Exponentiation

"Exponent" redirects here. For other uses, seeExponent (disambiguation).

Inmathematics,exponentiation,denotedbⁿ,is anoperationinvolving two numbers: thebase,b,and theexponentorpower,n.Whennis a positiveinteger,exponentiation corresponds to repeatedmultiplicationof the base: that is,bⁿis theproductof multiplyingnbases: $$b^{n}=\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}.$$In particular,$b^{1}=b$.

Graphs ofy=b^xfor various basesb:base10,basee,base2,base⁠1/2⁠.Each curve passes through the point(0, 1)because any nonzero number raised to the power of0is1.Atx= 1,the value ofyequals the base because any number raised to the power of1is the number itself.

The exponent is usually shown as asuperscriptto the right of the base asbⁿor in computer code asb^n.Thisbinary operationis often read as "bto the powern";it may also be referred to as"braised to thenth power "," thenth power ofb",or, most briefly, "bto then".

The above definition of$b^{n}$immediately implies several properties, in particular the multiplication rule:

$${\begin{aligned}b^{n}\times b^{m}&=\underbrace {b\times \dots \times b} _{n{\text{ times}}}\times \underbrace {b\times \dots \times b} _{m{\text{ times}}}\\[1ex]&=\underbrace {b\times \dots \times b} _{n+m{\text{ times}}}\ =\ b^{n+m}.\end{aligned}}$$

That is, when multiplying a base raised to one power times the same base raised to another power, the powers add. Extending this rule to the power zero gives$b^{0}\times b^{n}=b^{0+n}=b^{n}$,and dividing both sides by$b^{n}$gives$b^{0}=b^{n}/b^{n}=1$.That is, the multiplication rule implies the definition$$b^{0}=1.$$A similar argument implies the definition for negative integer powers:$$b^{-n}=1/b^{n}.$$That is, extending the multiplication rule gives$b^{-n}\times b^{n}=b^{-n+n}=b^{0}=1$.Dividing both sides by$b^{n}$gives$b^{-n}=1/b^{n}$.This also implies the definition for fractional powers:$$b^{n/m}={\sqrt[{m}]{b^{n}}}.$$For example,$b^{1/2}\times b^{1/2}=b^{1/2\,+\,1/2}=b^{1}=b$,meaning$(b^{1/2})^{2}=b$,which is the definition of square root:$b^{1/2}={\sqrt {b}}$.

The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define$b^{x}$for any positive real base$b$and any real number exponent$x$.More involved definitions allowcomplexbase and exponent, as well as certain types ofmatricesas base or exponent.

Exponentiation is used extensively in many fields, includingeconomics,biology,chemistry,physics,andcomputer science,with applications such ascompound interest,population growth,chemical reaction kinetics,wavebehavior, andpublic-key cryptography.

Etymology

The termexponentoriginates from theLatinexponentem,thepresent participleofexponere,meaning "to put forth".The termpower(Latin:potentia, potestas, dignitas) is a mistranslationof theancient Greekδύναμις (dúnamis,here: "amplification") used by theGreekmathematicianEuclidfor the square of a line,followingHippocrates of Chios.

History

Antiquity

The Sand Reckoner

Main article:The Sand Reckoner

InThe Sand Reckoner,Archimedesproved the law of exponents,10^a· 10^b= 10^a+b,necessary to manipulate powers of10.He then used powers of10to estimate the number of grains of sand that can be contained in the universe.

Islamic Golden Age

Mālandkaʿbah( "square" and "cube" )

In the 9th century, the Persian mathematicianAl-Khwarizmiused the terms مَال (māl,"possessions", "property" ) for asquare—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"—and كَعْبَة (Kaʿbah,"cube" ) for acube,which laterIslamicmathematicians represented inmathematical notationas the lettersmīm(m) andkāf(k), respectively, by the 15th century, as seen in the work ofAbu'l-Hasan ibn Ali al-Qalasadi.

15th–18th century

Introducing exponents

Nicolas Chuquetused a form of exponential notation in the 15th century, for example12²to represent12x².This was later used byHenricus GrammateusandMichael Stifelin the 16th century. In the late 16th century,Jost Bürgiwould use Roman numerals for exponents in a way similar to that of Chuquet, for exampleiii4for4x³.

"Exponent"; "square" and "cube"

The wordexponentwas coined in 1544 by Michael Stifel.In the 16th century,Robert Recordeused the terms "square", "cube", "zenzizenzic" (fourth power), "sursolid" (fifth), "zenzicube" (sixth), "second sursolid" (seventh), and "zenzizenzizenzic"(eighth)."Biquadrate" has been used to refer to the fourth power as well.

Modern exponential notation

In 1636,James Humeused in essence modern notation, when inL'algèbre de Viètehe wroteAⁱⁱⁱforA³.Early in the 17th century, the first form of our modern exponential notation was introduced byRené Descartesin his text titledLa Géométrie;there, the notation is introduced in Book I.

I designate...aa,ora²in multiplyingaby itself; anda³in multiplying it once more again bya,and thus to infinity.

— René Descartes, La Géométrie

Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would writepolynomials,for example, asax+bxx+cx³+d.

"Indices"

Samuel Jeakeintroduced the termindicesin 1696.The terminvolutionwas used synonymously with the termindices,but had declined in usageand should not be confused withits more common meaning.

Variable exponents, non-integer exponents

In 1748,Leonhard Eulerintroduced 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 notalgebraic functions,since in those the exponents must be constant.

20th century

As calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation. For exampleKonrad Zuseintroducedfloating point arithmeticin his 1938 computer Z1. Oneregistercontained representation of leading digits, and a second contained representation of the exponent of 10. EarlierLeonardo Torres QuevedocontributedEssays on Automation(1914) which had suggested the floating-point representation of numbers. The more flexibledecimal floating-pointrepresentation was introduced in 1946 with aBell Laboratoriescomputer. Eventually educators and engineers adoptedscientific notationof numbers, consistent with common reference toorder of magnitudein aratio scale.

For instance, in 1961 theSchool Mathematics Study Groupdeveloped the notation in connection with units used in themetric system.

Exponents also came to be used to describeunits of measurementandquantity dimensions.For instance, sinceforceis mass times acceleration, it is measured in kg m/sec².Using M for mass, L for length, and T for time, the expression M L T^–2is used indimensional analysisto describe force.

Terminology

The expressionb²=b·bis called "thesquareofb"or"bsquared ", because the area of a square with side-lengthbisb².(It is true that it could also be called "bto the second power ", but" the square ofb"and"bsquared "are more traditional)

Similarly, the expressionb³=b·b·bis called "thecubeofb"or"bcubed ", because the volume of a cube with side-lengthbisb³.

When an exponent is apositive integer,that exponent indicates how many copies of the base are multiplied together. For example,3⁵= 3 · 3 · 3 · 3 · 3 = 243.The base3appears5times in the multiplication, because the exponent is5.Here,243is the5th power of 3,or3 raised to the 5th power.

The word "raised" is usually omitted, and sometimes "power" as well, so3⁵can be simply read "3 to the 5th", or "3 to the 5".

Integer exponents

The exponentiation operation with integer exponents may be defined directly from elementaryarithmetic operations.

Positive exponents

The definition of the exponentiation as an iterated multiplication can beformalizedby usinginduction,and this definition can be used as soon as one has anassociativemultiplication:

The base case is

$b^{1}=b$

and therecurrenceis

$b^{n+1}=b^{n}\cdot b.$

The associativity of multiplication implies that for any positive integersmandn,

$b^{m+n}=b^{m}\cdot b^{n},$

and

$(b^{m})^{n}=b^{mn}.$

Zero exponent

As mentioned earlier, a (nonzero) number raised to the0power is1:

$b^{0}=1.$

This value is also obtained by theempty productconvention, which may be used in everyalgebraic structurewith a multiplication that has anidentity.This way the formula

$b^{m+n}=b^{m}\cdot b^{n}$

also holds for$n=0$.

The case of0⁰is controversial. In contexts where only integer powers are considered, the value1is generally assigned to0⁰but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context.For more details, seeZero to the power of zero.

Negative exponents

Exponentiation with negative exponents is defined by the following identity, which holds for any integernand nonzerob:

$b^{-n}={\frac {1}{b^{n}}}$.

Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity ($\infty$).

This definition of exponentiation with negative exponents is the only one that allows extending the identity$b^{m+n}=b^{m}\cdot b^{n}$to negative exponents (consider the case$m=-n$).

The same definition applies toinvertible elementsin a multiplicativemonoid,that is, analgebraic structure,with an associative multiplication and amultiplicative identitydenoted1(for example, thesquare matricesof a given dimension). In particular, in such a structure, the inverse of aninvertible elementxis standardly denoted$x^{-1}.$

Identities and properties

"Laws of Indices" redirects here. For the horse, seeLaws of Indices (horse).

The followingidentities,often calledexponent rules,hold for all integer exponents, provided that the base is non-zero:

${\begin{aligned}b^{m}\cdot b^{n}&=b^{m+n}\\\left(b^{m}\right)^{n}&=b^{m\cdot n}\\b^{n}\cdot c^{n}&=(b\cdot c)^{n}\end{aligned}}$

Unlike addition and multiplication, exponentiation is notcommutative:for example,$2^{3}=8$,but reversing the operands gives the different value$3^{2}=9$.Also unlike addition and multiplication, exponentiation is notassociative:for example,(2³)²= 8²= 64,whereas2⁽³²⁾= 2⁹= 512.Without parentheses, the conventionalorder of operationsforserial exponentiationin superscript notation is top-down (orright-associative), not bottom-up(orleft-associative). That is,

$b^{p^{q}}=b^{\left(p^{q}\right)},$

which, in general, is different from

$\left(b^{p}\right)^{q}=b^{pq}.$

Powers of a sum

The powers of a sum can normally be computed from the powers of the summands by thebinomial formula

$(a+b)^{n}=\sum _{i=0}^{n}{\binom {n}{i}}a^{i}b^{n-i}=\sum _{i=0}^{n}{\frac {n!}{i!(n-i)!}}a^{i}b^{n-i}.$

However, this formula is true only if the summands commute (i.e. thatab=ba), which is implied if they belong to astructurethat iscommutative.Otherwise, ifaandbare, say,square matricesof the same size, this formula cannot be used. It follows that incomputer algebra,manyalgorithmsinvolving integer exponents must be changed when the exponentiation bases do not commute. Some general purposecomputer algebra systemsuse a different notation (sometimes^^instead of^) for exponentiation with non-commuting bases, which is then callednon-commutative exponentiation.

Combinatorial interpretation

For nonnegative integersnandm,the value ofn^mis the number offunctionsfrom asetofmelements to a set ofnelements (seecardinal exponentiation). Such functions can be represented asm-tuplesfrom ann-element set (or asm-letter words from ann-letter alphabet). Some examples for particular values ofmandnare given in the following table:

nm	Thenmpossiblem-tuples of elements from the set{1,...,n}
05= 0	none
14= 1	(1, 1, 1, 1)
23= 8	(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)
32= 9	(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)
41= 4	(1), (2), (3), (4)
50= 1	()

Particular bases

Powers of ten

See also:Scientific notation

Main article:Power of 10

In the base ten (decimal) number system, integer powers of10are written as the digit1followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example,10³=1000and10⁻⁴=0.0001.

Exponentiation with base10is used inscientific notationto denote large or small numbers. For instance,299792458m/s(thespeed of lightin vacuum, inmetres per second) can be written as2.99792458×10⁸m/sand thenapproximatedas2.998×10⁸m/s.

SI prefixesbased on powers of10are also used to describe small or large quantities. For example, the prefixkilomeans10³=1000,so a kilometre is1000 m.

Powers of two

Main article:Power of two

The first negative powers of2have special names:$2^{-1}$is ahalf;$2^{-2}$is aquarter.

Powers of2appear inset theory,since a set withnmembers has apower set,the set of all of itssubsets,which has2ⁿmembers.

Integer powers of2are important incomputer science.The positive integer powers2ⁿgive the number of possible values for ann-bitintegerbinary number;for example, abytemay take2⁸= 256different values. Thebinary number systemexpresses any number as a sum of powers of2,and denotes it as a sequence of0and1,separated by abinary point,where1indicates a power of2that appears in the sum; the exponent is determined by the place of this1:the nonnegative exponents are the rank of the1on the left of the point (starting from0), and the negative exponents are determined by the rank on the right of the point.

Powers of one

Every power of one equals:1ⁿ= 1.

Powers of zero

For a positive exponentn> 0,thenth power of zero is zero:0ⁿ= 0.For a negative\ exponent,$0^{-n}=1/0^{n}=1/0$is undefined.

The expression0⁰is eitherdefined as$\lim _{x\to 0}x^{x}=1$,or it is left undefined.

Powers of negative one

Since a negative number times another negative is positive, we have:

$(-1)^{n}=\left\{{\begin{array}{rl}1&{\text{for even }}n,\\-1&{\text{for odd }}n.\\\end{array}}\right.$

Because of this, powers of−1are useful for expressing alternatingsequences.For a similar discussion of powers of the complex numberi,see§ nth roots of a complex number.

Large exponents

Thelimit of a sequenceof powers of a number greater than one diverges; in other words, the sequence grows without bound:

bⁿ→ ∞asn→ ∞whenb> 1

This can be read as "bto the power ofntends to+∞asntends to infinity whenbis greater than one ".

Powers of a number withabsolute valueless than one tend to zero:

bⁿ→ 0asn→ ∞when|b| < 1

Any power of one is always one:

bⁿ= 1for allnforb= 1

Powers of a negative number$b\leq -1$alternate between positive and negative asnalternates between even and odd, and thus do not tend to any limit asngrows.

If the exponentiated number varies while tending to1as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is

(1 + 1/n)ⁿ→easn→ ∞

See§ Exponential functionbelow.

Other limits, in particular those of expressions that take on anindeterminate form,are described in§ Limits of powersbelow.

Power functions

Main article:Power law

Power functions forn= 1, 3, 5

Power functions forn= 2, 4, 6

Real functions of the form$f(x)=cx^{n}$,where$c\neq 0$,are sometimes called power functions.When$n$is anintegerand$n\geq 1$,two primary families exist: for$n$even, and for$n$odd. In general for$c>0$,when$n$is even$f(x)=cx^{n}$will tend towards positiveinfinitywith increasing$x$,and also towards positive infinity with decreasing$x$.All graphs from the family of even power functions have the general shape of$y=cx^{2}$,flattening more in the middle as$n$increases.Functions with this kind ofsymmetry($f(-x)=f(x)$)are calledeven functions.

When$n$is odd,$f(x)$'sasymptoticbehavior reverses from positive$x$to negative$x$.For$c>0$,$f(x)=cx^{n}$will also tend towards positiveinfinitywith increasing$x$,but towards negative infinity with decreasing$x$.All graphs from the family of odd power functions have the general shape of$y=cx^{3}$,flattening more in the middle as$n$increases and losing all flatness there in the straight line for$n=1$.Functions with this kind of symmetry($f(-x)=-f(x)$)are calledodd functions.

For$c<0$,the opposite asymptotic behavior is true in each case.

Table of powers of decimal digits

n	n2	n3	n4	n5	n6	n7	n8	n9	n10
1	1	1	1	1	1	1	1	1	1
2	4	8	16	32	64	128	256	512	1024
3	9	27	81	243	729	2187	6561	19683	59049
4	16	64	256	1024	4096	16384	65536	262144	1048576
5	25	125	625	3125	15625	78125	390625	1953125	9765625
6	36	216	1296	7776	46656	279936	1679616	10077696	60466176
7	49	343	2401	16807	117649	823543	5764801	40353607	282475249
8	64	512	4096	32768	262144	2097152	16777216	134217728	1073741824
9	81	729	6561	59049	531441	4782969	43046721	387420489	3486784401
10	100	1000	10000	100000	1000000	10000000	100000000	1000000000	10000000000

Rational exponents

From top to bottom:x^1/8,x^1/4,x^1/2,x¹,x²,x⁴,x⁸.

Ifxis a nonnegativereal number,andnis a positive integer,$x^{1/n}$or${\sqrt[{n}]{x}}$denotes the unique nonnegative realnth rootofx,that is, the unique nonnegative real numberysuch that$y^{n}=x.$

Ifxis a positive real number, and${\frac {p}{q}}$is arational number,withpandq > 0integers, then${\textstyle x^{p/q}}$is defined as

$x^{\frac {p}{q}}=\left(x^{p}\right)^{\frac {1}{q}}=(x^{\frac {1}{q}})^{p}.$

The equality on the right may be derived by setting$y=x^{\frac {1}{q}},$and writing$(x^{\frac {1}{q}})^{p}=y^{p}=\left((y^{p})^{q}\right)^{\frac {1}{q}}=\left((y^{q})^{p}\right)^{\frac {1}{q}}=(x^{p})^{\frac {1}{q}}.$

Ifris a positive rational number,0^r= 0,by definition.

All these definitions are required for extending the identity$(x^{r})^{s}=x^{rs}$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 realnth root, which is negative, ifnisodd,and no real root ifnis even. In the latter case, whichever complexnth root one chooses for$x^{\frac {1}{n}},$the identity$(x^{a})^{b}=x^{ab}$cannot be satisfied. For example,

$\left((-1)^{2}\right)^{\frac {1}{2}}=1^{\frac {1}{2}}=1\neq (-1)^{2\cdot {\frac {1}{2}}}=(-1)^{1}=-1.$

See§ Real exponentsand§ Non-integer powers of complex numbersfor 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 (§ Limits of rational exponents,below), or in terms of thelogarithmof the base and theexponential function(§ Powers via logarithms,below). The result is always a positive real number, and theidentities and propertiesshown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly tocomplexexponents.

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. One may choose one of these values, called theprincipal value,but there is no choice of the principal value for which the identity

$\left(b^{r}\right)^{s}=b^{rs}$

is true; see§ Failure of power and logarithm identities.Therefore, exponentiation with a basis that is not a positive real number is generally viewed as amultivalued function.

Limits of rational exponents

The limit ofe^1/nise⁰= 1whenntends to the infinity.

Since anyirrational numbercan be expressed as thelimit of a sequenceof rational numbers, exponentiation of a positive real numberbwith an arbitrary real exponentxcan be defined bycontinuitywith the rule

$b^{x}=\lim _{r(\in \mathbb {Q} )\to x}b^{r}\quad (b\in \mathbb {R} ^{+},\,x\in \mathbb {R} ),$

where the limit is taken over rational values ofronly. This limit exists for every positiveband every realx.

For example, ifx=π,thenon-terminating decimalrepresentationπ= 3.14159...and themonotonicityof the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain$b^{\pi }:$

$\left[b^{3},b^{4}\right],\left[b^{3.1},b^{3.2}\right],\left[b^{3.14},b^{3.15}\right],\left[b^{3.141},b^{3.142}\right],\left[b^{3.1415},b^{3.1416}\right],\left[b^{3.14159},b^{3.14160}\right],\ldots$

So, the upper bounds and the lower bounds of the intervals form twosequencesthat have the same limit, denoted$b^{\pi }.$

This defines$b^{x}$for every positiveband realxas acontinuous functionofbandx.See alsoWell-defined expression.

Exponential function

Main article:Exponential function

Theexponential functionmay be defined as$x\mapsto e^{x},$where$e\approx 2.718$isEuler's number,but to avoidcircular reasoning,this definition cannot be used here. Rather, we give an independent definition of the exponential function$\exp(x),$and of$e=\exp(1)$,relying only on positive integer powers (repeated multiplication). Then we sketch the proof that this agrees with the previous definition:$\exp(x)=e^{x}.$

There aremany equivalent ways to define the exponential function,one of them being

$\exp(x)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}.$

One has$\exp(0)=1,$and theexponential identity(or multiplication rule)$\exp(x)\exp(y)=\exp(x+y)$holds as well, since

$\exp(x)\exp(y)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}\left(1+{\frac {y}{n}}\right)^{n}=\lim _{n\rightarrow \infty }\left(1+{\frac {x+y}{n}}+{\frac {xy}{n^{2}}}\right)^{n},$

and the second-order term${\frac {xy}{n^{2}}}$does not affect the limit, yielding$\exp(x)\exp(y)=\exp(x+y)$.

Euler's number can be defined as$e=\exp(1)$.It follows from the preceding equations that$\exp(x)=e^{x}$whenxis an integer (this results from the repeated-multiplication definition of the exponentiation). Ifxis real,$\exp(x)=e^{x}$results from the definitions given in preceding sections, by using the exponential identity ifxis rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for everycomplexvalue ofx,and therefore it can be used to extend the definition of$\exp(z)$,and thus$e^{z},$from the real numbers to any complex argumentz.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 ofe^xas the exponential function allows definingb^xfor every positive real numbersb,in terms of exponential andlogarithmfunction. Specifically, the fact that thenatural logarithmln(x)is theinverseof the exponential functione^xmeans that one has

$b=\exp(\ln b)=e^{\ln b}$

for everyb> 0.For preserving the identity$(e^{x})^{y}=e^{xy},$one must have

$b^{x}=\left(e^{\ln b}\right)^{x}=e^{x\ln b}$

So,$e^{x\ln b}$can be used as an alternative definition ofb^xfor any positive realb.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

Ifbis a positive real number, exponentiation with basebandcomplexexponentzis defined by means of the exponential function with complex argument (see the end of§ Exponential function,above) as

$b^{z}=e^{(z\ln b)},$

where$\ln b$denotes thenatural logarithmofb.

This satisfies the identity

$b^{z+t}=b^{z}b^{t},$

In general, ${\textstyle \left(b^{z}\right)^{t}}$is not defined, sinceb^zis not a real number. If a meaning is given to the exponentiation of a complex number (see§ Non-integer powers of complex numbers,below), one has, in general,

$\left(b^{z}\right)^{t}\neq b^{zt},$

unlesszis real ortis an integer.

Euler's formula,

$e^{iy}=\cos y+i\sin y,$

allows expressing thepolar formof$b^{z}$in terms of thereal and imaginary partsofz,namely

$b^{x+iy}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)),$

where theabsolute valueof thetrigonometricfactor is one. This results from

$b^{x+iy}=b^{x}b^{iy}=b^{x}e^{iy\ln b}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)).$

Non-integer exponents with a complex base

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 ofnth roots, that is, of exponents$1/n,$wherenis a positive integer. Although the general theory of exponentiation with non-integer exponents applies tonth roots, this case deserves to be considered first, since it does not need to usecomplex logarithms,and is therefore easier to understand.

nth roots of a complex number

Every nonzero complex numberzmay be written inpolar formas

$z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),$

where$\rho$is theabsolute valueofz,and$\theta$is itsargument.The argument is definedup toan integer multiple of2π;this means that, if$\theta$is the argument of a complex number, then$\theta +2k\pi$is also an argument of the same complex number for every integer$k$.

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 annth root of a complex number can be obtained by taking thenth root of the absolute value and dividing its argument byn:

$\left(\rho e^{i\theta }\right)^{\frac {1}{n}}={\sqrt[{n}]{\rho }}\,e^{\frac {i\theta }{n}}.$

If$2\pi$is added to$\theta$,the complex number is not changed, but this adds$2i\pi /n$to the argument of thenth root, and provides a newnth root. This can be donentimes, and provides thennth roots of the complex number.

It is usual to choose one of thennth root as theprincipal root.The common choice is to choose thenth root for which$-\pi <\theta \leq \pi,$that is, thenth root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principalnth root acontinuous functionin the whole complex plane, except for negative real values of theradicand.This function equals the usualnth root for positive real radicands. For negative real radicands, and odd exponents, the principalnth root is not real, although the usualnth root is real.Analytic continuationshows that the principalnth root is the uniquecomplex differentiablefunction that extends the usualnth 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$2\pi,$the complex number comes back to its initial position, and itsnth roots arepermuted circularly(they are multiplied by$e^{2i\pi /n}$). This shows that it is not possible to define anth root function that is continuous in the whole complex plane.

Roots of unity

Main article:Root of unity

The three third roots of1

Thenth roots of unity are thencomplex numbers such thatwⁿ= 1,wherenis a positive integer. They arise in various areas of mathematics, such as indiscrete Fourier transformor algebraic solutions of algebraic equations (Lagrange resolvent).

Thennth roots of unity are thenfirst powers of$\omega =e^{\frac {2\pi i}{n}}$,that is$1=\omega ^{0}=\omega ^{n},\omega =\omega ^{1},\omega ^{2},\omega ^{n-1}.$Thenth roots of unity that have this generating property are calledprimitiventh roots of unity;they have the form$\omega ^{k}=e^{\frac {2k\pi i}{n}},$withkcoprimewithn.The unique primitive square root of unity is$-1;$the primitive fourth roots of unity are$i$and$-i.$

Thenth roots of unity allow expressing allnth roots of a complex numberzas thenproducts of a givennth roots ofzwith anth root of unity.

Geometrically, thenth roots of unity lie on theunit circleof thecomplex planeat the vertices of aregularn-gonwith one vertex on the real number 1.

As the number$e^{\frac {2k\pi i}{n}}$is the primitiventh root of unity with the smallest positiveargument,it is called theprincipal primitiventh root of unity,sometimes shortened asprincipalnth root of unity,although this terminology can be confused with theprincipal valueof$1^{1/n}$,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$z^{w}$.So, either aprincipal valueis defined, which is not continuous for the values ofzthat are real and nonpositive, or$z^{w}$is defined as amultivalued function.

In all cases, thecomplex logarithmis used to define complex exponentiation as

$z^{w}=e^{w\log z},$

where$\log z$is the variant of the complex logarithm that is used, which is a function or amultivalued functionsuch that

$e^{\log z}=z$

for everyzin itsdomain of definition.

Principal value

Theprincipal valueof thecomplex logarithmis the unique continuous function, commonly denoted$\log,$such that, for every nonzero complex numberz,

$e^{\log z}=z,$

and theargumentofzsatisfies

$-\pi <\operatorname {Arg} z\leq \pi.$

The principal value of the complex logarithm is not defined for$z=0,$it isdiscontinuousat negative real values ofz,and it isholomorphic(that is, complex differentiable) elsewhere. Ifzis real and positive, the principal value of the complex logarithm is the natural logarithm:$\log z=\ln z.$

The principal value of$z^{w}$is defined as $z^{w}=e^{w\log z},$ where$\log z$is the principal value of the logarithm.

The function$(z,w)\to z^{w}$is holomorphic except in the neighbourhood of the points wherezis real and nonpositive.

Ifzis real and positive, the principal value of$z^{w}$equals its usual value defined above. If$w=1/n,$wherenis 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$\log z$and$z^{w}$at the negative real values ofz.In this case, it is useful to consider these functions asmultivalued functions.

If$\log z$denotes one of the values of the multivalued logarithm (typically its principal value), the other values are$2ik\pi +\log z,$wherekis any integer. Similarly, if$z^{w}$is one value of the exponentiation, then the other values are given by

$e^{w(2ik\pi +\log z)}=z^{w}e^{2ik\pi w},$

wherekis any integer.

Different values ofkgive different values of$z^{w}$unlesswis arational number,that is, there is an integerdsuch thatdwis an integer. This results from theperiodicityof the exponential function, more specifically, that$e^{a}=e^{b}$if and only if$a-b$is an integer multiple of$2\pi i.$

If$w={\frac {m}{n}}$is a rational number withmandncoprime integerswith$n>0,$then$z^{w}$has exactlynvalues. In the case$m=1,$these values are the same as those described in§nth roots of a complex number.Ifwis an integer, there is only one value that agrees with that of§ Integer exponents.

The multivalued exponentiation is holomorphic for$z\neq 0,$in the sense that itsgraphconsists of several sheets that define each a holomorphic function in the neighborhood of every point. Ifzvaries continuously along a circle around0,then, after a turn, the value of$z^{w}$has changed of sheet.

Computation

Thecanonical form$x+iy$of$z^{w}$can be computed from the canonical form ofzandw.Although this can be described by a single formula, it is clearer to split the computation in several steps.

• Polar formofz.If$z=a+ib$is the canonical form ofz(aandbbeing real), then its polar form is$$z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),$$with${\textstyle \rho ={\sqrt {a^{2}+b^{2}}}}$and$\theta =\operatorname {atan2} (b,a)$,where⁠$\operatorname {atan2}$⁠is thetwo-argument arctangentfunction.
• Logarithmofz.Theprincipal valueof this logarithm is$\log z=\ln \rho +i\theta,$where$\ln$denotes thenatural logarithm.The other values of the logarithm are obtained by adding$2ik\pi$for any integerk.
• Canonical form of$w\log z.$If$w=c+di$withcanddreal, the values of$w\log z$are$$w\log z=(c\ln \rho -d\theta -2dk\pi )+i(d\ln \rho +c\theta +2ck\pi ),$$the principal value corresponding to$k=0.$
• Final result.Using the identities$e^{x+y}=e^{x}e^{y}$and$e^{y\ln x}=x^{y},$one gets$$z^{w}=\rho ^{c}e^{-d(\theta +2k\pi )}\left(\cos(d\ln \rho +c\theta +2ck\pi )+i\sin(d\ln \rho +c\theta +2ck\pi )\right),$$with$k=0$for the principal value.

Examples

• $i^{i}$
  The polar form ofiis$i=e^{i\pi /2},$and the values of$\log i$are thus$$\log i=i\left({\frac {\pi }{2}}+2k\pi \right).$$It follows that$$i^{i}=e^{i\log i}=e^{-{\frac {\pi }{2}}}e^{-2k\pi }.$$So, all values of$i^{i}$are real, the principal one being$$e^{-{\frac {\pi }{2}}}\approx 0.2079.$$
• $(-2)^{3+4i}$
  Similarly, the polar form of−2is$-2=2e^{i\pi }.$So, the above described method gives the values$${\begin{aligned}(-2)^{3+4i}&=2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2+3(\pi +2k\pi ))+i\sin(4\ln 2+3(\pi +2k\pi )))\\&=-2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2)+i\sin(4\ln 2)).\end{aligned}}$$In this case, all the values have the same argument$4\ln 2,$and different absolute values.

In both examples, all values of$z^{w}$have the same argument. More generally, this is true if and only if thereal partofwis 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 definedas single-valued functions.For example:

• The identitylog(b^x) =x⋅ log bholds wheneverbis a positive real number andxis a real number. But for theprincipal branchof the complex logarithm one has

  $$\log((-i)^{2})=\log(-1)=i\pi \neq 2\log(-i)=2\log(e^{-i\pi /2})=2\,{\frac {-i\pi }{2}}=-i\pi$$

  Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that: $$\log w^{z}\equiv z\log w{\pmod {2\pi i}}$$

  This identity does not hold even when considering log as a multivalued function. The possible values oflog(w^z)contain those ofz⋅ log was aproper subset.UsingLog(w)for the principal value oflog(w)andm,nas any integers the possible values of both sides are:

  $${\begin{aligned}\left\{\log w^{z}\right\}&=\left\{z\cdot \operatorname {Log} w+z\cdot 2\pi in+2\pi im\mid m,n\in \mathbb {Z} \right\}\\\left\{z\log w\right\}&=\left\{z\operatorname {Log} w+z\cdot 2\pi in\mid n\in \mathbb {Z} \right\}\end{aligned}}$$
• The identities(bc)^x=b^xc^xand(b/c)^x=b^x/c^xare valid whenbandcare positive real numbers andxis a real number. But, for the principal values, one has $$(-1\cdot -1)^{\frac {1}{2}}=1\neq (-1)^{\frac {1}{2}}(-1)^{\frac {1}{2}}=i\cdot i=i^{2}=-1$$ and $$\left({\frac {1}{-1}}\right)^{\frac {1}{2}}=(-1)^{\frac {1}{2}}=i\neq {\frac {1^{\frac {1}{2}}}{(-1)^{\frac {1}{2}}}}={\frac {1}{i}}=-i$$ On the other hand, whenxis an integer, the identities are valid for all nonzero complex numbers. If exponentiation is considered as a multivalued function then the possible values of(−1 ⋅ −1)^1/2are{1, −1}.The identity holds, but saying{1} = {(−1 ⋅ −1)^1/2}is incorrect.
• The identity(e^x)^y=e^xyholds for real numbersxandy,but assuming its truth for complex numbers leads to the followingparadox,discovered in 1827 byClausen: For any integern,we have:
  1. $e^{1+2\pi in}=e^{1}e^{2\pi in}=e\cdot 1=e$
  2. $\left(e^{1+2\pi in}\right)^{1+2\pi in}=e\qquad$(taking the$(1+2\pi in)$-th power of both sides)
  3. $e^{1+4\pi in-4\pi ^{2}n^{2}}=e\qquad$(using$\left(e^{x}\right)^{y}=e^{xy}$and expanding the exponent)
  4. $e^{1}e^{4\pi in}e^{-4\pi ^{2}n^{2}}=e\qquad$(using$e^{x+y}=e^{x}e^{y}$)
  5. $e^{-4\pi ^{2}n^{2}}=1\qquad$(dividing bye)
  but this is false when the integernis nonzero. The error is the following: by definition,$e^{y}$is a notation for$\exp(y),$a true function, and$x^{y}$is a notation for$\exp(y\log x),$which is a multi-valued function. Thus the notation is ambiguous whenx=e.Here, before expanding the exponent, the second line should be $$\exp \left((1+2\pi in)\log \exp(1+2\pi in)\right)=\exp(1+2\pi in).$$ Therefore, when expanding the exponent, one has implicitly supposed that$\log \exp z=z$for complex values ofz,which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity(e^x)^y=e^xymust be replaced by the identity $$\left(e^{x}\right)^{y}=e^{y\log e^{x}},$$ which is a true identity between multivalued functions.

Irrationality and transcendence

Main article:Gelfond–Schneider theorem

Ifbis a positive realalgebraic number,andxis a rational number, thenb^xis an algebraic number. This results from the theory ofalgebraic extensions.This remains true ifbis any algebraic number, in which case, all values ofb^x(as amultivalued function) are algebraic. Ifxisirrational(that is,not rational), and bothbandxare algebraic, Gelfond–Schneider theorem asserts that all values ofb^xaretranscendental(that is, not algebraic), except ifbequals0or1.

In other words, ifxis irrational and$b\not \in \{0,1\},$then at least one ofb,xandb^xis transcendental.

Integer powers in algebra

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to anyassociative operationdenoted as a multiplication.The definition ofx⁰requires further the existence of amultiplicative identity.

Analgebraic structureconsisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by1is amonoid.In such a monoid, exponentiation of an elementxis defined inductively by

• $x^{0}=1,$
• $x^{n+1}=xx^{n}$for every nonnegative integern.

Ifnis a negative integer,$x^{n}$is defined only ifxhas amultiplicative inverse.In this case, the inverse ofxis denotedx⁻¹,andxⁿis defined as$\left(x^{-1}\right)^{-n}.$

Exponentiation with integer exponents obeys the following laws, forxandyin the algebraic structure, andmandnintegers:

${\begin{aligned}x^{0}&=1\\x^{m+n}&=x^{m}x^{n}\\(x^{m})^{n}&=x^{mn}\\(xy)^{n}&=x^{n}y^{n}\quad {\text{if }}xy=yx,{\text{and, in particular, if the multiplication is commutative.}}\end{aligned}}$

These definitions are widely used in many areas of mathematics, notably forgroups,rings,fields,square matrices(which form a ring). They apply also tofunctionsfrom asetto itself, which form a monoid underfunction composition.This includes, as specific instances,geometric transformations,andendomorphismsof anymathematical structure.

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, iffis areal functionwhose valued can be multiplied,$f^{n}$denotes the exponentiation with respect of multiplication, and$f^{\circ n}$may denote exponentiation with respect offunction composition.That is,

$(f^{n})(x)=(f(x))^{n}=f(x)\,f(x)\cdots f(x),$

and

$(f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots )).$

Commonly,$(f^{n})(x)$is denoted$f(x)^{n},$while$(f^{\circ n})(x)$is denoted$f^{n}(x).$

In a group

Amultiplicative groupis a set with asassociative operationdenoted as multiplication, that has anidentity element,and such that every element has an inverse.

So, ifGis a group,$x^{n}$is defined for every$x\in G$and every integern.

The set of all powers of an element of a group form asubgroup.A group (or subgroup) that consists of all powers of a specific elementxis thecyclic groupgenerated byx.If all the powers ofxare distinct, the group isisomorphicto theadditive group$\mathbb {Z}$of the integers. Otherwise, the cyclic group isfinite(it has a finite number of elements), and its number of elements is theorderofx.If the order ofxisn,then$x^{n}=x^{0}=1,$and the cyclic group generated byxconsists of thenfirst powers ofx(starting indifferently from the exponent0or1).

Order of elements play a fundamental role ingroup theory.For example, the order of an element in a finite group is always a divisor of the number of elements of the group (theorderof the group). The possible orders of group elements are important in the study of the structure of a group (seeSylow theorems), and in theclassification of finite simple groups.

Superscript notation is also used forconjugation;that is,g^h=h⁻¹gh,wheregandhare 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$(g^{h})^{k}=g^{hk}$and$(gh)^{k}=g^{k}h^{k}.$

In a ring

In aring,it may occur that some nonzero elements satisfy$x^{n}=0$for some integern.Such an element is said to benilpotent.In acommutative ring,the nilpotent elements form anideal,called thenilradicalof the ring.

If the nilradical is reduced to thezero ideal(that is, if$x\neq 0$implies$x^{n}\neq 0$for every positive integern), the commutative ring is said to bereduced.Reduced rings are important inalgebraic geometry,since thecoordinate ringof anaffine algebraic setis always a reduced ring.

More generally, given an idealIin a commutative ringR,the set of the elements ofRthat have a power inIis an ideal, called theradicalofI.The nilradical is the radical of thezero ideal.Aradical idealis an ideal that equals its own radical. In apolynomial ring$k[x_{1},\ldots,x_{n}]$over afieldk,an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence ofHilbert's Nullstellensatz).

Matrices and linear operators

IfAis a square matrix, then the product ofAwith itselfntimes is called thematrix power.Also$A^{0}$is defined to be the identity matrix,and ifAis invertible, then$A^{-n}=\left(A^{-1}\right)^{n}$.

Matrix powers appear often in the context ofdiscrete dynamical systems,where the matrixAexpresses a transition from a state vectorxof some system to the next stateAxof the system.This is the standard interpretation of aMarkov chain,for example. Then$A^{2}x$is the state of the system after two time steps, and so forth:$A^{n}x$is the state of the system afterntime steps. The matrix power$A^{n}$is the transition matrix between the state now and the state at a timensteps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by usingeigenvalues and eigenvectors.

Apart from matrices, more generallinear operatorscan also be exponentiated. An example is thederivativeoperator of calculus,$d/dx$,which is a linear operator acting on functions$f(x)$to give a new function$(d/dx)f(x)=f'(x)$.Thenth power of the differentiation operator is thenth derivative:

$\left({\frac {d}{dx}}\right)^{n}f(x)={\frac {d^{n}}{dx^{n}}}f(x)=f^{(n)}(x).$

These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory ofsemigroups.Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving theheat equation,Schrödinger equation,wave equation,and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called thefractional derivativewhich, together with thefractional integral,is one of the basic operations of thefractional calculus.

Finite fields

Main article:Finite field

Afieldis an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication isassociativeand every nonzero element has amultiplicative inverse.This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of0.Common examples are the field ofcomplex numbers,thereal numbersand therational numbers,considered earlier in this article, which are allinfinite.

Afinite fieldis a field with afinite numberof elements. This number of elements is either aprime numberor aprime power;that is, it has the form$q=p^{k},$wherepis a prime number, andkis a positive integer. For every suchq,there are fields withqelements. The fields withqelements are allisomorphic,which allows, in general, working as if there were only one field withqelements, denoted$\mathbb {F} _{q}.$

One has

$x^{q}=x$

for every$x\in \mathbb {F} _{q}.$

Aprimitive elementin$\mathbb {F} _{q}$is an elementgsuch that the set of theq− 1first powers ofg(that is,$\{g^{1}=g,g^{2},\ldots,g^{p-1}=g^{0}=1\}$) equals the set of the nonzero elements of$\mathbb {F} _{q}.$There are$\varphi (p-1)$primitive elements in$\mathbb {F} _{q},$where$\varphi$isEuler's totient function.

In$\mathbb {F} _{q},$thefreshman's dreamidentity

$(x+y)^{p}=x^{p}+y^{p}$

is true for the exponentp.As$x^{p}=x$in$\mathbb {F} _{q},$It follows that the map

${\begin{aligned}F\colon {}&\mathbb {F} _{q}\to \mathbb {F} _{q}\\&x\mapsto x^{p}\end{aligned}}$

islinearover$\mathbb {F} _{q},$and is afield automorphism,called theFrobenius automorphism.If$q=p^{k},$the field$\mathbb {F} _{q}$haskautomorphisms, which are thekfirst powers (undercomposition) ofF.In other words, theGalois groupof$\mathbb {F} _{q}$iscyclicof orderk,generated by the Frobenius automorphism.

TheDiffie–Hellman key exchangeis an application of exponentiation in finite fields that is widely used forsecure communications.It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, thediscrete logarithm,is computationally expensive. More precisely, ifgis a primitive element in$\mathbb {F} _{q},$then$g^{e}$can be efficiently computed withexponentiation by squaringfor anye,even ifqis large, while there is no known computationally practical algorithm that allows retrievingefrom$g^{e}$ifqis sufficiently large.

Powers of sets

TheCartesian productof twosetsSandTis the set of theordered pairs$(x,y)$such that$x\in S$and$y\in T.$This operation is not properlycommutativenorassociative,but has these propertiesup tocanonicalisomorphisms,that allow identifying, for example,$(x,(y,z)),$$((x,y),z),$and$(x,y,z).$

This allows defining thenth power$S^{n}$of a setSas the set of alln-tuples$(x_{1},\ldots,x_{n})$of elements ofS.

WhenSis endowed with some structure, it is frequent that$S^{n}$is naturally endowed with a similar structure. In this case, the term "direct product"is generally used instead of" Cartesian product ", and exponentiation denotes product structure. For example$\mathbb {R} ^{n}$(where$\mathbb {R}$denotes the real numbers) denotes the Cartesian product ofncopies of$\mathbb {R},$as well as their direct product asvector space,topological spaces,rings,etc.

Sets as exponents

See also:Function (mathematics) § Set exponentiation

An-tuple$(x_{1},\ldots,x_{n})$of elements ofScan be considered as afunctionfrom$\{1,\ldots,n\}.$This generalizes to the following notation.

Given two setsSandT,the set of all functions fromTtoSis denoted$S^{T}$.This exponential notation is justified by the following canonical isomorphisms (for the first one, seeCurrying):

$(S^{T})^{U}\cong S^{T\times U},$

$S^{T\sqcup U}\cong S^{T}\times S^{U},$

where$\times$denotes the Cartesian product, and$\sqcup$thedisjoint union.

One can use sets as exponents for other operations on sets, typically fordirect sumsofabelian groups,vector spaces,ormodules.For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example,$\mathbb {R} ^{\mathbb {N} }$denotes the vector space of theinfinite sequencesof real numbers, and$\mathbb {R} ^{(\mathbb {N} )}$the vector space of those sequences that have a finite number of nonzero elements. The latter has abasisconsisting of the sequences with exactly one nonzero element that equals1,while theHamel basesof the former cannot be explicitly described (because their existence involvesZorn's lemma).

In this context,2can represents the set$\{0,1\}.$So,$2^{S}$denotes thepower setofS,that is the set of the functions fromSto$\{0,1\},$which can be identified with the set of thesubsetsofS,by mapping each function to theinverse imageof1.

This fits in with theexponentiation of cardinal numbers,in the sense that|S^T| = |S|^|T|,where|X|is the cardinality ofX.

In category theory

Main article:Cartesian closed category

In thecategory of sets,themorphismsbetween setsXandYare the functions fromXtoY.It results that the set of the functions fromXtoYthat is denoted$Y^{X}$in the preceding section can also be denoted$\hom(X,Y).$The isomorphism$(S^{T})^{U}\cong S^{T\times U}$can be rewritten

$\hom(U,S^{T})\cong \hom(T\times U,S).$

This means the functor "exponentiation to the powerT "is aright adjointto the functor "direct product withT ".

This generalizes to the definition ofexponentiation in a categoryin which finitedirect productsexist: in such a category, the functor$X\to X^{T}$is, if it exists, a right adjoint to the functor$Y\to T\times Y.$A category is called aCartesian closed category,if direct products exist, and the functor$Y\to X\times Y$has a right adjoint for everyT.

Repeated exponentiation

Main articles:TetrationandHyperoperation

Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 ortetration.Iterating tetration leads to another operation, and so on, a concept namedhyperoperation.This sequence of operations is expressed by theAckermann functionandKnuth's up-arrow notation.Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at(3, 3),the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and7625597484987(=3²⁷= 3³³=³3) respectively.

Limits of powers

Zero to the power of zerogives a number of examples of limits that are of theindeterminate form0⁰.The limits in these examples exist, but have different values, showing that the two-variable functionx^yhas no limit at the point(0, 0).One may consider at what points this function does have a limit.

More precisely, consider the function$f(x,y)=x^{y}$defined on$D=\{(x,y)\in \mathbf {R} ^{2}:x>0\}$.ThenDcan be viewed as a subset ofR²(that is, the set of all pairs(x,y)withx,ybelonging to theextended real number lineR= [−∞, +∞],endowed with theproduct topology), which will contain the points at which the functionfhas a limit.

In fact,fhas a limit at allaccumulation pointsofD,except for(0, 0),(+∞, 0),(1, +∞)and(1, −∞).Accordingly, this allows one to define the powersx^yby continuity whenever0 ≤x≤ +∞,−∞ ≤ y ≤ +∞,except for0⁰,(+∞)⁰,1^+∞and1^−∞,which remain indeterminate forms.

Under this definition by continuity, we obtain:

• x^+∞= +∞andx^−∞= 0,when1 <x≤ +∞.
• x^+∞= 0andx^−∞= +∞,when0 ≤x< 1.
• 0^y= 0and(+∞)^y= +∞,when0 <y≤ +∞.
• 0^y= +∞and(+∞)^y= 0,when−∞ ≤y< 0.

These powers are obtained by taking limits ofx^yforpositivevalues ofx.This method does not permit a definition ofx^ywhenx< 0,since pairs(x,y)withx< 0are not accumulation points ofD.

On the other hand, whennis an integer, the powerxⁿis already meaningful for all values ofx,including negative ones. This may make the definition0ⁿ= +∞obtained above for negativenproblematic whennis odd, since in this casexⁿ→ +∞asxtends to0through positive values, but not negative ones.

Efficient computation with integer exponents

Computingbⁿusing iterated multiplication requiresn− 1multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute2¹⁰⁰,applyHorner's ruleto the exponent 100 written in binary:

$100=2^{2}+2^{5}+2^{6}=2^{2}(1+2^{3}(1+2))$.

Then compute the following terms in order, reading Horner's rule from right to left.

22= 4

2 (22) = 23= 8

(23)2= 26= 64

(26)2= 212=4096

(212)2= 224=16777216

2 (224) = 225=33554432

(225)2= 250=1125899906842624

(250)2= 2100=1267650600228229401496703205376

This series of steps only requires 8 multiplications instead of 99.

In general, the number of multiplication operations required to computebⁿcan be reduced to$\sharp n+\lfloor \log _{2}n\rfloor -1,$by usingexponentiation by squaring,where$\sharp n$denotes the number of1s in thebinary representationofn.For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimaladdition-chain exponentiation.Finding theminimalsequence of multiplications (the minimal-length addition chain for the exponent) forbⁿis a difficult problem, for which no efficient algorithms are currently known (seeSubset sum problem), but many reasonably efficient heuristic algorithms are available.However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.

Iterated functions

See also:Iterated function

Function compositionis abinary operationthat is defined onfunctionssuch that thecodomainof the function written on the right is included in thedomainof the function written on the left. It is denoted$g\circ f,$and defined as

$(g\circ f)(x)=g(f(x))$

for everyxin the domain off.

If the domain of a functionfequals its codomain, one may compose the function with itself an arbitrary number of time, and this defines thenth power of the function under composition, commonly called thenth iterateof the function. Thus$f^{n}$denotes generally thenth iterate off;for example,$f^{3}(x)$means$f(f(f(x))).$

When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, thepointwise multiplication,which induces another exponentiation. When usingfunctional notation,the two kinds of exponentiation are generally distinguished by placing the exponent of the functional iterationbeforethe parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplicationafterthe parentheses. Thus$f^{2}(x)=f(f(x)),$and$f(x)^{2}=f(x)\cdot f(x).$When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example$f^{\circ 3}=f\circ f\circ f,$and$f^{3}=f\cdot f\cdot f.$For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically thetrigonometric functions.So,$\sin ^{2}x$and$\sin ^{2}(x)$both mean$\sin(x)\cdot \sin(x)$and not$\sin(\sin(x)),$which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.

In this context, the exponent$-1$denotes always theinverse function,if it exists. So$\sin ^{-1}x=\sin ^{-1}(x)=\arcsin x.$For themultiplicative inversefractions are generally used as in$1/\sin(x)={\frac {1}{\sin x}}.$

In programming languages

Programming languagesgenerally express exponentiation either as an infixoperatoror as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is thecaret(^). Theoriginal version of ASCIIincluded an uparrow symbol (↑), intended for exponentiation, but this wasreplaced by the caretin 1967, so the caret became usual in programming languages. The notations include:

• x ^ y:AWK,BASIC,J,MATLAB,Wolfram Language(Mathematica),R,Microsoft Excel,Analytica,TeX(and its derivatives),TI-BASIC,bc(for integer exponents),Haskell(for nonnegative integer exponents),Lua,and mostcomputer algebra systems.
• x ** y.TheFortrancharacter set did not include lowercase characters or punctuation symbols other than+-*/()&=.,'and so used**for exponentiation(the initial version useda xx binstead.). Many other languages followed suit:Ada,Z shell,KornShell,Bash,COBOL,CoffeeScript,Fortran,FoxPro,Gnuplot,Groovy,JavaScript,OCaml,ooRexx,F#,Perl,PHP,PL/I,Python,Rexx,Ruby,SAS,Seed7,Tcl,ABAP,Mercury,Haskell (for floating-point exponents),Turing,andVHDL.
• x ↑ y:Algol Reference language,Commodore BASIC,TRS-80 Level II/III BASIC.
• x ^^ y:Haskell (for fractional base, integer exponents),D.
• x⋆y:APL.

In most programming languages with an infix exponentiation operator, it isright-associative,that is,a^b^cis interpreted asa^(b^c).This is because(a^b)^cis equal toa^(b*c)and thus not as useful. In some languages, it is left-associative, notably inAlgol,MATLAB,and theMicrosoft Excelformula language.

Other programming languages use functional notation:

• (expt x y):Common Lisp.
• pown x y:F#(for integer base, integer exponent).

Still others only provide exponentiation as part of standardlibraries:

• pow(x, y):C,C++(inmathlibrary).
• Math.Pow(x, y):C#.
• math:pow(X, Y):Erlang.
• Math.pow(x, y):Java.
• [Math]::Pow(x, y):PowerShell.

In somestatically typedlanguages that prioritizetype safetysuch asRust,exponentiation is performed via a multitude of methods:

• x.pow(y)forxandyas integers
• x.powf(y)forxandyas floating point numbers
• x.powi(y)forxas a float andyas an integer

See also

• Double exponential function
• Exponential decay
• Exponential field
• Exponential growth
• Hyperoperation
• Tetration
• Pentation
• List of exponential topics
• Modular exponentiation
• Scientific notation
• Unicode subscripts and superscripts
• x^y=y^x
• Zero to the power of zero