Exponentiation

[[File:Expo02.svg|thumb|Graphs of for various bases :

Each curve passes through the point because any nonzero number raised to the power of is . At , the value of equals the base because any number raised to the power of is the number itself.]]

In mathematics, exponentiation, denoted , is an operation involving two numbers: the base, , and the exponent or power, . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases: or, most briefly, " to the ".

The above definition of <math>b^n</math> immediately implies several properties, in particular the multiplication rule:

<math display="block">\begin{align}

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{align}</math>

That is, when multiplying a base raised to one power times the same base raised to another power, the powers add.

Exponentiation can also be extended to powers that are not positive integers. When is non-zero, the definition <math display="block">

b^0=1

</math>

is compatible with the multiplication rule: <math>b^0 \times b^n = b^{0+n} = b^n</math>. A similar argument suggests the definition

<math display="block">

b^{-n} = 1/b^n,

</math>

for negative integer powers, and in particular <math>b^{-1} = \frac{1}{b}</math> for any nonzero number , and also the definition

<math display="block">b^{n/m} = \sqrt[m]{b^n}</math>

for fractional powers (when and are both integers). For example, <math> b^{1/2} \times b^{1/2} = b^{1/2 + 1/2} = b^1 = b </math>, meaning <math> (b^{1/2})^2 = b </math>, which is the definition of square root: <math>b^{1/2} = \sqrt{b} </math>.

The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define <math>b^x</math> for any positive real base <math>b</math> and any real number exponent <math>x</math>. More involved definitions allow complex base and exponent, as well as certain types of matrices as base or exponent.

Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.

Etymology

The term exponent originates from the Latin exponentem, the present participle of exponere, meaning "to put forth". The term power () is a mistranslation of the ancient Greek δύναμις (dúnamis, here: "amplification"

The word exponent was coined in 1544 by Michael Stifel. In the 16th century, Robert Recorde used 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.

History

In The Sand Reckoner, Archimedes proved the law of exponents, , necessary to manipulate powers of . He then used powers of to estimate the number of grains of sand that can be contained in the universe.

In the 9th century, the Persian mathematician Al-Khwarizmi used the terms مَال (māl, "possessions", "property") for a square—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"

Nicolas Chuquet used a form of exponential notation in the 15th century, for example to represent . This was later used by Henricus Grammateus and Michael 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 example for .

In 1636, James Hume used in essence modern notation, when in L'algèbre de Viète he wrote for . Early in the 17th century, the first form of our modern exponential notation was introduced by René Descartes in his text titled La Géométrie; there, the notation is introduced in Book I.

Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as .

Samuel Jeake introduced the term indices in 1696. The term involution was used synonymously with the term indices, but had declined in usage and should not be confused with its more common meaning.

In 1748, Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing: = \frac{1}{i} = -i</math>

On the other hand, when 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 are . The identity holds, but saying is incorrect.

| The identity holds for real numbers and , but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:

For any integer , we have:

1. <math>e^{1 + 2 \pi i n} = e^1 e^{2 \pi i n} = e \cdot 1 = e</math>
2. <math>\left(e^{1 + 2\pi i n}\right)^{1 + 2 \pi i n} = e\qquad</math> (taking the <math>(1 + 2 \pi i n)</math>-th power of both sides)
3. <math>e^{1 + 4 \pi i n - 4 \pi^2 n^2} = e\qquad</math> (using <math>\left(e^x\right)^y = e^{xy}</math> and expanding the exponent)
4. <math>e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e\qquad</math> (using <math>e^{x+y} = e^x e^y</math>)
5. <math>e^{-4 \pi^2 n^2} = 1\qquad</math> (dividing by )

but this is false when the integer is nonzero.

The error is the following: by definition, <math>e^y</math> is a notation for <math>\exp(y),</math> a true function, and <math>x^y</math> is a notation for <math>\exp(y\log x),</math> which is a multi-valued function. Thus the notation is ambiguous when . Here, before expanding the exponent, the second line should be

<math display="block">\exp\left((1 + 2\pi i n)\log \exp(1 + 2\pi i n)\right) = \exp(1 + 2\pi i n).</math>

Therefore, when expanding the exponent, one has implicitly supposed that <math>\log \exp z =z</math> for complex values of , which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity must be replaced by the identity

<math display="block">\left(e^x\right)^y = e^{y\log e^x},</math>

which is a true identity between multivalued functions.

Irrationality and transcendence

If is a positive real algebraic number, and is a rational number, then is an algebraic number. This results from the theory of algebraic extensions. This remains true if is any algebraic number, in which case, all values of (as a multivalued function) are algebraic. If is irrational (that is, not rational), and both and are algebraic, Gelfond–Schneider theorem asserts that all values of are transcendental (that is, not algebraic), except if equals or .

In other words, if is irrational and <math>b\not\in \{0,1\},</math> then at least one of , and is transcendental.

Integer powers in algebra

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication. The definition of requires further the existence of a multiplicative identity.

An algebraic structure consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by is a monoid. In such a monoid, exponentiation of an element is defined inductively by

• <math>x^0 = 1,</math>
• <math>x^{n+1} = x x^n</math> for every nonnegative integer .

If is a negative integer, <math>x^n</math> is defined only if has a multiplicative inverse. In this case, the inverse of is denoted , and is defined as <math>\left(x^{-1}\right)^{-n}.</math>

Exponentiation with integer exponents obeys the following laws, for and in the algebraic structure, and and integers:

: <math>\begin{align}

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{align}</math>

These definitions are widely used in many areas of mathematics, notably for groups, rings, fields, square matrices (which form a ring). They apply also to functions from a set to itself, which form a monoid under function composition. This includes, as specific instances, geometric transformations, and endomorphisms of any mathematical structure.

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if is a real function whose valued can be multiplied, <math>f^n</math> denotes the exponentiation with respect of multiplication, and <math>f^{\circ n}</math> may denote exponentiation with respect of function composition. That is,

: <math>(f^n)(x)=(f(x))^n=f(x) \,f(x) \cdots f(x),</math>

and

: <math>(f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots)).</math>

Commonly, <math>(f^n)(x)</math> is denoted <math>f(x)^n,</math> while <math>(f^{\circ n})(x)</math> is denoted <math>f^n(x).</math>

In a group

A multiplicative group is a set with as associative operation denoted as multiplication, that has an identity element, and such that every element has an inverse.

So, if is a group, <math>x^n</math> is defined for every <math>x\in G</math> and every integer .

The set of all powers of an element of a group form a subgroup. A group (or subgroup) that consists of all powers of a specific element is the cyclic group generated by . If all the powers of are distinct, the group is isomorphic to the additive group <math>\Z</math> of the integers. Otherwise, the cyclic group is finite (it has a finite number of elements), and its number of elements is the order of . If the order of is , then <math>x^n=x^0=1,</math> and the cyclic group generated by consists of the first powers of (starting indifferently from the exponent or ).

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

Superscript notation is also used for conjugation; that is, , where and 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 <math>(g^h)^k=g^{hk}</math> and <math>(gh)^k=g^kh^k.</math>

In a ring

In a ring, it may occur that some nonzero elements satisfy <math>x^n=0</math> for some integer . Such an element is said to be nilpotent. In a commutative ring, the nilpotent elements form an ideal, called the nilradical of the ring.

If the nilradical is reduced to the zero ideal (that is, if <math>x\neq 0</math> implies <math>x^n\neq 0</math> for every positive integer ), the commutative ring is said to be reduced. Reduced rings are important in algebraic geometry, since the coordinate ring of an affine algebraic set is always a reduced ring.

More generally, given an ideal in a commutative ring , the set of the elements of that have a power in is an ideal, called the radical of . The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring <math>k[x_1, \ldots, x_n]</math> over a field , 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 of Hilbert's Nullstellensatz).

Matrices and linear operators

If is a square matrix, then the product of with itself times is called the matrix power. Also <math>A^0</math> is defined to be the identity matrix, and if is invertible, then <math>A^{-n} = \left(A^{-1}\right)^n</math>.

Matrix powers appear often in the context of discrete dynamical systems, where the matrix expresses a transition from a state vector of some system to the next state of the system. This is the standard interpretation of a Markov chain, for example. Then <math>A^2x</math> is the state of the system after two time steps, and so forth: <math>A^nx</math> is the state of the system after time steps. The matrix power <math>A^n</math> is the transition matrix between the state now and the state at a time 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 using eigenvalues and eigenvectors.

Apart from matrices, more general linear operators can also be exponentiated. An example is the derivative operator of calculus, <math>d/dx</math>, which is a linear operator acting on functions <math>f(x)</math> to give a new function <math>(d/dx)f(x) = f'(x)</math>. The th power of the differentiation operator is the th derivative:

: <math>\left(\frac{d}{dx}\right)^nf(x) = \frac{d^n}{dx^n}f(x) = f^{(n)}(x).</math>

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 of semigroups. 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 the heat 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 the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.

Finite fields

A field is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is associative and every nonzero element has a multiplicative inverse. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of . Common examples are the field of complex numbers, the real numbers and the rational numbers, considered earlier in this article, which are all infinite.

A finite field is a field with a finite number of elements. This number of elements is either a prime number or a prime power; that is, it has the form <math>q=p^k,</math> where is a prime number, and is a positive integer. For every such , there are fields with elements. The fields with elements are all isomorphic, which allows, in general, working as if there were only one field with elements, denoted <math>\mathbb F_q.</math>

One has

: <math>x^q=x</math>

for every <math>x\in \mathbb F_q.</math>

A primitive element in <math>\mathbb F_q</math> is an element such that the set of the first powers of (that is, <math>\{g^1=g, g^2, \ldots, g^{p-1}=g^0=1\}</math>) equals the set of the nonzero elements of <math>\mathbb F_q.</math> There are <math>\varphi (p-1)</math> primitive elements in <math>\mathbb F_q,</math> where <math>\varphi</math> is Euler's totient function.

In <math>\mathbb F_q,</math> the freshman's dream identity

: <math>(x+y)^p = x^p+y^p</math>

is true for the exponent . As <math>x^p=x</math> in <math>\mathbb F_q,</math> It follows that the map

: <math>\begin{align}

F\colon{} & \mathbb F_q \to \mathbb F_q\\

& x\mapsto x^p

\end{align}</math>

is linear over <math>\mathbb F_q,</math> and is a field automorphism, called the Frobenius automorphism. If <math>q=p^k,</math> the field <math>\mathbb F_q</math> has automorphisms, which are the first powers (under composition) of . In other words, the Galois group of <math>\mathbb F_q</math> is cyclic of order , generated by the Frobenius automorphism.

The Diffie–Hellman key exchange is an application of exponentiation in finite fields that is widely used for secure communications. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the discrete logarithm, is computationally expensive. More precisely, if is a primitive element in <math>\mathbb F_q,</math> then <math>g^e</math> can be efficiently computed with exponentiation by squaring for any , even if is large, while there is no known computationally practical algorithm that allows retrieving from <math>g^e</math> if is sufficiently large.

Powers of sets

The Cartesian product of two sets and is the set of the ordered pairs <math>(x,y)</math> such that <math>x\in S</math> and <math>y\in T.</math> This operation is not properly commutative nor associative, but has these properties up to canonical isomorphisms, that allow identifying, for example, <math>(x,(y,z)),</math> <math>((x,y),z),</math> and <math>(x,y,z).</math>

This allows defining the th power <math>S^n</math> of a set as the set of all -tuples <math>(x_1, \ldots, x_n)</math> of elements of .

When is endowed with some structure, it is frequent that <math>S^n</math> 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 <math>\R^n</math> (where <math>\R</math> denotes the real numbers) denotes the Cartesian product of copies of <math>\R,</math> as well as their direct product as vector space, topological spaces, rings, etc.

Sets as exponents

A -tuple <math>(x_1, \ldots, x_n)</math> of elements of can be considered as a function from <math>\{1,\ldots, n\}.</math> This generalizes to the following notation.

Given two sets and , the set of all functions from to is denoted <math>S^T</math>. This exponential notation is justified by the following canonical isomorphisms (for the first one, see Currying):

: <math>(S^T)^U\cong S^{T\times U},</math>

: <math>S^{T\sqcup U}\cong S^T\times S^U,</math>

where <math>\times</math> denotes the Cartesian product, and <math>\sqcup</math> the disjoint union.

One can use sets as exponents for other operations on sets, typically for direct sums of abelian groups, vector spaces, or modules. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, <math>\R^\N</math> denotes the vector space of the infinite sequences of real numbers, and <math>\R^{(\N)}</math> the vector space of those sequences that have a finite number of nonzero elements. The latter has a basis consisting of the sequences with exactly one nonzero element that equals , while the Hamel bases of the former cannot be explicitly described (because their existence involves Zorn's lemma).

In this context, can represents the set <math>\{0,1\}.</math> So, <math>2^S</math> denotes the power set of , that is the set of the functions from to <math>\{0,1\},</math> which can be identified with the set of the subsets of , by mapping each function to the inverse image of .

This fits in with the exponentiation of cardinal numbers, in the sense that , where is the cardinality of .

In category theory

In the category of sets, the morphisms between sets and are the functions from to . It results that the set of the functions from to that is denoted <math>Y^X</math> in the preceding section can also be denoted <math>\hom(X,Y).</math> The isomorphism <math>(S^T)^U\cong S^{T\times U}</math> can be rewritten

:<math>\hom(U,S^T)\cong \hom(T\times U,S).</math>

This means the functor "exponentiation to the power " is a right adjoint to the functor "direct product with ".

This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor <math>X\to X^T</math> is, if it exists, a right adjoint to the functor <math>Y\to T\times Y.</math> A category is called a Cartesian closed category, if direct products exist, and the functor <math>Y\to X\times Y</math> has a right adjoint for every .

Repeated exponentiation

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 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation. This sequence of operations is expressed by the Ackermann function and Knuth'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 , the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and () respectively.

Limits of powers

Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 0⁰. The limits in these examples exist, but have different values, showing that the two-variable function has no limit at the point . One may consider at what points this function does have a limit.

More precisely, consider the function <math>f(x,y) = x^y</math> defined on <math> D = \{(x, y) \in \mathbf{R}^2 : x > 0 \}</math>. Then can be viewed as a subset of (that is, the set of all pairs with , belonging to the extended real number line , endowed with the product topology), which will contain the points at which the function has a limit.

In fact, has a limit at all accumulation points of , except for , , and . Accordingly, this allows one to define the powers by continuity whenever , , except for , , and , which remain indeterminate forms.

Under this definition by continuity, we obtain:

• and , when .
• and , when .
• and , when .
• and , when .

These powers are obtained by taking limits of for positive values of . This method does not permit a definition of when , since pairs with are not accumulation points of .

On the other hand, when is an integer, the power is already meaningful for all values of , including negative ones. This may make the definition obtained above for negative problematic when is odd, since in this case as tends to through positive values, but not negative ones.

Efficient computation with integer exponents

Computing using iterated multiplication requires multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute , apply Horner's rule to the exponent 100 written in binary:

: <math>100 = 2^2 +2^5 + 2^6 = 2^2(1+2^3(1+2))</math>.

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

{| class="wikitable sortable static-row-numbers" style="text-align:right;"

|-

| 2² = 4

|-

| 2 (2²) = 2³ = 8

|-

| (2³)² = 2⁶ = 64

|-

| (2⁶)² = 2¹² =

|-

| (2¹²)² = 2²⁴ =

|-

| 2 (2²⁴) = 2²⁵ =

|-

| (2²⁵)² = 2⁵⁰ =

|-

| (2⁵⁰)² = 2¹⁰⁰ =

|}

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

In general, the number of multiplication operations required to compute can be reduced to <math>\sharp n +\lfloor \log_{2} n\rfloor -1,</math> by using exponentiation by squaring, where <math>\sharp n</math> denotes the number of s in the binary representation of . For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal addition-chain exponentiation. Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for is a difficult problem, for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available. However, in practical computations, exponentiation by squaring is efficient enough, and much easier to implement.

Iterated functions

Function composition is a binary operation that is defined on functions such that the codomain of the function written on the right is included in the domain of the function written on the left. It is denoted <math>g\circ f,</math> and defined as

: <math>(g\circ f)(x)=g(f(x))</math>

for every in the domain of .

If the domain of a function equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the th power of the function under composition, commonly called the th iterate of the function. Thus <math>f^n</math> denotes generally the th iterate of ; for example, <math>f^3(x)</math> means <math>f(f(f(x))).</math>

The notations include:

• <code>x ^ y</code>: 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 most computer algebra systems.
• <code>x ** y</code>. The Fortran character set did not include lowercase characters or punctuation symbols other than <code>+-*/()&amp;=.,'</code> and so used <code>**</code> for exponentiation (the initial version used <code>a xx b</code> instead.
• <code>x ^^ y</code>: Haskell (for fractional base, integer exponents), D.
• <code>x⋆y</code>: APL.

In most programming languages with an infix exponentiation operator, it is right-associative, that is, <code>a^b^c</code> is interpreted as <code>a^(b^c)</code>. This is because <code>(a^b)^c</code> is equal to <code>a^(b*c)</code> and thus not as useful. In some languages, it is left-associative, notably in Algol, MATLAB, and the Microsoft Excel formula language.

Other programming languages use functional notation:

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

Still others only provide exponentiation as part of standard libraries:

• <code>pow(x, y)</code>: C, C++ (in <code>math</code> library).
• <code>Math.Pow(x, y)</code>: C#.
• <code>math:pow(X, Y)</code>: Erlang.
• <code>Math.pow(x, y)</code>: Java.
• <code>[Math]::Pow(x, y)</code>: PowerShell.

In some statically typed languages that prioritize type safety such as Rust, exponentiation is performed via a multitude of methods:

• <code>x.pow(y)</code> for <code>x</code> and <code>y</code> as integers
• <code>x.powf(y)</code> for <code>x</code> and <code>y</code> as floating-point numbers
• <code>x.powi(y)</code> for <code>x</code> as a float and <code>y</code> as an integer

See also

<!-- Please keep entries in alphabetical order & add a short description WP:SEEALSO -->

Notes

References