Exponentiation

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

Exponentiation is a mathematical operation, written as aⁿ, involving two numbers, the base a and the exponent (or power) n. When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors of a (the product itself can also be called power):

$a^n = \underbrace{a \times \cdots \times a}_n,$

just as multiplication by a positive integer corresponds to repeated addition:

$a \times n = \underbrace{a + \cdots + a}_n.$

The exponent is usually shown as a superscript to the right of the base. The exponentiation aⁿ can be read as: a raised to the n-th power, a raised to the power [of] n, or possibly a raised to the exponent [of] n, or more briefly as a to the n. Some exponents have their own pronunciation: for example, a² is usually read as a squared and a³ as a cubed. When superscripts cannot be used, as in plain ASCII text, common alternative formats include a^n and a**n.

The power aⁿ can be defined also when n is a negative integer, for nonzero a. No natural extension to all real a and n exists, but when the base a is a positive real number, aⁿ can be defined for all real and even complex exponents n via the exponential function e^z. Trigonometric functions can be expressed in terms of complex exponentiation.

Exponentiation where the exponent is a matrix is used for solving systems of linear differential equations.

Exponentiation is used pervasively in many other fields as well, 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.

Graphs of y=a^x for various bases a: base 10 (green), base e (red), base 2 (blue), and base ½ (cyan). Each curve passes through the point (0,1) because any nonzero number raised to the power 0 is 1. At x=1, the y-value equals the base because any number raised to the power 1 is itself.

Terminology

When this article refers to 'an odd power' of a number it means the exponent is an odd number, not that the result is odd. For instance 2³ which is 8 is an odd power of 2 because the exponent is 3. This is the usual usage and applies to any similar form like an even power, negative power, or positive power.

Integer exponents

The exponentiation operation with integer exponents requires only elementary algebra.

Positive integer exponents

The expression a² = a·a is called the square of a because the area of a square with side-length a is a².

The expression a³ = a·a·a is called the cube, because the volume of a cube with side-length a is a³.

So 3² is pronounced "three squared", and 2³ is "two cubed".

The exponent says 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".

Formally, powers with positive integer exponents may be defined by the initial condition a¹ = a and the recurrence relation aⁿ⁺¹ = a·aⁿ.

Exponents one and zero

Notice that a¹ is the "product" of only one a, which is defined to be a. Also note that a^{n − 1} = aⁿ/a. Assuming n = 1, we get a⁰ = 1. Another way of saying this is that when n, m, and n − m are positive (and if a is not equal to zero), one can see that

$\frac{a^n}{a^m} = a^{n - m}.$

Extended to the special case when n and m are equal, the equality would read

$1 = \frac{a^n}{a^n} = a^{n - n} = a^0$

since both the numerator and the denominator are equal. Therefore we take this as the definition of a⁰. This leads to the following rule:

• Any number raised to the power 1 is the number itself.
• Any nonzero number raised to the power 0 is 1; one interpretation of these powers is as empty products. The case of 0⁰ is discussed below.

Combinatorial interpretation

For nonnegative integers n and m, the power n^m equals the cardinality of the set of m-tuples from an n-element set, or the number of 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 empty tuple.

See also exponentiation over sets.

Negative integer exponents

By definition, raising a nonzero number to the −1 power produces its reciprocal:

$a^{-1} = \frac{1}{a}.$

One also defines

$a^{-n} = \frac{1}{a^n}$

for any nonzero a and any positive integer n. Raising 0 to a negative power would imply division by 0, so it is left undefined.

The definition of a⁻ⁿ for nonzero a is made so that the identity a^maⁿ = a^m+n, initially true only for nonnegative integers m and n, holds for arbitrary integers m and n. In particular, requiring this identity for m = −n is requiring

$a^{-n} \, a^{n} = a^{-n\,+\,n} = a^0 = 1,$

where a⁰ is defined above, and this motivates the definition a⁻ⁿ = ¹/_aⁿ shown above.

Exponentiation to a negative integer power can alternatively be seen as repeated division of 1 by the base. For instance,

$3^{-4} = (((1/3)/3)/3)/3 = \frac{1}{81} = \frac{1}{3^{4}}$.

Identities and properties

The most important identity satisfied by integer exponentiation is

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

This identity has the consequence

$a^{m - n} =\frac{a^m}{a^n}$

for a ≠ 0, and

$(a^m)^n = a^{m\cdot n}$

Another basic identity is

$(a \cdot b)^n = a^n \cdot b^n$

While addition and multiplication are commutative (for example, 2+3 = 5 = 3+2 and 2·3 = 6 = 3·2), exponentiation is not commutative: 2³ = 8, but 3² = 9.

Similarly, while addition and multiplication are associative (for example, (2+3)+4 = 9 = 2+(3+4) and (2·3)·4 = 24 = 2·(3·4), exponentiation is not associative either: 2³ to the 4th power is 8⁴ or 4096, but 2 to the 3⁴ power is 2⁸¹ or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, the order is usually understood to be top-down, not bottom-up:

$a^{b^c}=a^{(b^c)}\ne (a^b)^c=a^{(b\cdot c)}=a^{b\cdot c}.$

Powers of ten

See Scientific notation

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, 10³ = 1000 and 10⁻⁴ = 0.0001.

Exponentiation with base 10 is used in scientific notation to describe large or small numbers. For instance, 299,792,458 meters/second (the speed of light in a vacuum, in meters per second) can be written as 2.99792458·10⁸ m/s and then approximated as 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 10³ = 1000, so a kilometre is 1000 metres.

Powers of two

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

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 1000 in binary.

Powers of 1

The integer powers of one are 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.

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 the section on Powers of complex numbers.

Large exponents

The limit of a sequence of powers of a number greater than one diverges, in other words they grow without bound:

aⁿ → ∞ as n → ∞ when a > 1 .

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

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

aⁿ → 0 as n → ∞ when |a| < 1 .

Any power of one is always itself:

aⁿ = 1 for all n if a = 1 .

If the number a 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 section below Powers of e.

Other limits, in particular of those tending to indeterminate forms, are described in limits of powers below.

Real powers of positive numbers

Raising a positive real number to a power that is not an integer can be accomplished in two ways.

• Rational number exponents can be defined in terms of n^th roots, and arbitrary nonzero exponents can then be defined by continuity.
• The natural logarithm can be used to define real exponents using the exponential function.

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

$(a^r)^s = a^{r\cdot s}$

cannot be extended consistently to where a is a negative real number, see negative nth roots. The failure of this identity is the basis for the problems with complex number powers detailed under failure of power and logarithm identities.

Principal n-th root

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

Main article: n-th root

An n-th root of a number a is a number x such that xⁿ = a.

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

When one speaks of the n-th root of a positive real number a, one usually means the principal n-th root.

Rational exponents

A power of a positive real number a with a rational exponent m/n in lowest terms satisfies

$a^{m/n} = \left(a^m\right)^{1/n} = \sqrt[n]{a^m}$

where m is an integer and n is a positive integer.

Powers of e

Main article: 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. It provides a path for defining exponentiation with noninteger exponents. It is defined as the following limit where the power goes to infinity as the base tends to one:

$e =\lim_{n \rightarrow \infty} \left(1+\frac 1 n \right)^n .$

The exponential function, defined by

$e^x =\lim_{n \rightarrow \infty} \left(1+\frac x n \right)^n ,$

has the x written as a power as it satisfies the basic exponential identity

$e^{x+y} = e^{x} \cdot e^{y} .$

The exponential function is defined for all integer, fractional, real, and complex values of x. It can even be used to extend exponentiation to some nonnumerical entities such as square matrices; however, the exponential identity only holds when x and y commute.

A short proof that e to a positive integer power k is the same as e^k is:

$\begin{align} (e)^k &= \left[\lim_{n \rightarrow \infty} \left(1+\frac{1}{n} \right) ^n\right]^k = \lim_{n \rightarrow \infty} \left[\left(1+\frac{1}{n} \right) ^n\right]^k \\ &= \lim_{n \rightarrow \infty} \left(1+\frac k {n\cdot k} \right)^{n \cdot k} = \lim_{n \cdot k \rightarrow \infty} \left(1+\frac k {n\cdot k} \right)^{n \cdot k} \\ &= \lim_{m \rightarrow \infty} \left(1+\frac k m \right)^m = e^k \end{align}$

This proof shows also that e^x+y satisfies the exponential identity when x and y are positive integers. These results are in fact generally true for all numbers, not just for the positive integers.

Real exponents

Since any real number can be approximated by rational numbers, exponentiation to an arbitrary real exponent x can be defined by continuity with the rule

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

where the limit as r gets close to x is taken only over rational values of r.

For example, if

$x \approx 1.732$

then

$5^x \approx 5^{1.732} =5^{433/250}=\sqrt[250]{5^{433}} \approx 16.241 .$

Exponentiation by a real power is normally accomplished using logarithms instead of using limits of rational powers.

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

$b = e^{\ln b}.\,$

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

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

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.

Negative nth roots

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.

If n is even, then xⁿ = a has two real solutions if a is positive, which are the positive and negative nth roots. The equation has no solution in real numbers if a is negative.

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

Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for negative a if m and n are odd, and can be either sign if a is positive and n is even. (−27)^1/3 = −3, (−27)^2/3 = 9, and 4^3/2 has two roots 8 and −8. Since there is no real number x such that x² = −1, the definition of a^m/n when a is negative and n is even must use the imaginary unit i, as described more fully in the section Powers of complex numbers.

Neither the logarithm method nor the rational exponent method can be used to define a^r as a real number for a negative real number a and an arbitrary real number r. Indeed, e^r is positive for every real number r, so ln(a) is not defined as a real number for a ≤ 0. (On the other hand, arbitrary complex powers of negative numbers a can be defined by choosing a complex logarithm of a.)

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

For example, consider a = −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.

Care needs to be taken when applying the power law 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.

Complex powers of positive real numbers

Imaginary powers of e

Main article: Exponential function

The exponential function e^z can be defined as the limit of (1 + z/N)^N, as N approaches infinity, and thus e^iπ is the limit of (1 + iπ/N)^N. In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ/N)^N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + iπ/N)^N. It can be seen that as N gets larger (1 + iπ/N)^N approaches a limit of −1. Therefore, e^iπ = −1, which is known as Euler's identity.

The geometric interpretation of the operations on complex numbers and the definition of powers of e is the clue to understanding e^ix for real x. Consider the right triangle (0, 1, 1 + ix/n). For big values of n the triangle is almost a circular sector with a small central angle equal to x/n radians. The triangles (0, (1 + ix/n)^k, (1 + ix/n)^k+1) are mutually similar for all values of k. So for large values of n the limiting point of (1 + ix/n)ⁿ is the point on the unit circle whose angle from the positive real axis is x radians. The polar coordinates of this point are (r, θ) = (1, x), and the cartesian coordinates are (cos x, sin x). So e ^ix = cos x + isin x, and 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:

$\{ z : e^z = 1 \} = \{ 2k\pi i : k \in \mathbb{Z} \}.$

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

$\{ z : e^z = a \} = \{ b + 2k\pi i : k \in \mathbb{Z} \}.$

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

Main article: Euler's formula

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

$\cos(z) = \frac{e^{i\cdot z} + e^{-i\cdot z}}{2}; \qquad \sin(z) = \frac{e^{i\cdot z} - e^{-i\cdot z}}{2\cdot i}.\,$

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

$e^{i\cdot (x+y)}=e^{i\cdot x}\cdot e^{i\cdot y}.\,$

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

Complex powers of e

The power z = e^x+i·y can be computed as e^x · e^i·y. The real factor e^x is the absolute value of z and the complex factor e^i·y identifies the direction of z.

Complex powers of positive real numbers

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

2ⁱ = e ^i·ln(2) = cos(ln(2)) + i·sin(ln(2)) ≈ 0.76924 + 0.63896i

eⁱ ≈ 0.54030 + 0.84147i

10ⁱ ≈ −0.66820 + 0.74398i

(e^2π)ⁱ ≈ 535.49ⁱ ≈ 1

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 powers of positive real numbers 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 (see below). 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 for the corresponding real numbers.

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. When dealing with complex numbers the complex number operation is normally used instead.

Complex power of a complex number

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

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

a^b = e^blog a

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

If b is an integer, then the value of a^b is independent of the choice of log a, and it agrees with the earlier definition of exponentation with an integer exponent.

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

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

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

A similar construction is employed in quaternions.

Complex roots of unity

Main article: Root of unity

The three 3rd roots of 1

A complex number a such that aⁿ = 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 zⁿ = 1 but z^k ≠ 1 for all natural numbers k such that 0 < k < n, then z 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 4-th roots of unity; the other one is −i.

The number e^{2πi (1/n)} is the primitive nth root of unity with the smallest positive complex 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.^[1])

The other nth roots of unity are given by

$\left( e^{ 2 \pi i / n } \right) ^k = e^{2 \pi i k / n}$

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 a^z in the important special case where z = 1/n and n is a positive integer. These are the nth roots of a; they are solutions of the equation xⁿ = a. 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 a^1/n as the principal value of the root. If a is a positive real number, it is also conventional to select a positive real number as the principal value of the root a^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 a is obtained by multiplying the principal value a^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

$z = re^{i\theta} = e^{\ln(r) + i\theta} \,,$

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 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 a^b, write a in polar form:

$a = r e^{i\theta} \,$.

Then

$\log a = \log r + i \theta \,,$

and thus

$a^b = e^{b \log a} = e^{b(\log r + i\theta)}. \,$

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

$\left( r^c e^{-d\theta} \right) e^{i (d \log r + c\theta)} = \left( r^c e^{-d\theta} \right) \left[ \cos(d \log r + c\theta) + i \sin(d \log r + c\theta) \right].$

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:

$i = 1 \cdot e^{i \pi/2},\,$

$i = 0 + 1i.\,$

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

$\ i^i = \left( 1^0 e^{-\pi/2} \right) e^{i(1\cdot \log 1 + 0 \cdot \pi/2)} = e^{-\pi/2} \approx 0.2079.$

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

$-2 = 2e^{i \pi} \, ,$

and use the formula above to compute

$(-2)^{3+4i} = \left( 2^3 e^{-4\pi} \right) e^{i(4\log(2) + 3\pi)} \approx (2.602 - 1.006 i) \cdot 10^{-5}.$

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

$i = 1 \cdot e^{i \pi/2 + i 2 \pi k}, \ \text{where } k \text{ is an integer},\,$

$i^i = e^{i \left(i \pi/2 + i 2 \pi k\right)} ,$

$= e^{-\left(\pi/2 + 2 \pi k\right)} .$

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. For example:

• The identity log(a^b) = b · log a holds whenever a is a positive real number and b is a real number. But for the principal branch of the complex logarithm one has

  $i\pi = \log(-1) = \log((-i)^2) \,\neq\, 2\log(-i) = 2(-i\pi/2) = -i\pi.$

  Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:

  $\log(a^b) \equiv b \cdot \log(a) \pmod{2 \pi i} .$

  This identity does not hold even when considering log as a multivalued function. The possible values of log(a^b) contain those of b · log a as a subset. Using Log(a) for the principal value of log(a) and m, n as any integers the possible values of both sides are:

  $\left\{\log(a^b)\right\} = \left\{ b \cdot \operatorname{Log}(a) + b \cdot 2 \pi i n + 2 \pi i m \right\} ,$

  $\left\{b \cdot \log(a)\right\} = \left\{ b \cdot \operatorname{Log}(a) + b \cdot 2 \pi i n \right\} .$

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

  $1 = (-1\times -1)^{1/2} \not = (-1)^{1/2}(-1)^{1/2} = -1,$

  and

  $i = (-1)^{1/2} = \left (\frac{1}{-1}\right )^{1/2} \not = \frac{1^{1/2}}{(-1)^{1/2}} = \frac{1}{i} = -i.$

  On the other hand, when c 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^a)^b = e^ab holds for real numbers a and b, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:^[3]

  For any integer n, we have:

  1.   $e^{1 + 2 \pi i n} = e^{1} e^{2 \pi i n} = e \cdot 1 = e \,;$
  2.   $\left( e^{1+2\pi i n} \right)^{1 + 2 \pi i n} = e \,;$
  3.   $e^{1 + 4 \pi i n - 4 \pi^{2} n^{2}} = e \,;$
  4.   $e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e \,;$
  5.   $e^{-4 \pi^2 n^2} = 1 \,.$

  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, it is implicit in the first line and not obvious. It 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.

Zero to the zero power

Plot of z = abs(x)^y with red curves yielding different limits as (x,y) approaches (0,0). The green curves all yield a limit of 1.

Most authors agree with the statements related to 0⁰ in the two lists below, but make different decisions when it comes to defining 0⁰ or not: see the next subsection.

In most settings not involving continuity in the exponent, interpreting 0⁰ as 1 simplifies formulas and eliminates the need for special cases in theorems. (See the next paragraph for some settings that do involve continuity.) For example:

• Regarding a⁰ as an empty product assigns it the value 1, even when a=0.
• The combinatorial interpretation of 0⁰ is the number of empty tuples of elements from the empty set. There is exactly one empty tuple.
• 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.^[4]
• The notation $\textstyle \sum a_nx^n$ for polynomials and power series rely on defining 0⁰ = 1. Identities like $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ and $e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ and the binomial theorem $(1+x)^n = \sum_{k = 0}^n \binom{n}{k} x^k$ are not valid for x = 0 unless 0⁰ = 1.^[5]
• In differential calculus, the power rule $\textstyle\frac{d}{dx} x^n = nx^{n-1}$ is not valid for n = 1 at x = 0 unless 0⁰ = 1.

On the other hand, when 0⁰ arises from a limit of the form $\lim_{(x,y)\rarr(0,0)} x^y$, it must be handled as an indeterminate form.

• 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.^[6] 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 be undefined. 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:

$\lim_{t \to 0^+} {t}^{t} = 1, \quad \lim_{t \to 0^+} (e^{-1/t^2})^t = 0, \quad \lim_{t \to 0^+} (e^{-1/t^2})^{-t} = +\infty, \quad \lim_{t \to 0^+} (e^{-1/t})^{at} = e^{-a}$.

So 0⁰ is an indeterminate form. This behavior shows that 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 0⁰ is defined.^[7] However, under certain conditions, such as when f and g are both analytic functions and f is nonnegative, the limit approaching from the right is always 1.^[8][9][10]

• In the complex domain, the function z^w is defined for nonzero z by choosing a branch of log z and setting z^w := e^w log z, but there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.^[11]

History of differing points of view

Different authors interpret the situation above in different ways:

• Some argue that the best value for 0⁰ depends on context, and hence that defining it once and for all is problematic.^[12] According to Benson (1999), "The choice whether to define 0⁰ is based on convenience, not on correctness."^[13]
• Others argue that 0⁰ is 1. According to p. 408 of Knuth (1992), it "has to be 1", although he goes on to say that "Cauchy had good reason to consider 0⁰ as an undefined limiting form" and that "in this much stronger sense, the value of 0⁰ is less defined than, say, the value of 0+0" (emphases in original).^[14]

The debate has been going on at least since the early 19th century. At that time, most mathematicians agreed that 0⁰ = 1, until in 1821 Cauchy^[15] listed 0⁰ along with expressions like ⁰⁄₀ in a table of undefined forms. In the 1830s Libri^[16][17] published an unconvincing argument for 0⁰ = 1, and Möbius^[18] sided with him, erroneously claiming that $\lim_{t \to 0^+} f(t)^{g(t)} = 1$ whenever $\lim_{t \to 0^+} f(t) = \lim_{t \to 0^+} g(t) = 0.$ A commentator who signed his name simply as "S" provided the counterexample of (e^−1/t)^t, and this quieted the debate for some time, with the apparent conclusion of this episode being that 0⁰ should be undefined. More details can be found in Knuth (1992).^[14]

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 different functions for computing a power:^[19]

• 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^[20] and C++ standards describe this as the normative behaviour. The Java standard^[21] mandates this behavior. The .NET Framework method System.Math.Pow also treats 0⁰ as 1.^[22]

Mathematics software

• Sage simplifies a⁰ to 1, even if no constraints are placed on a.^[23] It does not simplify 0^a, and it takes 0⁰ to be 1.
• Maple simplifies a⁰ to 1 and 0^a to 0, even if no constraints are placed on a (the latter simplification is only valid for a > 0), and evaluates 0⁰ to 1.^{[citation needed]}
• Macsyma also simplifies a⁰ to 1 and 0^a to 0, even if no constraints are placed on a, but issues an error for 0⁰.^{[citation needed]}
• Mathematica and Wolfram Alpha simplify a⁰ into 1, even if no constraints are placed on a.^[24] While Mathematica does not simplify 0^a, Wolfram Alpha returns two results, 0 and indeterminate.^[25] Both Mathematica and Wolfram Alpha take 0⁰ to be an indeterminate form.^[26]

Limits of powers

The section zero to the zero power gives a number of examples of limits which 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 ask 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,−∞).^[27] 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:

• a^+∞ = +∞ and a^−∞ = 0, when 1 < a ≤ +∞.
• a^+∞ = 0 and a^−∞ = +∞, when 0 ≤ a < 1.
• 0^b = 0 and (+∞)^b = +∞, when 0 < b ≤ +∞.
• 0^b = +∞ and (+∞)^b = 0, when −∞ ≤ b < 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 tⁿ → +∞ as t tends to 0 through positive values, but not negative ones.

Efficient computation of integer powers

The simplest method of computing aⁿ 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 aⁿ 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 aⁿ is a difficult problem for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available.^[28]

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.

Generalizations

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

$x^1=x\,$

$x^n=x^{n-1}x \quad\hbox{for }n>1$

One has the following properties

• $\ (x^i x^j) x^k = x^i (x^j x^k)$ (power-associative property),
• $\ x^{m+n}=x^m x^n$
• $\ (x^m)^n=x^{mn}$

If the operation has a two-sided identity element 1 (often denoted by e), then x⁰ is defined to be equal to 1 for any x.

• $\ x1 = 1x = x,$ Two sided identity
• $\ x^0=1$

If the operation also has two-sided inverses, and multiplication 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.

• $\ x x^{-1} = x^{-1} x = 1,$ Two sided inverse
• $\ (x y) z = x (y z),$ Associative
• $\ x^{-n}=\left(x^{-1}\right)^n$
• $\ x^{m-n}=x^m x^{-n}$

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

• $\ (xy)^n = x^n y^n$

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 one has several operations around, any of which might be repeated using exponentiation, 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

Main article: Cartesian product

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

$\bigoplus_{i \in \mathbb{N}} V_{i},$

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 Euclidean 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:

$S^N \equiv \{ f\colon N \to S \}.\,$

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

Main article: Cartesian closed category

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

Main articles: cardinal arithmetic and ordinal arithmetic

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.^[4] 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.

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 tetration. Iterating tetration leads to another operation, and so on. 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, tetration yield 6, 9, 27, and 7,625,597,484,987 respectively.

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, TeX (and its derivatives), TI-BASIC, bc (for integer exponents), Haskell (for nonnegative integer exponents), Lua, ASP and most computer algebra systems
• x ^^ y: Haskell (for fractional base, integer exponents), D
• x ** y: Ada, Bash, COBOL, Fortran, FoxPro, Gnuplot, OCaml, Perl, PL/I, Python, Rexx, Ruby, SAS, Tcl, ABAP, Haskell (for floating-point exponents), Turing, VHDL
• x⋆y: APL
• Power(x, y): Microsoft Excel, Delphi/Pascal (declared in "Math"-unit)
• pow(x, y): C, C++, PHP, Tcl, Python
• math.pow(x, y): Python (always fractional results)
• Math.pow(x, y): Java, JavaScript, Modula-3, Standard ML
• Math.Pow(x, y) or BigInteger.Pow(x, y): C# (and other languages using the BCL)
• (expt x y): Common Lisp, Scheme
• math:pow(x, y): Erlang

In Bash, C, C++, C#, 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.

History of the notation

The term power was used by the Greek mathematician Euclid for the square of a line.^[29] In the 9th century, 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.^[30]

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. Samuel Jeake introduced the term indices in 1696.^[29] In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), surfolide (fifth), zenzicube (sixth), second surfolide (seventh) and Zenzizenzizenzic (eighth).^[31] 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,^[32] is now rare and should not be confused with its more common meaning.

See also

Complex logarithm Exponential decay Exponential function Exponential growth List of exponential topics

Mathematics portal Logarithm Modular exponentiation Nth root Tetration Unicode subscripts and superscripts

References

1. ^ This definition of a principal root of unity can be found in:
   • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2001). Introduction to Algorithms (second ed.). MIT Press. ISBN 0262032937.  Online resource
   • Paul Cull, Mary Flahive, and Robby Robson (2005). Difference Equations: From Rabbits to Chaos (Undergraduate Texts in Mathematics ed.). Springer. ISBN 0387232346.  Defined on page 351, available on Google books.
   • "Principal root of unity", MathWorld.
2. ^ Complex number to a complex power may be real at Cut The Knot gives some references to iⁱ
3. ^ Steiner J, Clausen T, Abel NH (1827). "Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen". Journal für die reine und angewandte Mathematik 2: 286–287. http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=270662. 
4. ^ ^{a b} N. Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5.
5. ^ "Some textbooks leave the quantity 0⁰ undefined, because the functions x⁰ and 0^x have different limiting values when x decreases to 0. But this is a mistake. We must define x⁰ = 1, for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = −y. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant".Ronald Graham, Donald Knuth, and Oren Patashnik (1989-01-05). "Binomial coefficients". Concrete Mathematics (1st ed.). Addison Wesley Longman Publishing Co. p. 162. ISBN 0-201-14236-8. 
6. ^ Malik, S. C.; Savita Arora (1992). Mathematical Analysis. New York: Wiley. p. 223. ISBN 978-8122403237. "In general the limit of φ(x)/ψ(x) when x=a in case the limits of both the functions exist is equal to the limit of the numerator divided by the denominator. But what happens when both limits are zero? The division (0/0) then becomes meaningless. A case like this is known as an indeterminate form. Other such forms are ∞/∞ 0×∞, ∞−∞, 0⁰, 1^∞ and ∞⁰." 
7. ^ L. J. Paige (March 1954). "A note on indeterminate forms". American Mathematical Monthly 61 (3): 189–190. doi:10.2307/2307224. JSTOR 2307224. 
8. ^ sci.math FAQ: What is 0^0?
9. ^ Rotando, Louis M.; Korn, Henry (1977). "The Indeterminate Form 0⁰". Mathematics Magazine (Mathematical Association of America) 50 (1): 41–42. doi:10.2307/2689754. JSTOR 2689754. 
10. ^ Lipkin, Leonard J. (2003). "On the Indeterminate Form 0⁰". The College Mathematics Journal (Mathematical Association of America) 34 (1): 55–56. doi:10.2307/3595845. JSTOR 3595845. 
11. ^ "... Let's start at x = 0. Here x^x is undefined." Mark D. Meyerson, The x^x Spindle, Mathematics Magazine 69, no. 3 (June 1996), 198-206.
12. ^ Examples include Edwards and Penny (1994). Calculus, 4th ed,, Prentice-Hall, p. 466, and Keedy, Bittinger, and Smith (1982). Algebra Two. Addison-Wesley, p. 32.
13. ^ Donald C. Benson, The Moment of Proof : Mathematical Epiphanies. New York Oxford University Press (UK), 1999. ISBN 978-0-19-511721-9
14. ^ ^{a b} Donald E. Knuth, Two notes on notation, Amer. Math. Monthly 99 no. 5 (May 1992), 403–422.
15. ^ Augustin-Louis Cauchy, Cours d'Analyse de l'École Royale Polytechnique (1821). In his Oeuvres Complètes, series 2, volume 3.
16. ^ Guillaume Libri, Note sur les valeurs de la fonction 0^0x, Journal für die reine und angewandte Mathematik 6 (1830), 67–72.
17. ^ Guillaume Libri, Mémoire sur les fonctions discontinues, Journal für die reine und angewandte Mathematik 10 (1833), 303–316.
18. ^ A. F. Möbius, Beweis der Gleichung 0⁰ = 1, nach J. F. Pfaff, Journal für die reine und angewandte Mathematik 12 (1834), 134–136.
19. ^ Handbook of Floating-Point Arithmetic. Birkhäuser Boston. 2009. p. 216. ISBN 978-0817647049. 
20. ^ John Benito (April 2003) (PDF). Rationale for International Standard—Programming Languages—C. Revision 5.10. p. 182. http://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf. 
21. ^ "Math (Java 2 Platform SE 1.4.2) pow". Oracle. http://download.oracle.com/javase/1.4.2/docs/api/java/lang/Math.html#pow%28double,%20double%29. 
22. ^ ".NET Framework Class Library Math.Pow Method". Microsoft. http://msdn.microsoft.com/en-us/library/system.math.pow.aspx. 
23. ^ "Sage worksheet calculating x^0". Jason Grout. http://sagenb.org/home/pub/2433/. 
24. ^ "Wolfram Alpha calculates a^0". Wolfram Alpha LLC, accessed July 24, 2011. http://www.wolframalpha.com/input/?i=a^0. 
25. ^ "Wolfram Alpha calculates 0^a". Wolfram Alpha LLC, accessed July 24, 2011. http://www.wolframalpha.com/input/?i=0^a. 
26. ^ "Wolfram Alpha calculates 0^0". Wolfram Alpha LLC, accessed July 24, 2011. http://www.wolframalpha.com/input/?i=0^0. 
27. ^ N. Bourbaki, Topologie générale, V.4.2.
28. ^ Gordon, D. M. 1998. A survey of fast exponentiation methods. J. Algorithms 27, 1 (Apr. 1998), 129-146. doi:http://dx.doi.org/10.1006/jagm.1997.0913
29. ^ ^{a b} O'Connor, John J.; Robertson, Edmund F., "Etymology of some common mathematical terms", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Miscellaneous/Mathematical_notation.html .
30. ^ O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Qalasadi.html .
31. ^ Quinion, Michael. "World Wide Words". http://www.worldwidewords.org/weirdwords/ww-zen1.htm. Retrieved 2010-03-19 
32. ^ This definition of "involution" appears in the OED second edition, 1989, and Merriam-Webster online dictionary [1]. The most recent usage in this sense cited by the OED is from 1806.

External links

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