more complicated. In contexts where only integer powers are considered, the value is generally assigned to
but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context.
Negative exponents
Exponentiation with negative exponents is defined by the following identity, which holds for any integer and nonzero :
:
.
Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (
).
This definition of exponentiation with negative exponents is the only one that allows extending the identity
to negative exponents (consider the case
).
The same definition applies to
invertible elements in a multiplicative
monoid, that is, an
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
, with an associative multiplication and a
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
denoted (for example, the
square matrices of a given dimension). In particular, in such a structure, the inverse of an
invertible element is standardly denoted
Identities and properties
The following
identities, often called , hold for all integer exponents, provided that the base is non-zero:
:
Unlike addition and multiplication, exponentiation is not
commutative. For example, . Also unlike addition and multiplication, exponentiation is not
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
. For example, , whereas . Without parentheses, the conventional
order of operations
In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.
For exampl ...
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 exampl ...
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
:
However, this formula is true only if the summands commute (i.e. that ), which is implied if they belong to a
structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
that is
commutative. Otherwise, if and are, say,
square matrices of the same size, this formula cannot be used. It follows that in
computer algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
, many
algorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose
computer algebra systems use a different notation (sometimes instead of ) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.
Combinatorial interpretation
For nonnegative integers and , the value of is the number of
functions from a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of elements to a set of elements (see
cardinal exponentiation). Such functions can be represented as -
tuples from an -element set (or as -letter words from an -letter alphabet). Some examples for particular values of and are given in the following table:
:
Particular bases
Powers of ten
In the base ten (
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
) 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 to denote large or small numbers. For instance, (the
speed of light in vacuum, in
metres per second) can be written as and then
approximated as .
SI prefix
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
es based on powers of are also used to describe small or large quantities. For example, the prefix
kilo means , so a kilometre is .
Powers of two
The first negative powers of are commonly used, and have special names, e.g.: ''
half'' 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 i ...
''.
Powers of appear in
set theory, since a set with members has a
power set, the set of all of its
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s, which has members.
Integer powers of are important in
computer science. The positive integer powers give the number of possible values for an -
bit integer
binary number
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one).
The base-2 numeral system is a positional notatio ...
; for example, a
byte 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, 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
sequences. For a similar discussion of powers of the complex number , see .
Large exponents
The
limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound:
: as when
This can be read as "''b'' to the power of ''n'' tends to
+∞ as ''n'' tends to infinity when ''b'' is greater than one".
Powers of a number with
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), 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 form, are described in below.
Power functions
Real functions of the form
, where
, are sometimes called power functions. When
is an
integer and
, two primary families exist: for
even, and for
odd. In general for
, when
is even
will tend towards positive
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
with increasing
, and also towards positive infinity with decreasing
. All graphs from the family of even power functions have the general shape of
, flattening more in the middle as
increases.
Functions with this kind of
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
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
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
behavior reverses from positive
to negative
. For
,
will also tend towards positive
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
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, 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, 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 of the base and the
exponential function (, 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 exponents.
On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values (see ). One may choose one of these values, called the
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive ...
, 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.
Limits of rational exponents
Since any
irrational number can be expressed as the
limit of a sequence 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 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 that have the same limit, denoted
This defines
for every positive and real as a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
of and . See also
Well-defined expression.
The exponential function
The ''exponential function'' is often defined as
where
is
Euler's number. For avoiding
circular reasoning
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fallacy
* Circular ...
, 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 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 function. Specifically, the fact that the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
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 ad ...
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 exponent is defined by means of the exponential function with complex argument (see the end of , above) as
:
where
denotes the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
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,
:
allows expressing the
polar form of
in terms of the
real and imaginary parts of , namely
:
where the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the
trigonometric factor is one. This results from
:
Non-integer powers of complex numbers
In the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only. For other bases, difficulties appear already with the apparently simple case of th roots, that is, of exponents
where is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to th roots, this case deserves to be considered first, since it does not need to use
complex logarithm
In mathematics
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 ...
s, and is therefore easier to understand.
th roots of a complex number
Every nonzero complex number may be written in
polar form as
:
where
is the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of , and
is its
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
. The argument is defined
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
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 mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
in the whole complex plane, except for negative real values of the
radicand. This function equals the usual th root for positive real radicands. For negative real radicands, and odd exponents, the principal th root is not real, although the usual th root is real.
Analytic continuation 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 or algebraic solutions of algebraic equations (
Lagrange resolvent).
The th roots of unity are the first powers of
, that is
The th roots of unity that have this generating property are called ''primitive th roots of unity''; they have the form
with
coprime 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 of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
at the vertices of a
regular -gon with one vertex on the real number 1.
As the number
is the primitive th root of unity with the smallest positive
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
, it is called the ''principal primitive th root of unity'', sometimes shortened as ''principal th root of unity'', although this terminology can be confused with the
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive ...
of
which is 1.
Complex exponentiation
Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for
. So, either a
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive ...
is defined, which is not continuous for the values of that are real and nonpositive, or
is defined as a
multivalued function.
In all cases, the
complex logarithm
In mathematics
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 ...
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 such that
:
for every in its
domain of definition.
Principal value
The
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive ...
of the
complex logarithm
In mathematics
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 ...
is the unique function, commonly denoted
such that, for every nonzero complex number ,
:
and the
imaginary part of satisfies
:
The principal value of the complex logarithm is not defined for
it is
discontinuous at negative real values of , and it is
holomorphic (that is, complex differentiable) elsewhere. If is real and positive, the principal value of the complex logarithm is the natural logarithm:
The principal value of
is defined as
where
is the principal value of the logarithm.
The function
is holomorphic except in the neighbourhood of the points where is real and nonpositive.
If is real and positive, the principal value of
equals its usual value defined above. If
where is an integer, this principal value is the same as the one defined above.
Multivalued function
In some contexts, there is a problem with the discontinuity of the principal values of
and
at the negative real values of . In this case, it is useful to consider these functions as
multivalued functions.
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, 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 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 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 of ''. If
is the canonical form of ( and being real), then its polar form is
where
and
(see
atan2
In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive for the definition of this function).
*''
Logarithm of ''. The
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive ...
of this logarithm is
where
denotes the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
. 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 of is an integer.
Failure of power and logarithm identities
Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined ''as single-valued functions''. For example:
Irrationality and transcendence
If is a positive real
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
, and is a rational number, then is an algebraic number. This results from the theory of
algebraic extensions. This remains true if is any algebraic number, in which case, all values of (as a
multivalued function) are algebraic. If is
irrational (that is, ''not rational''), and both and are algebraic, Gelfond–Schneider theorem asserts that all values of are
transcendental
Transcendence, transcendent, or transcendental may refer to:
Mathematics
* Transcendental number, a number that is not the root of any polynomial with rational coefficients
* Algebraic element or transcendental element, an element of a field exten ...
(that is, not algebraic), except if equals or .
In other words, if is irrational and
then at least one of , and is transcendental.
Integer powers in algebra
The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any
associative operation denoted as a multiplication.
[More generally, power associativity is sufficient for the definition.] The definition of
requires further the existence of a
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
.
An
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a
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 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 iden ...
,
rings,
fields,
square matrices (which form a ring). They apply also to
functions from a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
to itself, which form a monoid under
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
. This includes, as specific instances,
geometric transformations, and
endomorphisms of any
mathematical structure
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional ...
.
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 whose valued can be multiplied,
denotes the exponentiation with respect of multiplication, and
may denote exponentiation with respect of
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
. That is,
:
and
:
Commonly,
is denoted
while
is denoted
In a group
A
multiplicative group is a set with as
associative operation denoted as multiplication, that has an
identity element, 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. A group (or subgroup) that consists of all powers of a specific element is the
cyclic group generated by . If all the powers of are distinct, the group is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
additive group of the integers. Otherwise, the cyclic group is
finite (it has a finite number of elements), and its number of elements is the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
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. 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), and in the
classification of finite simple groups.
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 a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
, the nilpotent elements form an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
, called the
nilradical of the ring.
If the nilradical is reduced to the
zero ideal (that is, if
implies
for every positive integer ), the commutative ring is said
reduced. Reduced rings important in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, since the
coordinate ring of an
affine algebraic set is always a reduced ring.
More generally, given an ideal in a commutative ring , the set of the elements of that have a power in is an ideal, called the
radical
Radical may refer to:
Politics and ideology Politics
*Radical politics, the political intent of fundamental societal change
*Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
of . The nilradical is the radical of the
zero ideal. A
radical ideal is an ideal that equals its own radical. In a
polynomial ring