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

b^n = \underbrace{b \times \cdots \times b}_n,

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

b \times n = \underbrace{b + \cdots + b}_n.

The exponent is usually shown as a superscript to the right of the base. The exponentiation bn can be read as: b raised to the n-th power, b raised to the power of n, or possibly b raised to the exponent of n, most briefly as b to the n. Some exponents have their own pronunciation: for example, b2 is usually read as b squared and b3 as b cubed.

The power bn can be defined also when n is a negative integer, for nonzero b. No natural extension to all real b and n exists, but when the base b is a positive real number, bn can be defined for all real and even complex exponents n via the exponential function ez. 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, 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=bx for various bases b: base 10 (green), base e (red), base 2 (blue), and base  (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.
Graphs of y=bx for various bases b: base 10 (green), base e (red), base 2 (blue), and base  (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.


Background and terminology

The expression b2 = b b is called the square of b because the area of a square with side-length b is b2.

The expression b3 = b b b is called the cube, because the volume of a cube with side-length b is b3.

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".

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

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

Formally, powers with positive integer exponents may be defined by the initial condition

\,b^1 = b

and the recurrence relation

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

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

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

Arbitrary integer exponents

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

\,b^{n} = \frac{b^{n+1}}{b}.

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

\,b^0 = \frac{b^{1}}{b} = 1
\,b^{-1} = \frac{b^{0}}{b} = \frac{1}{b}

and more generally,

\,b^{-n} = \frac{1}{b^n}

for any nonzero b and any nonnegative integer n (and indeed any integer n).

The following observations may be made:

  • Any number raised to the exponent 1 is the number itself.
  • Any nonzero number raised to the exponent 0 is 1; one interpretation of these powers is as empty products.
  • These equations do not decide the value of 00. This is discussed below.
  • Raising 0 to a negative exponent would imply division by 0, so it is left undefined.

The identity

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

initially defined only for positive integers m and n, holds for arbitrary integers m and n, with the constraint that m and n must both be positive when b is zero.

Combinatorial interpretation

For nonnegative integers n and m, the power nm 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.

Identities and properties

The following identities hold, provided that the base is non-zero whenever the integer exponent is not positive:

b^{m + n} = b^m \cdot b^n
(b^m)^n = b^{m\cdot n}
(b \cdot c)^n = b^n \cdot c^n.

Exponentiation is not commutative. This contrasts with addition and multiplication, which are. For example, and , but , whereas .

Exponentiation is not associative either. Addition and multiplication are. For example, and , but 23 to the 4 is 84 or 4096, whereas 2 to the 34 is 281 or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up:

b^{p^q}=b^{(p^q)}\ne (b^p)^q=b^{(p\cdot q)}=b^{p\cdot q}.

(In the above formula, one could think that it is the (smaller) size of the last exponent q with respect to the p, which indicates the order of evaluation, but the convention also applies to the linearized writing b^p^q = b^(p^q) (b^p)^q, as it is used for example in computer algebra systems.)

Particular bases

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, = 1000 and = 0.0001.

Exponentiation with base 10 is used in scientific notation to denote large or small numbers. For instance, 299,792,458 m/s (the speed of light in vacuum, in metre per second) can be written as and then approximated as .

SI prefixes based on powers of 10 are also used to describe small or large quantities. For example, the prefix kilo means , 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 as 1000 in binary.

Powers of one

The integer powers of one are all one: .

Powers of zero

If the exponent is positive, the power of zero is zero: , where .

If the exponent is negative, the power of zero (0n, where n

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:

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|

Any power of one is always itself:

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)ne 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.

Rational powers

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

An n-th 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 n-th root of b. It is denoted n b, where is the radical symbol; alternatively, it may be written b1/n. For example: 41/2 = 2, 81/3 = 2,

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

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

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.

Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for negative b if m and n are odd, and can be either sign if b is positive and n is even. ( 27)1/3 =  3, ( 27)2/3 = 9, and 43/2 has two roots 8 and 8. Since there is no real number x such that x2 =  1, the definition of bm/n when b is negative and n is even must use the imaginary unit i, as described more fully in the section Powers of complex numbers.

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

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

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

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 = 93/2 = 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 powers

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

(b^r)^s = b^{r\cdot s}

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

The extension of exponentiation to real powers of positive real numbers can be done either by extending the rational powers to reals by continuity, or more usually by using the exponential function and its inverse the natural logarithm.

Limits of rational powers

Since any irrational number can be approximated by a rational number, exponentiation of a positive real number b to an arbitrary real exponent x can be defined by continuity with the rule[1]

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. This limit only exists for positive b. The ( , )-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 so all the rational powers in the interval are within the desired accuracy.

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

[b^3,b^4], [b^{3.1},b^{3.2}], [b^{3.14},b^{3.15}], [b^{3.141},b^{3.142}], [b^{3.1415},b^{3.1416}], [b^{3.14159},b^{3.14160}], ...

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

The exponential function

The important mathematical constant , 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 ex usually denotes a generalized exponentiation definition called the exponential function, exp(x), which can be defined in many equivalent ways, for example by:

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

Among other properties, exp satisfies the exponential identity:

\exp(x+y) = \exp(x) \cdot \exp(y)

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

Since \exp(1) is equal to and \exp(x) satisfies the exponential identity, it immediately follows that exp(x) coincides with the repeated-multiplication definition of ex for integer x, and it also follows that rational powers denote (positive) roots as usual, so exp(x) coincides with the ex 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 ex. It is defined for b > 0, and satisfies

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

If bx 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 bx 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 powers of negative numbers

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 41/2 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 br as a real number for a negative real number b and an arbitrary real number r. Indeed, er 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) = br has a unique continuous extension[1] from the rational numbers to the real numbers for each b > 0. But when b 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.

Complex powers of positive real numbers

Imaginary powers of e

The exponential function ez can be defined as the limit of <!-- nowrap -->, as N approaches infinity, and thus ei is the limit of <!-- nowrap -->. In this animation N takes various increasing values from 1 to 100. The computation of <!-- nowrap --> is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of <!-- nowrap -->. It can be seen that as N gets larger <!-- nowrap --> approaches a limit of 1. Therefore, <!-- nowrap --> which is known as Euler's identity.
The exponential function ez can be defined as the limit of , as N approaches infinity, and thus ei is the limit of . In this animation N takes various increasing values from 1 to 100. The computation of is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of . It can be seen that as N gets larger approaches a limit of 1. Therefore, 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 eix for real x. Consider the right triangle For big values of n the triangle is almost a circular sector with a small central angle equal to x/n radians. The triangles are mutually similar for all values of k. So for large values of n the limiting point of is the point on the unit circle whose angle from the positive real axis is x radians. The polar coordinates of this point are and the cartesian coordinates are (cos x, sin x). So and this is Euler's formula, connecting algebra to trigonometry by means of complex numbers.

The solutions to the equation ez = 1 are the integer multiples of 2 i:

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

More generally, if ev = w, then every solution to ez = w can be obtained by adding an integer multiple of 2 i to v:

\{ z : e^z = w \} = \{ v + 2k\pi i : k \in \mathbb{Z} \}.

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

More simply: ei = 1; ex + iy = ex(cos y + i sin y).

Trigonometric functions

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

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

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(x+y)}=e^{ix}\cdot e^{iy}.\,

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

Complex powers of e

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

Complex powers of positive real numbers

If b is a positive real number, and z is any complex number, the power bz is defined as ez ln(b), where x = ln(b) is the unique real solution to the equation ex = b. 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)i 535.49i 1

The identity (b^z)^u=b^{zu} is not generally valid for complex powers. A simple counterexample is given by:

(e^{2\pi i})^i=1^i=1\neq e^{-2\pi}=e^{2\pi i\cdot i}.

The identity is, however, valid when z is a real number, and also when u 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 ex 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 = z1/2 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 w and z with w   0, the notation wz is ambiguous in the same sense that log w is.

To obtain a value of wz, 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

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

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 wz is independent of the choice of log w, and it agrees with the earlier definition of exponentation with an integer exponent.

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

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

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

The three 3rd roots of 1 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 wk 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 4-th roots of unity; the other one is i.

The number e2 i (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.[2])

The other nth roots of unity are given by

\,( e^{ 2 \pi i / n } ) ^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 wq 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 w1/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 w1/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 w1/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 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, ) is the same point in polar coordinates. That is, r is the "radius" r2 = u2 + v2 and is the "angle" = atan2(v, u). The polar angle is ambiguous since any multiple of 2 could be added to without changing the location of the point. Each choice of 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 chosen in the interval ( , ]. 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 wz, write w in polar form:

w = r e^{i\theta} \,.


\log w = \log r + i \theta \,,

and thus

w^z = e^{z \log w} = e^{z(\log r + i\theta)}. \,

If z is decomposed as c + di, then the formula for wz 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 to be in the interval ( , ]. 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, = /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 = 1ei(5 /2) is used to compute i i, the power is found to be e 5 /2; the principal value of i i, computed above, is e /2. The set of all possible values for i i is given by:[3]

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 as single-valued functions. For example:

  • The identity log(bx) = 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
  • 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(w^z) \equiv z \cdot \log(w) \pmod{2 \pi i} .
  • This identity does not hold even when considering log as a multivalued function. The possible values of log(wz) 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:
  • \left\{\log(w^z)\right\} = \left\{ z \cdot \operatorname{Log}(w) + z \cdot 2 \pi i n + 2 \pi i m \right\} ,
  • \left\{z \cdot \log(w)\right\} = \left\{ z \cdot \operatorname{Log}(w) + z \cdot 2 \pi i n \right\} .
  • The identities (bc)x = bxcx and (b/c)x = bx/cx are valid when b and c are positive real numbers and x 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 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 (ex)y = exy holds for real numbers x and y, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:[4]
  • For any integer n, we have:
    1.   e^{1 + 2 \pi i n} = e^{1} e^{2 \pi i n} = e \cdot 1 = e \,;
    1.   \left( e^{1+2\pi i n} \right)^{1 + 2 \pi i n} = e \,;
    1.   e^{1 + 4 \pi i n - 4 \pi^{2} n^{2}} = e \,;
    1.   e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e \,;
    1.   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. Implicit in the first line is that e is a real number, whereas the result of e1+2 in 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. (e^z)^w \ne e^{z w}, but rather (e^z)^w = e^{(z + 2\pi i n) w}\, multivalued over integers n

Zero to the zero power

Plot of <!-- nowrap -->. The red curves (with z constant) yield different limits as (x,y) approaches (0,0). The green curves (of finite constant slope, <!-- nowrap -->) all yield a limit of 1.
Plot of . The red curves (with z constant) yield different limits as (x,y) approaches (0,0). The green curves (of finite constant slope, ) all yield a limit of 1.

For discrete exponents

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 b0 as an empty product assigns it the value 1, even when b = 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.[5]
  • The notation \textstyle \sum a_nx^n for polynomials and power series rely on defining 00 = 1. Identities like \textstyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n and \textstyle e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!} and the binomial theorem \textstyle (1+x)^n = \sum_{k = 0}^n \binom{n}{k} x^k are not valid for x = 0 unless 00 = 1.[6]
  • 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.

In analysis

On the other hand, when 00 arises when trying to determine a limit of the form \lim_{x\rarr 0} f(x)^{g(x)}, 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.[7] 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 xy, 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.[8] 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.[9][10][11]
  • In the complex domain, the function zw is defined for nonzero z by choosing a branch of log z and setting zw := ew log z, but there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.[12]

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.[13] According to Benson (1999), "The choice whether to define 00 is based on convenience, not on correctness."[14]
  • 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).[15]

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[16] listed 00 along with expressions like in a table of undefined forms. In the 1830s Libri[17][18] published an unconvincing argument for 00 = 1, and M bius[19] 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).[15]

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:[20]

  • 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 epower' 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[21] and C++ standards describe this as the normative behaviour. The Java standard[22] mandates this behavior. The .NET Framework method System.Math.Pow also treats 00 as 1.[23]

Mathematics software

  • Sage simplifies b0 to 1, even if no constraints are placed on b.[24] It does not simplify 0x, and it takes 00 to be 1.
  • Maple simplifies b0 to 1 and 0x to 0, even if no constraints are placed on b (the latter simplification is only valid for x > 0), and evaluates 00 to 1.
  • Macsyma also simplifies b0 to 1 and 0x to 0, even if no constraints are placed on b and x, but issues an error for 00.
  • Mathematica and Wolfram Alpha simplify b0 into 1, even if no constraints are placed on b.[25] While Mathematica does not simplify 0x, Wolfram Alpha returns two results, 0 and indeterminate.[26] Both Mathematica and Wolfram Alpha take 00 to be an indeterminate form.[27]

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 xy 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) = xy defined on D = {(x,y)   R2 : x > 0}. Then D can be viewed as a subset of 2 (that is, the set of all pairs (x,y) with x,y belonging to the extended real number line  = [ , + ], 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, ).[28] Accordingly, this allows one to define the powers xy 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
    • 0y = 0 and (+ )y = + , when 0 y + .
      • 0y = + and (+ )y = 0, when y xy for positive values of x. This method does not permit a definition of xy when x x,y) with x 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 of integer powers

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.[29]

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 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^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, gh = 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 (a0a1a2, ..., an 1) represents the function that sends i to ai.

For an infinite cardinal number and a set A, the notation A is also used to denote the set of all functions from a set of size to A. This is sometimes written A 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 Vi is a vector space. Then if Vi = V for each i, the resulting direct sum can be written in exponential notation as V N, or simply VN 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, SN 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 |SN| = |S||N|, where |X| is the cardinality of X. When "2" is defined as {0,1}, we have |2X| = 2|X|, where 2X, 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 0^0 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 and are cardinal numbers, the expression represents the cardinality of the set of functions from any set of cardinality to any set of cardinality .[5] If and 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 xy 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

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

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.[30] Archimedes discovered and proved the law of exponents, 10^a 10^b = 10^{a+b}, necessary to manipulate powers of 10.[31] In the 9th century, the Persian mathematician Muhammad ibn M s al-Khw rizm 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 Ab al-Hasan ibn Al al-Qalas d .[32]

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.[30] In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), surfolide (fifth), zenzicube (sixth), second surfolide (seventh) and Zenzizenzizenzic (eighth).[33] 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,[34] is now rare and should not be confused with its more common meaning.

See also

  • Modular exponentiation
  • Unicode subscripts and superscripts


External links

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