Graphs ofy =bx for various basesb:base 10,base e,base 2, base 1/2. Each curve passes through the point(0, 1) because any nonzero number raised to the power of0 is1. Atx = 1, the value ofy equals the base because any number raised to the power of1 is the number itself.
Inmathematics,exponentiation, denotedbn, is anoperation involving two numbers: thebase,b, and theexponent orpower,n.[1] Whenn is a positiveinteger, exponentiation corresponds to repeatedmultiplication of the base: that is,bn is theproduct of multiplyingn bases:[1]In particular,.
The exponent is usually shown as asuperscript to the right of the base asbn or in computer code asb^n. Thisbinary operation is often read as "b to the powern"; it may also be referred to as "b raised to thenth power", "thenth power ofb",[2] or, most briefly, "b to then".
The above definition of immediately implies several properties, in particular the multiplication rule:[nb 1]
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, and dividing both sides by gives. That is, the multiplication rule implies the definitionA similar argument implies the definition for negative integer powers:That is, extending the multiplication rule gives. Dividing both sides by gives. This also implies the definition for fractional powers:For example,, meaning, which is the definition of square root:.
The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define for any positive real base and any real number exponent. More involved definitions allowcomplex base and exponent, as well as certain types ofmatrices as base or exponent.
InThe Sand Reckoner,Archimedes proved the law of exponents,10a · 10b = 10a+b, necessary to manipulate powers of10.[8] He then used powers of10 to estimate the number of grains of sand that can be contained in the universe.
In the 9th century, the Persian mathematicianAl-Khwarizmi used 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"[9]—and كَعْبَة (Kaʿbah, "cube") for acube, which laterIslamic mathematicians represented inmathematical notation as the lettersmīm (m) andkāf (k), respectively, by the 15th century, as seen in the work ofAbu'l-Hasan ibn Ali al-Qalasadi.[10]
Nicolas Chuquet used a form of exponential notation in the 15th century, for example122 to represent12x2.[11] This was later used byHenricus Grammateus andMichael Stifel in the 16th century. In the late 16th century,Jost Bürgi would use Roman numerals for exponents in a way similar to that of Chuquet, for exampleiii4 for4x3.[12]
The wordexponent was coined in 1544 by Michael Stifel.[13][14] In the 16th century,Robert Recorde used the terms "square", "cube", "zenzizenzic" (fourth power), "sursolid" (fifth), "zenzicube" (sixth), "second sursolid" (seventh), and "zenzizenzizenzic" (eighth).[9] "Biquadrate" has been used to refer to the fourth power as well.
In 1636,James Hume used in essence modern notation, when inL'algèbre de Viète he wroteAiii forA3.[15] Early in the 17th century, the first form of our modern exponential notation was introduced byRené Descartes in his text titledLa Géométrie; there, the notation is introduced in Book I.[16]
I designate ...aa, ora2 in multiplyinga by itself; anda3 in 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.
Samuel Jeake introduced the termindices in 1696.[6] The terminvolution was used synonymously with the termindices, but had declined in usage[17] and should not be confused withits more common meaning.
In 1748,Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing:
Consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are notalgebraic functions, since in those the exponents must be constant.[18]
As calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation. For exampleKonrad Zuse introducedfloating-point arithmetic in his 1938 computer Z1. Oneregister contained representation of leading digits, and a second contained representation of the exponent of 10. EarlierLeonardo Torres Quevedo contributedEssays on Automation (1914) which had suggested the floating-point representation of numbers. The more flexibledecimal floating-point representation was introduced in 1946 with aBell Laboratories computer. Eventually educators and engineers adoptedscientific notation of numbers, consistent with common reference toorder of magnitude in aratio scale.[19]
Exponents also came to be used to describeunits of measurement andquantity dimensions. For instance, sinceforce is 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–2 is used indimensional analysis to describe force.[22][23]
The expressionb2 =b ·b is called "thesquare ofb" or "b squared", because the area of a square with side-lengthb isb2. (It is true that it could also be called "b to the second power", but "the square ofb" and "b squared" are more traditional)
Similarly, the expressionb3 =b ·b ·b is called "thecube ofb" or "b cubed", because the volume of a cube with side-lengthb isb3.
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 base3 appears5 times in the multiplication, because the exponent is5. Here,243 is the5th power of 3, or3 raised to the 5th power.
The word "raised" is usually omitted, and sometimes "power" as well, so35 can be simply read "3 to the 5th", or "3 to the 5".
The definition of the exponentiation as an iterated multiplication can beformalized by usinginduction,[24] and this definition can be used as soon as one has anassociative multiplication:
As mentioned earlier, a (nonzero) number raised to the0 power is1:[25][1]
This value is also obtained by theempty product convention, which may be used in everyalgebraic structure with a multiplication that has anidentity. This way the formula
also holds for.
The case of00 is controversial. In contexts where only integer powers are considered, the value1 is generally assigned to00 but, 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.
Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity ().[26]
This definition of exponentiation with negative exponents is the only one that allows extending the identity to negative exponents (consider the case).
The followingidentities, often calledexponent rules, hold for all integer exponents, provided that the base is non-zero:[1]
Unlike addition and multiplication, exponentiation is notcommutative: for example,, but reversing the operands gives the different value. Also unlike addition and multiplication, exponentiation is notassociative: for example,(23)2 = 82 = 64, whereas2(32) = 29 = 512. Without parentheses, the conventionalorder of operations forserial exponentiation in superscript notation is top-down (orright-associative), not bottom-up[27][28][29] (orleft-associative). That is,
The powers of a sum can normally be computed from the powers of the summands by thebinomial formula
However, this formula is true only if the summands commute (i.e. thatab =ba), which is implied if they belong to astructure that iscommutative. Otherwise, ifa andb are, say,square matrices of the same size, this formula cannot be used. It follows that incomputer algebra, manyalgorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general purposecomputer algebra systems use a different notation (sometimes^^ instead of^) for exponentiation with non-commuting bases, which is then callednon-commutative exponentiation.
For nonnegative integersn andm, the value ofnm is the number offunctions from aset ofm elements to a set ofn elements (seecardinal exponentiation). Such functions can be represented asm-tuples from ann-element set (or asm-letter words from ann-letter alphabet). Some examples for particular values ofm andn are given in the following table:
nm
Thenm possiblem-tuples of elements from the set{1, ...,n}
In the base ten (decimal) number system, integer powers of10 are written as the digit1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example,103 =1000 and10−4 =0.0001.
SI prefixes based on powers of10 are also used to describe small or large quantities. For example, the prefixkilo means103 =1000, so a kilometre is1000 m.
The first negative powers of2 have special names:is ahalf; is aquarter.
Powers of2 appear inset theory, since a set withn members has apower set, the set of all of itssubsets, which has2n members.
Integer powers of2 are important incomputer science. The positive integer powers2n give the number of possible values for ann-bit integerbinary number; for example, abyte may take28 = 256 different values. Thebinary number system expresses any number as a sum of powers of2, and denotes it as a sequence of0 and1, separated by abinary point, where1 indicates a power of2 that appears in the sum; the exponent is determined by the place of this1: the nonnegative exponents are the rank of the1 on the left of the point (starting from0), and the negative exponents are determined by the rank on the right of the point.
Since a negative number times another negative is positive, we have:
Because of this, powers of−1 are useful for expressing alternatingsequences. For a similar discussion of powers of the complex numberi, see§ nth roots of a complex number.
Thelimit of a sequence of 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 "b to the power ofn tends to+∞ asn tends to infinity whenb is greater than one".
Powers of a number withabsolute value less than one tend to zero:
bn → 0 asn → ∞ when|b| < 1
Any power of one is always one:
bn = 1 for alln forb = 1
Powers of a negative number alternate between positive and negative asn alternates between even and odd, and thus do not tend to any limit asn grows.
If the exponentiated number varies while tending to1 as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is
Real functions of the form, where, are sometimes called power functions.[30] When is aninteger and, two primary families exist: for even, and for odd. In general for, when is even will tend towards positiveinfinity 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.[31] Functions with this kind ofsymmetry() are calledeven functions.
When is odd,'sasymptotic behavior reverses from positive to negative. For, will also tend towards positiveinfinity 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 calledodd functions.
For, the opposite asymptotic behavior is true in each case.[31]
Ifx is a nonnegativereal number, andn is a positive integer, or denotes the unique nonnegative realnth root ofx, that is, the unique nonnegative real numbery such that
Ifx is a positive real number, and is arational number, withp andq > 0 integers, then is defined as
The equality on the right may be derived by setting and writing
Ifr is a positive rational number,0r = 0, 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 realnth root, which is negative, ifn isodd, and no real root ifn is even. In the latter case, whichever complexnth root one chooses for the identity cannot be satisfied. For example,
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 thelogarithm of the base and theexponential function (§ Powers via logarithms, below). The result is always a positive real number, and theidentities 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 tocomplex 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. One may choose one of these values, called theprincipal value, but there is no choice of the principal value for which the identity
The limit ofe1/n ise0 = 1 whenn tends to the infinity.
Since anyirrational number can be expressed as thelimit of a sequence of rational numbers, exponentiation of a positive real numberb with an arbitrary real exponentx can be defined bycontinuity with the rule[32]
where the limit is taken over rational values ofr only. This limit exists for every positiveb and every realx.
For example, ifx =π, thenon-terminating decimal representationπ = 3.14159... and themonotonicity 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 twosequences that have the same limit, denoted
Theexponential function may be defined as where isEuler's number, but to avoidcircular reasoning, this definition cannot be used here. Rather, we give an independent definition of the exponential function and of, relying only on positive integer powers (repeated multiplication). Then we sketch the proof that this agrees with the previous definition:
One has and theexponential identity (or multiplication rule) 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 whenx is an integer (this results from the repeated-multiplication definition of the exponentiation). Ifx is real, results from the definitions given in preceding sections, by using the exponential identity ifx is rational, and the continuity of the exponential function otherwise.
The limit that defines the exponential function converges for everycomplex value ofx, and therefore it can be used to extend the definition of, and thus 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.
The definition ofex as the exponential function allows definingbx for every positive real numbersb, in terms of exponential andlogarithm function. Specifically, the fact that thenatural logarithmln(x) is theinverse of the exponential functionex means that one has
for everyb > 0. For preserving the identity one must have
So, can be used as an alternative definition ofbx for 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.
Ifb is a positive real number, exponentiation with baseb andcomplex exponentz is defined by means of the exponential function with complex argument (see the end of§ Exponential function, above) as
In general, is not defined, sincebz is 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,
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 wheren is 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.
Every nonzero complex numberz may be written inpolar form as
where is theabsolute value ofz, and is itsargument. The argument is definedup to an integer multiple of2π; this means that, if is the argument of a complex number, then is also an argument of the same complex number for every integer.
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:
If is added to, the complex number is not changed, but this adds to the argument of thenth root, and provides a newnth root. This can be donen times, 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 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 function in 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 continuation shows that the principalnth root is the uniquecomplex differentiable function 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 the complex number comes back to its initial position, and itsnth roots arepermuted circularly (they are multiplied by). This shows that it is not possible to define anth root function that is continuous in the whole complex plane.
Thenth roots of unity are then complex numbers such thatwn = 1, wheren is a positive integer. They arise in various areas of mathematics, such as indiscrete Fourier transform or algebraic solutions of algebraic equations (Lagrange resolvent).
Thennth roots of unity are then first powers of, that is Thenth roots of unity that have this generating property are calledprimitiventh roots of unity; they have the form withkcoprime withn. The unique primitive square root of unity is the primitive fourth roots of unity are and
Thenth roots of unity allow expressing allnth roots of a complex numberz as then products of a givennth roots ofz with anth root of unity.
Geometrically, thenth roots of unity lie on theunit circle of thecomplex plane at the vertices of aregularn-gon with one vertex on the real number 1.
As the number 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 value of, which is 1.[34][35][36]
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 aprincipal value is defined, which is not continuous for the values ofz that are real and nonpositive, or is defined as amultivalued function.
In all cases, thecomplex logarithm is used to define complex exponentiation as
where is the variant of the complex logarithm that is used, which is a function or amultivalued function such that
The principal value of the complex logarithm is not defined for it isdiscontinuous at negative real values ofz, and it isholomorphic (that is, complex differentiable) elsewhere. Ifz is real and positive, the principal value of the complex logarithm is the natural logarithm:
The principal value of is defined aswhere is the principal value of the logarithm.
The function is holomorphic except in the neighbourhood of the points wherez is real and nonpositive.
Ifz is real and positive, the principal value of equals its usual value defined above. If wheren is an integer, this principal value is the same as the one defined above.
In some contexts, there is a problem with the discontinuity of the principal values of and at the negative real values ofz. In this case, it is useful to consider these functions asmultivalued functions.
If denotes one of the values of the multivalued logarithm (typically its principal value), the other values are wherek is any integer. Similarly, if is one value of the exponentiation, then the other values are given by
wherek is any integer.
Different values ofk give different values of unlessw is arational number, that is, there is an integerd such thatdw is an integer. This results from theperiodicity of the exponential function, more specifically, that if and only if is an integer multiple of
If is a rational number withm andncoprime integers with then has exactlyn values. In the case these values are the same as those described in§nth roots of a complex number. Ifw is an integer, there is only one value that agrees with that of§ Integer exponents.
The multivalued exponentiation is holomorphic for in the sense that itsgraph consists of several sheets that define each a holomorphic function in the neighborhood of every point. Ifz varies continuously along a circle around0, then, after a turn, the value of has changed of sheet.
Thecanonical form of can be computed from the canonical form ofz andw. Although this can be described by a single formula, it is clearer to split the computation in several steps.
Polar form ofz. If is the canonical form ofz (a andb being real), then its polar form is with and, where is thetwo-argument arctangent function.
Logarithm ofz. Theprincipal value of this logarithm is where denotes thenatural logarithm. The other values of the logarithm are obtained by adding for any integerk.
Canonical form of If withc andd real, the values of are the principal value corresponding to
Final result. Using the identities and one gets with for the principal value.
The polar form ofi is and the values of are thus It follows thatSo, all values of are real, the principal one being
Similarly, the polar form of−2 is So, the above described method gives the valuesIn 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 thereal part ofw is an integer.
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(bx) =x ⋅ log b holds wheneverb is a positive real number andx is a real number. But for theprincipal branch of the complex logarithm one has
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:
This identity does not hold even when considering log as a multivalued function. The possible values oflog(wz) contain those ofz ⋅ log w as aproper subset. UsingLog(w) for the principal value oflog(w) andm,n as any integers the possible values of both sides are:
The identities(bc)x =bxcx and(b/c)x =bx/cx are valid whenb andc are positive real numbers andx is a real number. But, for the principal values, one hasandOn the other hand, whenx is 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)1/2 are{1, −1}. The identity holds, but saying{1} = {(−1 ⋅ −1)1/2} is incorrect.
The identity(ex)y =exy holds for real numbersx andy, but assuming its truth for complex numbers leads to the followingparadox, discovered in 1827 byClausen:[37]For any integern, we have:
(taking the-th power of both sides)
(using and expanding the exponent)
(using)
(dividing bye)
but this is false when the integern is nonzero.The error is the following: by definition, is a notation for a true function, and is a notation for which is a multi-valued function. Thus the notation is ambiguous whenx =e. Here, before expanding the exponent, the second line should beTherefore, when expanding the exponent, one has implicitly supposed that for complex values ofz, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity(ex)y =exy must be replaced by the identitywhich is a true identity between multivalued functions.
Ifb is a positive realalgebraic number, andx is a rational number, thenbx is an algebraic number. This results from the theory ofalgebraic extensions. This remains true ifb is any algebraic number, in which case, all values ofbx (as amultivalued function) are algebraic. Ifx isirrational (that is,not rational), and bothb andx are algebraic, Gelfond–Schneider theorem asserts that all values ofbx aretranscendental (that is, not algebraic), except ifb equals0 or1.
In other words, ifx is irrational and then at least one ofb,x andbx is transcendental.
The definition of exponentiation with positive integer exponents as repeated multiplication may apply to anyassociative operation denoted as a multiplication.[nb 2] The definition ofx0 requires further the existence of amultiplicative identity.[38]
Analgebraic structure consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by1 is amonoid. In such a monoid, exponentiation of an elementx is defined inductively by
for every nonnegative integern.
Ifn is a negative integer, is defined only ifx has amultiplicative inverse.[39] In this case, the inverse ofx is denotedx−1, andxn is defined as
Exponentiation with integer exponents obeys the following laws, forx andy in the algebraic structure, andm andn integers:
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, iff is areal function whose valued can be multiplied, denotes the exponentiation with respect of multiplication, and may denote exponentiation with respect offunction composition. That is,
So, ifG is a group, is defined for every 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 elementx is thecyclic group generated byx. If all the powers ofx are distinct, the group isisomorphic to theadditive group of the integers. Otherwise, the cyclic group isfinite (it has a finite number of elements), and its number of elements is theorder ofx. If the order ofx isn, then and the cyclic group generated byx consists of then first powers ofx (starting indifferently from the exponent0 or1).
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 (theorder of 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,gh =h−1gh, whereg andh 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 aring, it may occur that some nonzero elements satisfy for some integern. Such an element is said to benilpotent. In acommutative ring, the nilpotent elements form anideal, called thenilradical of the ring.
More generally, given an idealI in a commutative ringR, the set of the elements ofR that have a power inI is an ideal, called theradical ofI. The nilradical is the radical of thezero ideal. Aradical ideal is an ideal that equals its own radical. In apolynomial ring 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).
IfA is a square matrix, then the product ofA with itselfn times is called thematrix power. Also is defined to be the identity matrix,[40] and ifA is invertible, then.
Matrix powers appear often in the context ofdiscrete dynamical systems, where the matrixA expresses a transition from a state vectorx of some system to the next stateAx of the system.[41] This is the standard interpretation of aMarkov chain, for example. Then is the state of the system after two time steps, and so forth: is the state of the system aftern time steps. The matrix power is the transition matrix between the state now and the state at a timen steps 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 operators can also be exponentiated. An example is thederivative operator of calculus,, which is a linear operator acting on functions to give a new function. Thenth power of the differentiation operator is thenth derivative:
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.[42] 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 derivative which, together with thefractional integral, is one of the basic operations of thefractional calculus.
Afield is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication isassociative and 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 numbers and therational numbers, considered earlier in this article, which are allinfinite.
Afinite field is a field with afinite number of elements. This number of elements is either aprime number or aprime power; that is, it has the form wherep is a prime number, andk is a positive integer. For every suchq, there are fields withq elements. The fields withq elements are allisomorphic, which allows, in general, working as if there were only one field withq elements, denoted
One has
for every
Aprimitive element in is an elementg such that the set of theq − 1 first powers ofg (that is,) equals the set of the nonzero elements of There are primitive elements in where isEuler's totient function.
TheDiffie–Hellman key exchange is 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, ifg is a primitive element in then can be efficiently computed withexponentiation by squaring for anye, even ifq is large, while there is no known computationally practical algorithm that allows retrievinge from ifq is sufficiently large.
This allows defining thenth power of a setS as the set of alln-tuples of elements ofS.
WhenS is endowed with some structure, it is frequent that 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 (where denotes the real numbers) denotes the Cartesian product ofn copies of as well as their direct product asvector space,topological spaces,rings, etc.
An-tuple of elements ofS can be considered as afunction from This generalizes to the following notation.
Given two setsS andT, the set of all functions fromT toS is denoted. This exponential notation is justified by the following canonical isomorphisms (for the first one, seeCurrying):
where denotes the Cartesian product, and thedisjoint union.
One can use sets as exponents for other operations on sets, typically fordirect sums ofabelian groups,vector spaces, ormodules. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, denotes the vector space of theinfinite sequences of real numbers, and the vector space of those sequences that have a finite number of nonzero elements. The latter has abasis consisting of the sequences with exactly one nonzero element that equals1, while theHamel bases of the former cannot be explicitly described (because their existence involvesZorn's lemma).
In this context,2 can represents the set So, denotes thepower set ofS, that is the set of the functions fromS to which can be identified with the set of thesubsets ofS, by mapping each function to theinverse image of1.
In thecategory of sets, themorphisms between setsX andY are the functions fromX toY. It results that the set of the functions fromX toY that is denoted in the preceding section can also be denoted The isomorphism can be rewritten
This means the functor "exponentiation to the powerT" is aright adjoint to the functor "direct product withT".
This generalizes to the definition ofexponentiation in a category in which finitedirect products exist: in such a category, the functor is, if it exists, a right adjoint to the functor A category is called aCartesian closed category, if direct products exist, and the functor has a right adjoint for everyT.
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 function andKnuth'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.
Zero to the power of zero gives a number of examples of limits that are of theindeterminate form 00. The limits in these examples exist, but have different values, showing that the two-variable functionxy has no limit at the point(0, 0). One may consider at what points this function does have a limit.
More precisely, consider the function defined on. ThenD can be viewed as a subset ofR2 (that is, the set of all pairs(x,y) withx,y belonging to theextended real number lineR = [−∞, +∞], endowed with theproduct topology), which will contain the points at which the functionf has a limit.
In fact,f has a limit at allaccumulation points ofD, except for(0, 0),(+∞, 0),(1, +∞) and(1, −∞).[43] Accordingly, this allows one to define the powersxy by 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+∞ = 0 andx−∞ = +∞, when0 <x < 1.
0y = 0 and(+∞)y = +∞, when0 <y ≤ +∞.
0y = +∞ and(+∞)y = 0, when−∞ ≤y < 0.
These powers are obtained by taking limits ofxy forpositive values ofx. This method does not permit a definition ofxy whenx < 0, since pairs(x,y) withx < 0 are not accumulation points ofD.
On the other hand, whenn is an integer, the powerxn is already meaningful for all values ofx, including negative ones. This may make the definition0n = +∞ obtained above for negativen problematic whenn is odd, since in this casexn → +∞ asx tends to0 through positive values, but not negative ones.
Computingbn using iterated multiplication requiresn − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute2100, applyHorner's rule to the exponent 100 written in binary:
.
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 computebn can be reduced to by usingexponentiation by squaring, where denotes the number of1s in thebinary representation ofn. 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 theminimal sequence of multiplications (the minimal-length addition chain for the exponent) forbn is a difficult problem, for which no efficient algorithms are currently known (seeSubset sum problem), but many reasonably efficient heuristic algorithms are available.[44] However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.
If the domain of a functionf equals 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 iterate of the function. Thus denotes generally thenth iterate off; for example, means[45]
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 iterationbefore the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplicationafter the parentheses. Thus and When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example and For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically thetrigonometric functions. So, and both mean and not which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.[46][47][48]
Programming languages generally express exponentiation either as an infixoperator or as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is thecaret (^). Theoriginal version of ASCII included an uparrow symbol (↑), intended for exponentiation, but this wasreplaced by the caret in 1967, so the caret became usual in programming languages.[49]The notations include:
In most programming languages with an infix exponentiation operator, it isright-associative, that is,a^b^c is interpreted asa^(b^c).[55] This is because(a^b)^c is equal toa^(b*c) and thus not as useful. In some languages, it is left-associative, notably inAlgol,MATLAB, and theMicrosoft Excel formula language.
Other programming languages use functional notation:
^There are three common notations formultiplication: is most commonly used for explicit numbers and at a very elementary level; is most common whenvariables are used; is used for emphasizing that one talks of multiplication or when omitting the multiplication sign would be confusing.
^Archimedes. (2009). THE SAND-RECKONER. In T. Heath (Ed.), The Works of Archimedes: Edited in Modern Notation with Introductory Chapters (Cambridge Library Collection - Mathematics, pp. 229-232). Cambridge: Cambridge University Press.doi:10.1017/CBO9780511695124.017.
^Descartes, René (1637). "La Géométrie".Discourse de la méthode [...]. Leiden: Jan Maire. p. 299.Etaa, oua2, pour multipliera par soy mesme; Eta3, pour le multiplier encore une fois para, & ainsi a l'infini (Andaa, ora2, in order to multiplya by itself; anda3, in order to multiply it once more bya, and thus to infinity).
^Euler, Leonhard (1748).Introductio in analysin infinitorum (in Latin). Vol. I. Lausanne: Marc-Michel Bousquet. pp. 69,98–99.Primum ergo considerandæ sunt quantitates exponentiales, seu Potestates, quarum Exponens ipse est quantitas variabilis. Perspicuum enim est hujusmodi quantitates ad Functiones algebraicas referri non posse, cum in his Exponentes non nisi constantes locum habeant.
^Knobloch, Eberhard (1994). "The infinite in Leibniz's mathematics – The historiographical method of comprehension in context". In Kostas Gavroglu; Jean Christianidis; Efthymios Nicolaidis (eds.).Trends in the Historiography of Science. Boston Studies in the Philosophy of Science. Vol. 151. Springer Netherlands. p. 276.doi:10.1007/978-94-017-3596-4_20.ISBN9789401735964.A positive power of zero is infinitely small, a negative power of zero is infinite.
^Richard Gillam (2003).Unicode Demystified: A Practical Programmer's Guide to the Encoding Standard. Addison-Wesley Professional. p. 33.ISBN0201700522.
