Exponentiation Explained

Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b 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 b (the product itself can also be called power):

bn=\underbrace{b x … x b}n,

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

b x n=\underbrace{b+ … +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.

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 expressionb3 = 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

b1=b

and the recurrence relation

bn+1=bnb.

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

bm+n=bmbn.

Arbitrary integer exponents

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

bn=

bn+1
b

.

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

b0=

b1
b

=1

b-1=

b0
b

=

1
b
and more generally,

b-n=

1
bn
for any nonzero b and any nonnegative integer n (and indeed any integer n).

The following observations may be made:

The identity

bm+n=bmbn,

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 = │ There is no 5-tuple from the empty set.
14 = │ │ = 1.There is one 4-tuple from a one-element set.
23 = │ │ = 8.There are eight 3-tuples from a two-element set.
32 = │ │ = 9.There are nine 2-tuples from a three-element set.
41 = │ │ = 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:

bm=bmbn

(bm)n=bm

(bc)n=bncn.

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:

pq
b
(pq)
=b

\ne(bp)q=b(p=bp.

Particular bases

Powers of ten

See Scientific notationIn 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 < 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:

bn → ∞ as n → ∞ when b > 1 .This can be read as "b to the power of n tends to +∞ as n tends to infinity when b is greater than one".

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

bn → 0 as n → ∞ when |b| < 1 .Any power of one is always 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

See main article: nth root.

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

bm/n=\left(bm\right)1/n=\sqrt[n]{bm}

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

(br)s=br

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]

bx=\limrbr(r\inQ,x\inR),

where the limit as r gets close to x is taken only over rational values of r. This limit only exists for positive b. The (ε, δ)-definition of limit is used, this involves showing that for any desired accuracy of the result

bx

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

[b3,b4]

,

[b3.1,b3.2]

,

[b3.14,b3.15]

,

[b3.141,b3.142]

,

[b3.1415,b3.1416]

,

[b3.14159,b3.14160]

, ...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)=bx

is thus defined for any real number .

The exponential function

See main article: 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)=\limn\left(1+

x
n

\right)n

Among other properties, exp satisfies the exponential identity:

\exp(x+y)=\exp(x)\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=eln.

If bx is to preserve the logarithm and exponent rules,then one must have

bx=(eln)x=ex

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 < 0, the function f is not even continuous on the set of rational numbers r for which it is defined.

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

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

Complex powers of positive real numbers

Imaginary powers of e

See main article: Exponential function.

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 2πi:

\{z:ez=1\}=\{2k\pii:k\inZ\}.

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

\{z:ez=w\}=\{v+2k\pii:k\inZ\}.

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

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

Trigonometric functions

See main article: Euler's formula. It follows from Euler's formula stated above that the trigonometric functions cosine and sine are

\cos(z)=

eiz+e-iz
2

;    \sin(z)=

eiz-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

ei(x+y)=eixeiy.

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

Complex powers of e

The power

z=ex

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

(bz)u=bzu

is not generally valid for complex powers. A simple counterexample is given by:

(e2\pi)i=1i=1 ≠ e-2\pi=e2\pi.

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 numbersabove. 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

wz=ez

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

See main article: Root of unity.

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

If wn = 1 but 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 e2πi (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)k=e2

for 2 ≤ kn.

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=rei\theta=eln(r),

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 wz, write w in polar form:

w=rei\theta

.Then

logw=logr+i\theta,

and thus

wz=ez=ez(log.

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

\left(rce-d\theta\right)ei=\left(rce-d\theta\right)\left[\cos(dlogr+c\theta)+i\sin(dlogr+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 ⋅ ei,

i=0+1i.

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

ii=\left(10e-\pi/2\right)ei(1 ⋅ =e-\pi/2 ≈ 0.2079.

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

-2=2ei,

and use the formula above to compute

(-2)3+4i=\left(23e-4\pi\right)ei(4log(2)(2.602-1.006i) ⋅ 10-5.

The value of a complex power depends on the branch used. For example, if the polar form i = 1ei(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:[3]

i=1 ⋅ ei, wherekisaninteger,

ii=ei,

=e-\left(\pi/2.

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:

i\pi=log(-1)=log((-i)2) ≠ 2log(-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(wz)\equivz ⋅ log(w)\pmod{2\pii}.

This identity does not hold even when considering log as a multivalued function. The possible values of log(wz) contain those of z · log&thinsp;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(wz)\right\}=\left\{z\operatorname{Log}(w)+z ⋅ 2\piin+2\piim\right\},

\left\{z ⋅ log(w)\right\}=\left\{z\operatorname{Log}(w)+z ⋅ 2\piin\right\}.

1=(-1 x -1)1/2\not=(-1)1/2(-1)1/2=-1,

and

i=(-1)1/2=\left(

1
-1

\right)1/2\not=

11/2
(-1)1/2

=

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 . The identity holds but saying

Notes and References

  1. Book: Denlinger, Charles G.. Elements of Real Analysis. Jones and Bartlett. 2011. 278–283. 978-0-7637-7947-4.
  2. This definition of a principal root of unity can be found in:
    • Book: Introduction to Algorithms. second. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. MIT Press. 2001. 0-262-03293-7. Online resource
    • Book: Difference Equations: From Rabbits to Chaos. Undergraduate Texts in Mathematics. Paul Cull, Mary Flahive, and Robby Robson. 2005. Springer. 0-387-23234-6. Defined on page 351, available on Google books.
    • "Principal root of unity", MathWorld.
  3. http://www.cut-the-knot.org/do_you_know/complex.shtml Complex number to a complex power may be real