Exponentiation

Exponentiation is a mathematical operation, written as bⁿ, involving two numbers, the base b and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bⁿ is the product of multiplying n bases:

In that case, bⁿ is called the n-th power of b, or b raised to the power n.

The exponent is usually shown as a superscript to the right of the base. Some common exponents have their own names: the exponent 2 (or 2nd power) is called the square of b (b²) or b squared; the exponent 3 (or 3rd power) is called the cube of b (b³) or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b.

When n is a negative integer and b is not zero, bⁿ is naturally defined as 1/b⁻ⁿ, preserving the property bⁿ × b^m = b^{n + m}.

The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.

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

History of the notation

The term power was used by the Greek mathematician Euclid for the square of a line.^[1] Archimedes discovered and proved the law of exponents, 10^a 10^b = 10^a+b, necessary to manipulate powers of 10.^[2] In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms mal for a square and kab for a cube, which later Islamic mathematicians represented in mathematical notation as m and k, respectively, by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī.^[3]

In the late 16th century, Jost Bürgi used Roman numerals for exponents.^[4]

Early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie; there, the notation is introduced in Book I.^[5]

Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel in the 16th century. The word "exponent" was coined in 1544 by Michael Stifel.^[6] Samuel Jeake introduced the term indices in 1696.^[1] In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth).^[7] Biquadrate has been used to refer to the fourth power as well.

Some mathematicians (e.g., Isaac Newton) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx³ + d.

Another historical synonym, involution,^[8] is now rare and should not be confused with its more common meaning.

In 1748 Leonard Euler wrote "consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant."^[9] With this introduction of transcendental functions, Euler laid the foundation for the modern introduction of natural logarithm as the inverse function for y = e^x.

Terminology

The expression b² = b ⋅ b is called the square of b because the area of a square with side-length b is b². It is pronounced "b squared".

The expression b³ = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b³. It is pronounced "b cubed".

The exponent indicates how many copies of the base are multiplied together. For example, 3⁵ = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the base, 5 is the exponent, and 243 is the power or, more specifically, the fifth power of 3, 3 raised to the fifth power, or 3 to the power of 5.

The word "raised" is usually omitted, and very often "power" as well, so 3⁵ is typically pronounced "three to the fifth" or "three to the five". The exponentiation bⁿ can be read as b raised to the n-th power, or b raised to the power of n, or b raised by the exponent of n, or most briefly as b to the n.

Exponentiation may be generalized from integer exponents to more general types of numbers.

Integer exponents

The exponentiation operation with integer exponents requires only elementary algebra.

Positive integer exponents

Formally, powers with positive integer exponents may be defined by the initial condition^[10]

and the recurrence relation

From the associativity of multiplication, it follows that for any positive integers m and n,

Zero exponent

Any nonzero number raised by the exponent 0 is 1;^[11] one interpretation of such a power is as an empty product. The case of 0⁰ is discussed below.

Negative exponents

The following identity holds for an arbitrary integer n and nonzero b:

Raising 0 by a negative exponent is left undefined.

The identity above may be derived through a definition aimed at extending the range of exponents to negative integers.

For non-zero b and positive n, the recurrence relation from the previous subsection can be rewritten as

By defining this relation as valid for all integer n and nonzero b, it follows that

and more generally for any nonzero b and any nonnegative integer n,

This is then readily shown to be true for every integer n.

Combinatorial interpretation

For nonnegative integers n and m, the power n^m is the number of functions from a set of m elements to a set of n elements (see cardinal exponentiation). Such functions can be represented as m-tuples from an n-element set (or as m-letter words from an n-letter alphabet).

05 = │ { } │ = 0	There is no 5-tuple from the empty set.
14 = │ { (1,1,1,1) } │ = 1	There is one 4-tuple from a one-element set.
23 = │ { (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2) } │ = 8	There are eight 3-tuples from a two-element set.
32 = │ { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) } │ = 9	There are nine 2-tuples from a three-element set.
41 = │ { (1), (2), (3), (4) } │ = 4	There are four 1-tuples from a four-element set.
50 = │ { () } │ = 1	There is exactly one 0-tuple.

Identities and properties

The following identities hold for all integer exponents, provided that the base is non-zero:

Exponentiation is not commutative. This contrasts with addition and multiplication, which are. For example, 2 + 3 = 3 + 2 = 5 and 2 ⋅ 3 = 3 ⋅ 2 = 6, but 2³ = 8, whereas 3² = 9.

Exponentiation is not associative either. Addition and multiplication are. For example, (2 + 3) + 4 = 2 + (3 + 4) = 9 and (2 ⋅ 3) ⋅ 4 = 2 ⋅ (3 ⋅ 4) = 24, but 2³ to the 4 is 8⁴ or 7003409600000000000♠4096, whereas 2 to the 3⁴ is 2⁸¹ or 7024241785163922925♠2417851639229258349412352. Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up:

Note that some computer programs (notably Microsoft Office Excel) associate to the left instead, i.e. a^b^c is evaluated as (a^b)^c.

Particular bases

Powers of ten

In the base ten (decimal) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, 7003100000000000000♠10³ = 7003100000000000000♠1000 and 6996100000000000000♠10⁻⁴ = 6996100000000000000♠0.0001.

Exponentiation with base 10 is used in scientific notation to denote large or small numbers. For instance, 7008299792458000000♠299792458 m/s (the speed of light in vacuum, in metres per second) can be written as 7008299792458000000♠2.99792458×10⁸ m/s and then approximated as 7008299800000000000♠2.998×10⁸ m/s.

SI prefixes based on powers of 10 are also used to describe small or large quantities. For example, the prefix kilo means 7003100000000000000♠10³ = 7003100000000000000♠1000, so a kilometre is 7003100000000000000♠1000 m.

Powers of two

The positive powers of 2 are important in computer science because there are 2ⁿ possible values for an n-bit binary register.

Powers of 2 are important in set theory since a set with n members has a power set, or set of all subsets of the original set, with 2ⁿ members.

The negative powers of 2 are commonly used, and the first two have special names: half, and quarter.

In the base 2 (binary) number system, integer powers of 2 are written as 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, two to the power of three is written as 1000 in binary.

Powers of one

The powers of one are all one: 1ⁿ = 1.

Powers of zero

If the exponent is positive, the power of zero is zero: 0ⁿ = 0, where n > 0.

If the exponent is negative, the power of zero (0ⁿ, where n < 0) is undefined, because division by zero is implied.

If the exponent is zero, some authors define 0⁰ = 1, whereas others leave it undefined, as discussed below under § Zero to the power of zero.

Powers of minus one

If n is an even integer, then (−1)ⁿ = 1.

If n is an odd integer, then (−1)ⁿ = −1.

Because of this, powers of −1 are useful for expressing alternating sequences. For a similar discussion of powers of the complex number i, see § Powers of complex numbers.

Large exponents

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

bⁿ → ∞ as n → ∞ when b > 1

This can be read as "b to the power of n tends to +∞ as n tends to infinity when b is greater than one".

Powers of a number with absolute value less than one tend to zero:

bⁿ → 0 as n → ∞ when |b| < 1

Any power of one is always one:

bⁿ = 1 for all n if b = 1

If the number b varies tending to 1 as the exponent tends to infinity then the limit is not necessarily one of those above. A particularly important case is

(1 + 1/n)ⁿ → e as n → ∞

See § The exponential function below.

Other limits, in particular of those that take on an indeterminate form, are described in § Limits of powers below.

Rational exponents

An nth root of a number b is a number x such that xⁿ = b.

If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to xⁿ = b. This solution is called the principal nth root of b. It is denoted ⁿ√b, where √   is the radical symbol; alternatively, the principal root may be written b^1/n. For example: 4^1/2 = 2, 8^1/3 = 2.

The fact that solves follows from noting that

If n is even, then xⁿ = b has two real solutions if b is positive, which are the positive and negative nth roots (the positive one being denoted ). If b is negative, the equation has no solution in real numbers for even n.

If n is odd, then xⁿ = b has one real solution. The solution is positive if b is positive and negative if b is negative.

The principal root of a positive real number b with a rational exponent u/v in lowest terms satisfies

where u is an integer and v is a positive integer.

Rational powers u/v, where u/v is in lowest terms, are positive if u is even (and hence v is odd) (because then b^u is positive), and negative for negative b if u and v are odd (because then b^u is negative). There are two roots, one of each sign, if b is positive and v is even (as exemplified by the case in which u = 1 and v = 2, whereby a positive b has two square roots); in this case the principal root is defined to be the positive one.

Thus we have (−27)^1/3 = −3 and (−27)^2/3 = 9. The number 4 has two 3/2th roots, namely 8 and −8; however, by convention 4^3/2 denotes the principal root, which is 8. Since there is no real number x such that x² = −1, the definition of b^u/v when b is negative and v is even must use the imaginary unit i, as described more fully in the section § Powers of complex numbers.

Care needs to be taken when applying the power identities with negative nth roots. For instance, −27 = (−27)^{((2/3)⋅(3/2))} = ((−27)^2/3)^3/2 = 9^3/2 = 27 is clearly wrong. The problem here occurs in taking the positive square root rather than the negative one at the last step, but in general the same sorts of problems occur as described for complex numbers in the section § Failure of power and logarithm identities.

Real exponents

The identities and properties shown above for integer exponents are true for positive real numbers with non-integer exponents as well. However the identity

cannot be extended consistently to cases where b is a negative real number (see § Real exponents with negative bases). The failure of this identity is the basis for the problems with complex number powers detailed under § Failure of power and logarithm identities.

Exponentiation to real powers of positive real numbers can be defined either by extending the rational powers to reals by continuity, or more usually as given in § Powers via logarithms below.

Limits of rational exponents

Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by continuity with the rule^[12]

where the limit as r gets close to x is taken only over rational values of r. This limit only exists for positive b. The (ε, δ)-definition of limit is used, this involves showing that for any desired accuracy of the result b^x one can choose a sufficiently small interval around x so all the rational powers in the interval are within the desired accuracy.

For example, if x = π, the nonterminating decimal representation π = 3.14159... can be used (based on strict monotonicity of the rational power) to obtain the intervals bounded by rational powers

, , , , , , ...

The bounded intervals converge to a unique real number, denoted by . This technique can be used to obtain any irrational power of a positive real number b. The function f_b(x) = b^x is thus defined for any real number x.

The exponential function

The important mathematical constant e, sometimes called Euler's number, is approximately equal to 2.718 and is the base of the natural logarithm. Although exponentiation of e could, in principle, be treated the same as exponentiation of any other real number, such exponentials turn out to have particularly elegant and useful properties. Among other things, these properties allow exponentials of e to be generalized in a natural way to other types of exponents, such as complex numbers or even matrices, while coinciding with the familiar meaning of exponentiation with rational exponents.

As a consequence, the notation e^x usually denotes a generalized exponentiation definition called the exponential function, exp(x), which can be defined in many equivalent ways, for example by:

Among other properties, exp satisfies the exponential identity:

The exponential function is defined for all integer, fractional, real, and complex values of x. In fact, the matrix exponential is well-defined for square matrices (in which case the exponential identity only holds when x and y commute), and is useful for solving systems of linear differential equations.

Since exp(1) is equal to e and exp(x) satisfies the exponential identity, it immediately follows that exp(x) coincides with the repeated-multiplication definition of e^x for integer x, and it also follows that rational powers denote (positive) roots as usual, so exp(x) coincides with the e^x definitions in the previous section for all real x by continuity.

Powers via logarithms

The natural logarithm ln(x) is the inverse of the exponential function e^x. It is defined for b > 0, and satisfies

If b^x is to preserve the logarithm and exponent rules, then one must have

for each real number x.

This can be used as an alternative definition of the real number power b^x and agrees with the definition given above using rational exponents and continuity. The definition of exponentiation using logarithms is more common in the context of complex numbers, as discussed below.

Real exponents with negative bases

Powers of a positive real number are always positive real numbers. The solution of x² = 4, however, can be either 2 or −2. The principal value of 4^1/2 is 2, but −2 is also a valid square root. If the definition of exponentiation of real numbers is extended to allow negative results then the result is no longer well behaved.

Neither the logarithm method nor the rational exponent method can be used to define b^r as a real number for a negative real number b and an arbitrary real number r. Indeed, e^r is positive for every real number r, so ln(b) is not defined as a real number for b ≤ 0.

The rational exponent method cannot be used for negative values of b because it relies on continuity. The function f(r) = b^r has a unique continuous extension^[12] from the rational numbers to the real numbers for each b > 0. But when b < 0, the function f is not even continuous on the set of rational numbers r for which it is defined.

For example, consider b = −1. The nth root of −1 is −1 for every odd natural number n. So if n is an odd positive integer, (−1)^(m/n) = −1 if m is odd, and (−1)^(m/n) = 1 if m is even. Thus the set of rational numbers q for which (−1)^q = 1 is dense in the rational numbers, as is the set of q for which (−1)^q = −1. This means that the function (−1)^q is not continuous at any rational number q where it is defined.

On the other hand, arbitrary complex powers of negative numbers b can be defined by choosing a complex logarithm of b.

Irrational exponents

If a is a positive algebraic number, and b is a rational number, it has been shown above that a^b is algebraic. This remains true even if one accepts any algebraic number for a, with the only difference that a^b may take several values (see below), all algebraic. Gelfond–Schneider theorem provides some information on the nature of a^b when b is irrational (that is not rational). It states:

If a is an algebraic number different from 0 and 1, and b an irrational algebraic number, then all the values of a^b are transcendental numbers (that is, not algebraic).

Complex exponents with positive real bases

Imaginary exponents with base e

A complex number is an expression of the form , where x and y are real numbers, and i is the so-called imaginary unit, a number that satisfies the rule ). A complex number can be visualized as a point in the (x,y) plane. The polar coordinates of a point in the (x,y) plane consist of a non-negative real number r and angle θ such that x = r cos θ and y = r sin θ. So

The product of two complex numbers z₁ = x₁ + iy₁, z₂ = x₂ + iy₂ is obtained by expanding out the product of the binomials and simplifying using the rule :

As a consequence of the angle sum formulas of trigonometry, if z₁ and z₂ have polar coordinates (r₁, θ₁), (r₂, θ₂), then their product z₁z₂ has polar coordinates equal to (r₁r₂, θ₁ + θ₂).

Consider the right triangle in the complex plane which has 0, 1, 1 + ix/n as vertices. For large values of n, the triangle is almost a circular sector with a radius of 1 and a small central angle equal to x/n radians. 1 + ix/n may then be approximated by the number with polar coordinates (1, x/n). So, in the limit as n approaches infinity, (1 + ix/n)ⁿ approaches (1, x/n)ⁿ = (1ⁿ, nx/n) = (1, x), the point on the unit circle whose angle from the positive real axis is x radians. The cartesian coordinates of this point are (cos x, sin x). So e ^ix = cos x + isin x; this is Euler's formula, connecting algebra to trigonometry by means of complex numbers.

The solutions to the equation e^z = 1 are the integer multiples of 2πi:

More generally, if e^v = w, then every solution to e^z = w can be obtained by adding an integer multiple of 2πi to v:

Thus the complex exponential function is a periodic function with period 2πi.

More simply: e^iπ = −1; e^{x + iy} = e^x(cos y + i sin y).

Trigonometric functions

It follows from Euler's formula stated above that the trigonometric functions cosine and sine are

Before the invention of complex numbers, cosine and sine were defined geometrically. The above formula reduces the complicated formulas for trigonometric functions of a sum into the simple exponentiation formula

Using exponentiation with complex exponents may reduce problems in trigonometry to algebra.

Complex exponents with base e

The power z = e^{x + iy} can be computed as e^x ⋅ e^iy. The real factor e^x is the absolute value of z and the complex factor e^iy identifies the direction of z.

Complex exponents with positive real bases

If b is a positive real number, and z is any complex number, the power b^z is defined as e^{z ⋅ ln(b)}, where x = ln(b) is the unique real solution to the equation e^x = b. So the same method working for real exponents also works for complex exponents.

For example:

The identity (b^z)^u=b^zu is not generally valid for complex powers. The power b^z is a complex number and any power of it has to follow the rules for powers of complex numbers below. A simple counterexample is given by:

The identity is, however, valid for arbitrary complex when is an integer.

Powers of complex numbers

Integer powers of nonzero complex numbers are defined by repeated multiplication or division as above. If i is the imaginary unit and n is an integer, then iⁿ equals 1, i, −1, or −i, according to whether the integer n is congruent to 0, 1, 2, or 3 modulo 4. Because of this, the powers of i are useful for expressing sequences of period 4.

Complex powers of positive reals are defined via e^x as in section Complex exponents with positive real bases above. These are continuous functions.

Trying to extend these functions to the general case of noninteger powers of complex numbers that are not positive reals leads to difficulties. Either we define discontinuous functions or multivalued functions. Neither of these options is entirely satisfactory.

The rational power of a complex number must be the solution to an algebraic equation. Therefore, it always has a finite number of possible values. For example, w = z^1/2 must be a solution to the equation w² = z. But if w is a solution, then so is −w, because (−1)² = 1. A unique but somewhat arbitrary solution called the principal value can be chosen using a general rule which also applies for nonrational powers.

Complex powers and logarithms are more naturally handled as single valued functions on a Riemann surface. Single valued versions are defined by choosing a sheet. The value has a discontinuity along a branch cut. Choosing one out of many solutions as the principal value leaves us with functions that are not continuous, and the usual rules for manipulating powers can lead us astray.

Any nonrational power of a complex number has an infinite number of possible values because of the multi-valued nature of the complex logarithm. The principal value is a single value chosen from these by a rule which, amongst its other properties, ensures powers of complex numbers with a positive real part and zero imaginary part give the same value as does the rule defined above for the corresponding real base.

Exponentiating a real number to a complex power is formally a different operation from that for the corresponding complex number. However, in the common case of a positive real number the principal value is the same.

The powers of negative real numbers are not always defined and are discontinuous even where defined. In fact, they are only defined when the exponent is a rational number with the denominator being an odd integer. When dealing with complex numbers the complex number operation is normally used instead.

Complex exponents with complex bases

For complex numbers w and z with w ≠ 0, the notation w^z is ambiguous in the same sense that log w is.

To obtain a value of w^z, first choose a logarithm of w; call it log w. Such a choice may be the principal value Log w (the default, if no other specification is given), or perhaps a value given by some other branch of log w fixed in advance. Then, using the complex exponential function one defines

because this agrees with the earlier definition in the case where w is a positive real number and the (real) principal value of log w is used.

If z is an integer, then the value of w^z is independent of the choice of log w, and it agrees with the earlier definition of exponentiation with an integer exponent.

If z is a rational number m/n in lowest terms with z > 0, then the countably infinitely many choices of log w yield only n different values for w^z; these values are the n complex solutions s to the equation sⁿ = w^m.

If z is an irrational number, then the countably infinitely many choices of log w lead to infinitely many distinct values for w^z.

The computation of complex powers is facilitated by converting the base w to polar form, as described in detail below.

A similar construction is employed in quaternions.

Complex roots of unity

A complex number w such that wⁿ = 1 for a positive integer n is an nth root of unity. Geometrically, the nth roots of unity lie on the unit circle of the complex plane at the vertices of a regular n-gon with one vertex on the real number 1.

If wⁿ = 1 but w^k ≠ 1 for all natural numbers k such that 0 < k < n, then w is called a primitive nth root of unity. The negative unit −1 is the only primitive square root of unity. The imaginary unit i is one of the two primitive 4th roots of unity; the other one is −i.

The number e^2πi/n is the primitive nth root of unity with the smallest positive argument. (It is sometimes called the principal nth root of unity, although this terminology is not universal and should not be confused with the principal value of ⁿ√1, which is 1.^[13])

The other nth roots of unity are given by

for 2 ≤ k ≤ n.

Roots of arbitrary complex numbers

Although there are infinitely many possible values for a general complex logarithm, there are only a finite number of values for the power w^q in the important special case where q = 1/n and n is a positive integer. These are the nth roots of w; they are solutions of the equation zⁿ = w. As with real roots, a second root is also called a square root and a third root is also called a cube root.

It is conventional in mathematics to define w^1/n as the principal value of the root. If w is a positive real number, it is also conventional to select a positive real number as the principal value of the root w^1/n. For general complex numbers, the nth root with the smallest argument is often selected as the principal value of the nth root operation, as with principal values of roots of unity.

The set of nth roots of a complex number w is obtained by multiplying the principal value w^1/n by each of the nth roots of unity. For example, the fourth roots of 16 are 2, −2, 2i, and −2i, because the principal value of the fourth root of 16 is 2 and the fourth roots of unity are 1, −1, i, and −i.

Computing complex powers

It is often easier to compute complex powers by writing the number to be exponentiated in polar form. Every complex number z can be written in the polar form

where r is a nonnegative real number and θ is the (real) argument of z. The polar form has a simple geometric interpretation: if a complex number u + iv is thought of as representing a point (u, v) in the complex plane using Cartesian coordinates, then (r, θ) is the same point in polar coordinates. That is, r is the "radius" r² = u² + v² and θ is the "angle" θ = atan2(v, u). The polar angle θ is ambiguous since any integer multiple of 2π could be added to θ without changing the location of the point. Each choice of θ gives in general a different possible value of the power. A branch cut can be used to choose a specific value. The principal value (the most common branch cut), corresponds to θ chosen in the interval (−π, π]. For complex numbers with a positive real part and zero imaginary part using the principal value gives the same result as using the corresponding real number.

In order to compute the complex power w^z, write w in polar form:

Then

and thus

If z is decomposed as c + di, then the formula for w^z can be written more explicitly as

This final formula allows complex powers to be computed easily from decompositions of the base into polar form and the exponent into Cartesian form. It is shown here both in polar form and in Cartesian form (via Euler's identity).

The following examples use the principal value, the branch cut which causes θ to be in the interval (−π, π]. To compute iⁱ, write i in polar and Cartesian forms:

Then the formula above, with r = 1, θ = π/2, c = 0, and d = 1, yields:

Similarly, to find (−2)^{3 + 4i}, compute the polar form of −2,

and use the formula above to compute

The value of a complex power depends on the branch used. For example, if the polar form i = 1e^5πi/2 is used to compute iⁱ, the power is found to be e^−5π/2; the principal value of iⁱ, computed above, is e^−π/2. The set of all possible values for iⁱ is given by:^[14]

So there is an infinity of values which are possible candidates for the value of iⁱ, one for each integer k. All of them have a zero imaginary part so one can say iⁱ has an infinity of valid real values.

Failure of power and logarithm identities

Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions. For example:

• The identity log(b^x) = x ⋅ log b holds whenever b is a positive real number and x is a real number. But for the principal 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 of log(w^z) contain those of z ⋅ log w as a subset. Using Log(w) for the principal value of log(w) and m, n as any integers the possible values of both sides are:

  

• The identities (bc)^x = b^xc^x and (b/c)^x = b^x/c^x are valid when b and c are positive real numbers and x is a real number. But a calculation using principal branches shows that

  

  and

  

  On the other hand, when x 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 wrong.

• The identity (e^x)^y = e^xy holds for real numbers x and y, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:^[15]

  For any integer n, we have:

  1. 
  2. 
  3. 
  4. 
  5. 

  but this is false when the integer n is nonzero.

  There are a number of problems in the reasoning:

  The major error is that changing the order of exponentiation in going from line two to three changes what the principal value chosen will be.

  From the multi-valued point of view, the first error occurs even sooner. Implicit in the first line is that e is a real number, whereas the result of e^1+2πin is a complex number better represented as e+0i. Substituting the complex number for the real on the second line makes the power have multiple possible values. Changing the order of exponentiation from lines two to three also affects how many possible values the result can have. , but rather multivalued over integers n.

Generalizations

Monoids

Exponentiation can be defined in any monoid.^[16] A monoid is an algebraic structure consisting of a set X together with a rule for composition ("multiplication") satisfying an associative law and a multiplicative identity, denoted by 1. Exponentiation is defined inductively by:

• for all
• for all and non-negative integers n

Monoids include many structures of importance in mathematics, including groups and rings (under multiplication), with more specific examples of the latter being matrix rings and fields.

Matrices and linear operators

If A is a square matrix, then the product of A with itself n times is called the matrix power. Also is defined to be the identity matrix,^[17] and if A is invertible, then .

Matrix powers appear often in the context of discrete dynamical systems, where the matrix A expresses a transition from a state vector x of some system to the next state Ax of the system.^[18] This is the standard interpretation of a Markov chain, for example. Then is the state of the system after two time steps, and so forth: is the state of the system after n time steps. The matrix power is the transition matrix between the state now and the state at a time n 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, , which is a linear operator acting on functions to give a new function . The n-th power of the differentiation operator is the n-th 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 of semigroups.^[19] 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 all well-defined and satisfy their familiar properties. The real numbers, for example, form a field, as do the complex numbers and rational numbers. Unlike these familiar examples of fields, which are all infinite sets, some fields have only finitely many elements. The simplest example is the field with two elements with addition defined by and , and multiplication and .

Exponentiation in finite fields has applications in public key cryptography. For example, the Diffie–Hellman key exchange uses the fact that exponentiation is computationally inexpensive in finite fields, whereas the discrete logarithm (the inverse of exponentiation) is computationally expensive.

Any finite field F has the property that there is a unique prime number p such that for all x in F; that is, x added to itself p times is zero. For example, in , the prime number p = 2 has this property. This prime number is called the characteristic of the field. Suppose that F is a field of characteristic p, and consider the function that raises each element of F to the power p. This is called the Frobenius automorphism of F. It is an automorphism of the field because of the Freshman's dream identity . The Frobenius automorphism is important in number theory because it generates the Galois group of F over its prime subfield.

In abstract algebra

Exponentiation for integer exponents can be defined for quite general structures in abstract algebra.

Let X be a set with a power-associative binary operation which is written multiplicatively. Then xⁿ is defined for any element x of X and any nonzero natural number n as the product of n copies of x, which is recursively defined by

One has the following properties

If the operation has a two-sided identity element 1, then x⁰ is defined to be equal to 1 for any x.

If the operation also has two-sided inverses and is associative, then the magma is a group. The inverse of x can be denoted by x⁻¹ and follows all the usual rules for exponents.

If the multiplication operation is commutative (as for instance in abelian groups), then the following holds:

If the binary operation is written additively, as it often is for abelian groups, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition". Thus, each of the laws of exponentiation above has an analogue among laws of multiplication.

When there are several power-associative binary operations defined on a set, any of which might be iterated, it is common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, x^∗n is x ∗ ... ∗ x, while x^#n is x # ... # x, whatever the operations ∗ and # might be.

Superscript notation is also used, especially in group theory, to indicate conjugation. That is, g^h = h⁻¹gh, where g and h are elements of some group. Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role.

Over sets

If n is a natural number and A is an arbitrary set, the expression Aⁿ is often used to denote the set of ordered n-tuples of elements of A. This is equivalent to letting Aⁿ denote the set of functions from the set {0, 1, 2, ..., n−1} to the set A; the n-tuple (a₀, a₁, a₂, ..., a_n−1) represents the function that sends i to a_i.

For an infinite cardinal number κ and a set A, the notation A^κ is also used to denote the set of all functions from a set of size κ to A. This is sometimes written ^κA to distinguish it from cardinal exponentiation, defined below.

This generalized exponential can also be defined for operations on sets or for sets with extra structure. For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of

where each V_i is a vector space.

Then if V_i = V for each i, the resulting direct sum can be written in exponential notation as V^⊕N, or simply V^N with the understanding that the direct sum is the default. We can again replace the set N with a cardinal number n to get Vⁿ, although without choosing a specific standard set with cardinality n, this is defined only up to isomorphism. Taking V to be the field R of real numbers (thought of as a vector space over itself) and n to be some natural number, we get the vector space that is most commonly studied in linear algebra, the real vector space Rⁿ.

If the base of the exponentiation operation is a set, the exponentiation operation is the Cartesian product unless otherwise stated. Since multiple Cartesian products produce an n-tuple, which can be represented by a function on a set of appropriate cardinality, S^N becomes simply the set of all functions from N to S in this case:

This fits in with the exponentiation of cardinal numbers, in the sense that |S^N| = |S|^|N|, where |X| is the cardinality of X. When "2" is defined as {0, 1}, we have |2^X| = 2^|X|, where 2^X, usually denoted by P(X), is the power set of X; each subset Y of X corresponds uniquely to a function on X taking the value 1 for x ∈ Y and 0 for x ∉ Y.

In category theory

In a Cartesian closed category, the exponential operation can be used to raise an arbitrary object to the power of another object. This generalizes the Cartesian product in the category of sets. If 0 is an initial object in a Cartesian closed category, then the exponential object 0⁰ is isomorphic to any terminal object 1.

Of cardinal and ordinal numbers

In set theory, there are exponential operations for cardinal and ordinal numbers.

If κ and λ are cardinal numbers, the expression κ^λ represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ.^[20] If κ and λ are finite, then this agrees with the ordinary arithmetic exponential operation. For example, the set of 3-tuples of elements from a 2-element set has cardinality 8 = 2³. In cardinal arithmetic, κ⁰ is always 1 (even if κ is an infinite cardinal or zero).

Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.

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 (3, 3), the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and 7012762559748498700♠7625597484987 (= 3²⁷ = 3³³ = ³3) respectively.

Zero to the power of zero

Discrete exponents

There are many widely used formulas having terms involving natural-number exponents that require 0⁰ to be evaluated to 1. For example, regarding b⁰ as an empty product assigns it the value 1, even when b = 0. Alternatively, the combinatorial interpretation of b⁰ is the number of empty tuples of elements from a set with b elements. There is exactly one empty tuple, even if b = 0. Equivalently, the set-theoretic interpretation of 0⁰ is the number of functions from the empty set to the empty set. There is exactly one such function, the empty function.^[20]

Polynomials and power series

Likewise, when working with polynomials, it is often necessary to assign the value 1. A polynomial is an expression of the form where x is an indeterminate, and the coefficients are real numbers (or, more generally, elements of some ring). The set of all real polynomials in x is denoted by . Polynomials are added termwise, and multiplied by the applying the usual rules for exponents in the indeterminate x (see Cauchy product). With these algebraic rules for manipulation, polynomials form a polynomial ring. The polynomial is the identity element of the polynomial ring, meaning that it is the (unique) element such that the product of with any polynomial is just .^[21] Polynomials can be evaluated by specializing the indeterminate x to be a real number. More precisely, for any given real number there is a unique unital ring homomorphism such that .^[22] This is called the evaluation homomorphism. Because it is a unital homomorphism, we have That is, for all specializations of x to a real number (including zero).

This perspective is significant for many polynomial identities appearing in combinatorics. For example, the binomial theorem is not valid for x = 0 unless 0⁰ = 1.^[23] Similarly, rings of power series require to be true for all specializations of x. Thus identities like and are only true as functional identities (including at x = 0) if 0⁰ = 1.

In differential calculus, the power rule is not valid for n = 1 at x = 0 unless 0⁰ = 1.

Continuous exponents

Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form.^[24] In fact, when f(t) and g(t) are real-valued functions both approaching 0 (as t approaches a real number or ±∞), with f(t) > 0, the function f(t)^g(t) need not approach 1; depending on f and g, the limit of f(t)^g(t) can be any nonnegative real number or +∞, or it can diverge. For example, the functions below are of the form f(t)^g(t) with f(t), g(t) → 0 as t → 0⁺, but the limits are different:

.

Thus, the two-variable function x^y, though continuous on the set {(x, y) : x > 0}, cannot be extended to a continuous function on any set containing (0, 0), no matter how one chooses to define 0⁰.^[25] However, under certain conditions, such as when f and g are both analytic functions and f is positive on the open interval (0, b) for some positive b, the limit approaching from the right is always 1.^[26][27][28]

Complex exponents

In the complex domain, the function z^w may be defined for nonzero z by choosing a branch of log z and defining z^w as e^{w log z}. This does not define 0^w since there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.^[29][30][31]

History of differing points of view

The debate over the definition of has been going on at least since the early 19th century. At that time, most mathematicians agreed that , until in 1821 Cauchy^[32] listed along with expressions like in a table of indeterminate forms. In the 1830s Libri^[33][34] published an unconvincing argument for , and Möbius^[35] sided with him, erroneously claiming that whenever . A commentator who signed his name simply as "S" provided the counterexample of , and this quieted the debate for some time. More historical details can be found in Knuth (1992).^[36]

More recent authors interpret the situation above in different ways:

• Some argue that the best value for depends on context, and hence that defining it once and for all is problematic.^[37] According to Benson (1999), "The choice whether to define is based on convenience, not on correctness. If we refrain from defining then certain assertions become unnecessarily awkward. The consensus is to use the definition , although there are textbooks that refrain from defining ."^[38]
• Others argue that should be defined as 1. Knuth (1992) contends strongly that "has to be 1", drawing a distinction between the value , which should equal 1 as advocated by Libri, and the limiting form (an abbreviation for a limit of where ), which is necessarily an indeterminate form as listed by Cauchy: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."^[36]

Treatment on computers

IEEE floating point standard

The IEEE 754-2008 floating point standard is used in the design of most floating point libraries. It recommends a number of functions for computing a power:^[39]

• pow treats 0⁰ as 1. This is the oldest defined version. If the power is an exact integer the result is the same as for pown, otherwise the result is as for powr (except for some exceptional cases).
• pown treats 0⁰ as 1. The power must be an exact integer. The value is defined for negative bases; e.g., pown(−3,5) is −243.
• powr treats 0⁰ as NaN (Not-a-Number – undefined). The value is also NaN for cases like powr(−3,2) where the base is less than zero. The value is defined by e^{power×log(base)}.

Programming languages

Most programming language with a power function are implemented using the IEEE pow function and therefore evaluate 0⁰ as 1. The later C^[40] and C++ standards describe this as the normative behaviour. The Java standard^[41] mandates this behavior. The .NET Framework method System.Math.Pow also treats 0⁰ as 1.^[42]

Mathematics software

• Sage simplifies b⁰ to 1, even if no constraints are placed on b.^[43] It takes 0⁰ to be 1, but does not simplify 0^x for other x.
• Maple distinguishes between integers 0, 1, ... and the corresponding floats 0.0, 1.0, ... (usually denoted 0., 1., ...). If x does not evaluates to a number, then x⁰ and x^0.0 are respectively evaluated to 1 (integer) and 1.0 (float); on the other hand, 0^x is evaluated to the integer 0, while 0.0^x is evaluated as 0.^x. If both the base and the exponent are zero (or are evaluated to zero), the result is Float(undefined) if the exponent is the float 0.0; with an integer as exponent, the evaluation of 0⁰ results in the integer 1, while that of 0.⁰ results in the float 1.0.
• Macsyma also simplifies b⁰ to 1 even if no constraints are placed on b, but issues an error for 0⁰. For x>0, it simplifies 0^x to 0.
• Mathematica and Wolfram Alpha simplify b⁰ into 1, even if no constraints are placed on b.^[44] While Mathematica does not simplify 0^x, Wolfram Alpha returns two results, 0 for x > 0, and "indeterminate" for real x.^[45] Both Mathematica and Wolfram Alpha take 0⁰ to be "(indeterminate)".^[46]
• Matlab, Python, Magma, GAP, singular, PARI/GP and the Google and iPhone calculators evaluate 0⁰ as 1.

Limits of powers

The section § 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 x^y 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 f(x, y) = x^y defined on D = {(x, y) ∈ R² : x > 0}. Then D can be viewed as a subset of R² (that is, the set of all pairs (x, y) with x, y belonging to the extended real number line R = [−∞, +∞], endowed with the product topology), which will contain the points at which the function f has a limit.

In fact, f has a limit at all accumulation points of D, except for (0, 0), (+∞, 0), (1, +∞) and (1, −∞).^[47] Accordingly, this allows one to define the powers x^y by continuity whenever 0 ≤ x ≤ +∞, −∞ ≤ y ≤ +∞, except for 0⁰, (+∞)⁰, 1^+∞ and 1^−∞, which remain indeterminate forms.

Under this definition by continuity, we obtain:

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

These powers are obtained by taking limits of x^y for positive values of x. This method does not permit a definition of x^y when x < 0, since pairs (x, y) with x < 0 are not accumulation points of D.

On the other hand, when n is an integer, the power xⁿ is already meaningful for all values of x, including negative ones. This may make the definition 0ⁿ = +∞ obtained above for negative n problematic when n is odd, since in this case xⁿ → +∞ as x tends to 0 through positive values, but not negative ones.

Efficient computation with integer exponents

The simplest method of computing bⁿ requires n − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2¹⁰⁰, note that 100 = 64 + 32 + 4. Compute the following in order:

1. 2² = 4
2. (2²)² = 2⁴ = 16
3. (2⁴)² = 2⁸ = 256
4. (2⁸)² = 2¹⁶ = 65,536
5. (2¹⁶)² = 2³² = 4,294,967,296
6. (2³²)² = 2⁶⁴ = 18,446,744,073,709,551,616
7. 2⁶⁴ 2³² 2⁴ = 2¹⁰⁰ = 1,267,650,600,228,229,401,496,703,205,376

This series of steps only requires 8 multiplication operations instead of 99 (since the last product above takes 2 multiplications).

In general, the number of multiplication operations required to compute bⁿ can be reduced to Θ(log n) by using exponentiation by squaring or (more generally) addition-chain exponentiation. Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for bⁿ is a difficult problem for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available.^[48]

Exponential notation for function names

Placing an integer superscript after the name or symbol of a function, as if the function were being raised to a power, commonly refers to repeated function composition rather than repeated multiplication. Thus f ³(x) may mean f(f(f(x))); in particular, f ⁻¹(x) usually denotes the inverse function of f. Iterated functions are of interest in the study of fractals and dynamical systems. Babbage was the first to study the problem of finding a functional square root f ^1/2(x).

However, for historical reasons, a special syntax applies to the trigonometric functions: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of −1 denotes the inverse function. That is, sin²x is just a shorthand way to write (sin x)² without using parentheses, whereas sin⁻¹x refers to the inverse function of the sine, also called arcsin x. There is no need for a shorthand for the reciprocals of trigonometric functions since each has its own name and abbreviation; for example, 1/(sin x) = (sin x)⁻¹ = csc x. A similar convention applies to logarithms, where log²x usually means (log x)², not log log x.

In programming languages

The superscript notation x^y is convenient in handwriting but inconvenient for typewriters and computer terminals that align the baselines of all characters on each line. Many programming languages have alternate ways of expressing exponentiation that do not use superscripts:

• x ↑ y: Algol, Commodore BASIC
• x ^ y: BASIC, J, MATLAB, 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
• x ^^ y: Haskell (for fractional base, integer exponents), D
• x ** y: Ada, Bash, COBOL, CoffeeScript, Fortran, FoxPro, Gnuplot, OCaml, F#, Perl, PHP, PL/I, Python, Rexx, Ruby, SAS, Seed7, Tcl, ABAP, Mercury, Haskell (for floating-point exponents), Turing, VHDL
• pown x y: F# (for integer base, integer exponent)
• x⋆y: APL

Many programming languages lack syntactic support for exponentiation, but provide library functions.

In Bash, C, C++, C#, Go, Java, JavaScript, Perl, PHP, Python and Ruby, the symbol ^ represents bitwise XOR. In Pascal, it represents indirection. In OCaml and Standard ML, it represents string concatenation.

List of whole-number exponentials

n	n2	n3	n4	n5	n6	n7	n8	n9	n10
2	4	8	16	32	64	128	256	512	1,024
3	9	27	81	243	729	2,187	6,561	19,683	59,049
4	16	64	256	1,024	4,096	16,384	65,536	262,144	1,048,576
5	25	125	625	3,125	15,625	78,125	390,625	1,953,125	9,765,625
6	36	216	1,296	7,776	46,656	279,936	1,679,616	10,077,696	60,466,176
7	49	343	2,401	16,807	117,649	823,543	5,764,801	40,353,607	282,475,249
8	64	512	4,096	32,768	262,144	2,097,152	16,777,216	134,217,728	1,073,741,824
9	81	729	6,561	59,049	531,441	4,782,969	43,046,721	387,420,489	3,486,784,401
10	100	1,000	10,000	100,000	1,000,000	10,000,000	100,000,000	1,000,000,000	10,000,000,000

See also

Exponential decay Exponential growth List of exponential topics

Modular exponentiation Scientific notation Unicode subscripts and superscripts

References

External links

• sci.math FAQ: What is 0⁰?
• Introducing 0th power at PlanetMath.org.
• Laws of Exponents with derivation and examples
• What does 0^0 (zero to the zeroth power) equal? on AskAMathematician.com