Exponentiation

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Exponentiation is a mathematical operation, written aⁿ, involving two numbers, the base a and the exponent n. When n is a whole number, exponentiation corresponds to repeated multiplication:

just as multiplication by a whole number corresponds to repeated addition:

Exponentiation can also be defined for exponents that are not whole numbers.

Exponentiation is also known as raising the number a to the power n, or a to the nth power. (Another historical synonym, involution,^[1] is now rare and should not be confused with its more common meaning.)

The exponent is usually shown as a superscript to the right of the base. Exponentiation is a basic mathematical tool that 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.

[edit] Exponentiation with integer exponents

The exponentiation operation with integer exponents only requires elementary algebra.

[edit] Positive integer exponents

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

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 or 3 raised to the fifth power.

The word "raised" is usually omitted, and most 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.

[edit] Exponents one and zero

The meaning of 3⁵ may also be viewed as 1·3·3·3·3·3 : the starting value 1 (the identity element of multiplication) is multiplied by the base as many times as indicated by the exponent. With this definition in mind, it is easy to see how to generalize exponentiation to exponents one and zero: 3¹ = 1·3 = 3 and 3⁰ = 1. This leads to the following rules:

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

[edit] Negative integer exponents

Raising a nonzero number to the −1 power produces its reciprocal.

Thus:

Raising 0 to a negative power would imply division by 0, and so is undefined.

A negative integer exponent can also be seen as repeated division by the base. Thus .

[edit] Identities and properties

The most important identity satisfied by integer exponentiation is:

This identity has the following consequences:

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.

[edit] Powers of ten

Powers of 10 are easily computed in the base ten (decimal) number system. For example, 10⁸ = 100000000.

Exponentiation with base 10 is used in scientific notation to describe large or small numbers. For instance, 299,792,458 (the speed of light in a vacuum, in meters per second) can be written as 2.99792458·10⁸ and then approximated as 2.998·10⁸, (or sometimes as 299.8·10⁶, or 299.8E+6, especially in computer software).

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.

[edit] Powers of two

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

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

[edit] 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 (0⁻ⁿ, where n > 0) remains 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.

[edit] Powers of minus one

The powers of minus one are useful for expressing alternating sequences.

If the exponent is even, the power of minus one is one: (−1)²ⁿ = 1.

If the exponent is odd, the power of minus one is minus one: (−1)²ⁿ⁺¹ = −1.

[edit] Powers of the imaginary unit

The powers of the imaginary unit i are useful for expressing sequences of period 4.

[edit] Powers of e

Main article: Exponential function

The number e, the base of the natural logarithm, is the limit of a sequence of positive or negative integer powers

and is approximately equal to 2.71828.

If x is a positive or negative integer then the power e^x can be computed as

The final equality

also holds when x=0. So it generalizes the meaning of e^x; x does not have to be an integer, but can be a fraction, a real number, a complex number, or a square matrix.

Notice in particular that when x is a real number, (1 + x/n) > 0 if n is large enough, and therefore e^x > 0 for every real number x.

[edit] Real powers of positive real numbers

Exponentiation with various bases; from top to bottom, base 10 (green), base e (red), base 2 (blue), base ½ (cyan). Note how all of the curves pass through the point (0, 1). This is because, in accordance with the properties of exponentiation, any non-zero number raised to the power 0 is 1. Also note that at x=1, the y value equals the base. This is because any number raised to the power 1 is that same number.

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

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 are true for non-integer exponents as well.

[edit] Rational exponents

For a given exponent, the inverse of exponentiation is extracting a root.

If a is a positive real number, and n is a positive integer, then the positive real solution to the equation

is called the n^th root of a, and is denoted .

.

For example: 8^1/3 = 2.

Exponentiation with a rational exponent m/n can be defined as

.

For example, 8^2/3 = 4.

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

where the limit is taken only over rational values of r.

For example, if

then

[edit] Logarithm method

The natural logarithm ln(x) is the inverse of the exponential function e^x. It is defined for every positive real number b and satisfies the equation

Assuming b^x is already defined, logarithm and exponent rules give the equality

This equality can be used to define exponentiation with any positive real base b as

This definition of the real number power b^x 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.

[edit] Powers of negative real numbers

Neither the logarithm method nor the fractional exponent method can be used to define a^k as a real number for a negative real number a and an arbitrary real number k. In some special cases, a definition is possible: integral powers of negative real numbers are real numbers, and rational powers of the form a^m/n where n is odd can be computed using roots. But since there is no real number x such that x² = −1, the definition of a^m/n when n is even and m is odd must use the imaginary unit i, as described more fully in the next section.

The logarithm method cannot be used to define a^k as a real number when a < 0 because e^x is nonnegative for every real number x, so log(a) cannot be a real number.

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, take 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.

[edit] Complex powers of complex numbers

While integer powers of complex numbers are simply given by repeated multiplication or division as above, the definition of non-integer powers involves more sophisticated concepts. For every complex number z the complex exponential function e^z is defined by

Euler's formula states that, for real x, e^ix is the point on the unit circle whose angle from the positive real axis is x radians, as in polar coordinates. Thus the set of solutions to the equation e^z = 1 is the set of integer multiples of 2πi:

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:

.

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

One solution, z=log a, to the equation e^z = a, is called the principal value of the complex logarithm. It is the unique solution whose imaginary part lies in the interval . For example, log(1) = 0, log(−1) = πi, log(i) = πi/2, and log(−i) = −πi/2. The principal value of the logarithm is known as a branch of the logarithm; other branches can be specified by choosing a different range for the imaginary part of the logarithm. The boundary between branches is known as a branch cut. The principal value has a branch cut extending from the origin along the negative real axis, and is discontinuous at each point of the branch cut.

The general complex power a^b of a nonzero complex number a is defined as

In the important special case where the exponent is a rational number, the power z=a^n/m is a solution to the equation z^m = aⁿ .

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

[edit] Complex roots of unity

Main article: Root of unity

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 z ⁿ = 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 , which is 1.^[2])

The other nth roots of unity are given by

for 2 ≤ k ≤ n.

[edit] 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.

[edit] Trigonometric functions

Main article: Euler's formula

The trigonometric functions cosine and sine can be defined, based on powers of e, as

When z is a real number, these definitions agree with the geometric definitions of sine and cosine in terms of angles (this is another consequence of Euler's formula). Historically, cosine and sine were first defined geometrically, and the geometric definition is usually the first one taught to students. The above definition in terms of complex exponentials, however, is useful both to extend their definitions to complex values of z and also to simplify axiomatic developments of mathematical analysis.

[edit] Computing complex powers

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

where r is a non-negative real number and θ is the (real) argument of z. The argument, like the complex logarithm, has many possible values for each z and so a branch cut is used to choose a specific value. The polar form has a simple geometric interpretation: if a complex number u+i v 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 branch cut corresponds to the notion that a polar angle θ is ambiguous, since any multiple of 2π could be added to θ without changing the location of the point. The principal value (the most common branch cut), as mentioned above, corresponds to θ chosen in the interval (−π, π].

In order to compute the complex power a^b, write a in polar form:

.

Then

and thus

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

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

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

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

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

and use the formula above to compute

The value of a complex power depends on the branch cut used. For example, if the polar form i = 1e^i(5π/2) is used to compute iⁱ, the power is found to be e^-5π/2; the principal value of iⁱ, computed above, is e^-π/2.

[edit] Failure of power and logarithm identities

Identities for powers and logarithms that hold for positive real numbers may fail when the positive real numbers are replaced by arbitrary complex numbers. There is no method to define complex powers or the complex logarithm as complex-valued functions while preserving the identities these operations possess in the positive real numbers.

An example involving logarithms concerns the rule log(a^b) = b·log(a), which holds whenever a is a positive real number and b is a real number. The following calculation shows that this identity does not hold in general for the principal value of the complex logarithm when a is not a positive real number:

Regardless of which branch of the logarithm is used, a similar failure of the identity will always exist.

An example involving power rules concerns the identities

These identities are valid when a and b are positive real numbers and c is a real number. But a calculation using principal values shows that

and

These examples illustrate that complex powers and logarithms do not behave the same way as their real counterparts, and so caution is required when working with the complex versions of these operations.

[edit] Solving polynomial equations

It was once conjectured that the roots of any polynomial could be expressed in terms of exponentiation with fractional exponents. The Abel-Ruffini theorem shows that this is not possible in general. For example, the solutions of the equation x⁵ = x+1 cannot be expressed in terms of fractional exponents. On the other hand, many root-finding algorithms are available that allow one to numerically compute the roots of arbitrary polynomials to any desired accuracy (given sufficient arithmetic precision).

[edit] Combinatorial interpretation

For non-negative integers n and m, the power n^m equals the cardinality of the set of m-tuples from an n-element set.

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 8 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 9 2-tuples from a three-element set.

4¹ = | { (1), (2), (3), (4) } | = 4. There are 4 1-tuples from a four-element set.

5⁰ = | { () } | = 1. There is exactly one empty tuple.

This leads to exponentiation over sets.

[edit] Zero to the zero power

The evaluation of 0⁰ presents a problem, because different mathematical reasoning leads to different results. The best choice for its value depends on the context. According to Benson (1999), "The choice whether to define 0⁰ is based on convenience, not on correctness."^[3] There are two principal treatments in practice, one from discrete mathematics and the other from analysis.

In many settings, especially in foundations and combinatorics, 0⁰ is defined to be 1. This definition arises in foundational treatments of the natural numbers as finite cardinals, and is useful for shortening combinatorial identities and removing special cases from theorems, as illustrated below. In many other settings, 0⁰ is left undefined. In calculus, 0⁰ is an indeterminate form, which must be analyzed rather than evaluated. In general, mathematical analysis treats 0⁰ as undefined^[4] in order that the exponential function be continuous.

Justifications for defining 0⁰ = 1 include:

• When 0⁰ is regarded as an empty product of zeros, its value is 1.
• 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.
• A power series identity with nonzero constant term, such as , is not valid for x = 0 unless 0⁰, which appears in the numerator of the first term of the series, is 1. Thus defining 0⁰ to be 1 allows this identity to be used instead of the longer identity .
• The binomial theorem is not valid for x = 0, unless 0⁰ = 1.^[5] By defining 0⁰ to be 1, a special case of the theorem can be eliminated.

In analysis, where quantities vary continuously, it is generally best to treat 0⁰ as an ill-defined quantity. Justifications for treating it as undefined include:

• The value 0⁰ often arises as the formal limit of exponentiated functions, f(x)^g(x), when f(x) and g(x) approach 0 as x approaches a (a constant or infinity). There, 0⁰ suggests [lim f(x)]^{lim g(x)}, which is a well defined quantity and is the correct value of lim f(x)^g(x) when f and g approach nonzero constants, but is not well defined when f and g approach 0. The same reasoning applies to certain powers involving infinity, and .

A more abstract way of saying this is the following: The real function x^y of the two nonnegative real variables x and y is not continuous at the point (x, y) = (0, 0), and so 0⁰ is not determined by continuity.^[6] That is, the function x^y has no continuous extension from the open first quadrant to include the point (0,0).^[7] A discontinuous extension would cause the function to lose a number of desirable properties. For example, it is ordinarily taken as a rule in calculus that whenever both sides of the equation are defined; this rule would fail if 0⁰ were defined.

• The function z^z, viewed as a function of a complex number variable z and defined as e^z log z is undefined at z = 0 because log z is undefined at z = 0. Moreover, because z^z has a logarithmic branch point at z = 0, it is not common to extend the domain of z^z to the origin in this context.^[8]

[edit] Treatment in programming languages

The evaluation of 0⁰ is possible in several computer programming languages. Many languages, including Java, Python, Ruby, Haskell, ML, Scheme, MATLAB, Microsoft Windows' Calculator, and others (especially when using IEEE floating-point arithmetic, but also for integer arithmetic), evaluate 0⁰ to be 1.^[9] Mathematica simplifies a⁰ to 1, even if no constraints are placed on a, but does not simplify 0^a, and takes 0⁰ to be an indeterminate form. Maple simplifies a⁰ to 1 and 0^a to 0, even if no constraints are placed on a, and evaluates 0⁰ to 1.

[edit] Powers with infinity

The expressions and arise in analysis for the same reason as 0⁰, and they are undefined for the same reason. That is, it is true that [lim f(x)]^{lim g(x)} = lim f(x)^g(x) when f and g approach nonzero finite constants, but not when they approach 0 or infinity; then, the limit of the power can be anything, not predictable from the limits of f and g.

There is one exception. If f and g both approach infinity as x approaches a, then lim f(x)^g(x) does equal infinity. Thus, it makes sense to say the expression is well defined and equals .

[edit] Efficiently computing a power

The simplest method of computing aⁿ requires n−1 multiplication operations, but it can be computed more efficiently as illustrated by the following example. To compute 2¹⁰⁰, note that 100 = 96 + 4 and 96 = 3*32. Compute the following in order:

2² = 4

(2²)² = 2⁴ = 16

(2⁴)² = 2⁸ = 256

(2⁸)² = 2¹⁶ = 65,536

(2¹⁶)² = 2³² = 4,294,967,296

2³² 2³² 2³²2⁴ = 2¹⁰⁰

This series of steps only requires 8 multiplication operations instead of 99.

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 an NP-hard problem, but many reasonably efficient heuristic algorithms are available.

[edit] 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 f³(x) may mean f(f(f(x))); in particular, f ^-1(x) usually denotes the inverse function of f.

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 is csc x. A similar convention applies to logarithms, where log²(x) = (log (x))² and there is no common abbreviation for log(log(x)).^{[citation needed]}

[edit] Generalizations of exponentiation

[edit] Exponentiation 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 we will write multiplicatively. In this very general situation, we can define xⁿ for any element x of X and any nonzero natural number n, by simply multiplying x by itself n times; by definition, power associativity means that it doesn't matter in which order we perform the multiplications.

Now additionally suppose that the operation has an identity element 1. Then we can define x⁰ to be equal to 1 for any x. Now xⁿ is defined for any natural number n, including 0.

Finally, suppose that the operation has inverses, and that the multiplication is associative (so that the magma is a group). Then we can define x⁻ⁿ to be the inverse of xⁿ when n is a natural number. Now xⁿ is defined for any integer n and any x in the group.

Exponentiation in this purely algebraic sense satisfies the following laws (whenever both sides are defined):

Here, we use a division slash ("/") to indicate multiplying by an inverse, in order to reserve the symbol x⁻¹ for raising x to the power −1, rather than the inverse of x. However, as one of the laws above states, x⁻¹ is always equal to the inverse of x, so the notation doesn't matter in the end.

If in addition the multiplication operation is commutative (so that the set X is an abelian group), then we have some additional laws:

• (xy)ⁿ = xⁿyⁿ
• (x/y)ⁿ = xⁿ/yⁿ

If we take this whole theory of exponentiation in an algebraic context but write the binary operation additively, 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.

[edit] Exponentiation over sets

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

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

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

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

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 N=2={0,1}, we have |2^X| = 2^|X|, where 2^X, usually denoted by PX, 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.

[edit] Exponentiation 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.

[edit] Exponentiation 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. If κ and λ are finite then this agrees with the ordinary exponential operation. For example, the set of 3-tuples of elements from a 2-element set has cardinality 8.

Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process. In the ordinal numbers, exponentiation is defined by transfinite induction. For ordinals α and β, the exponential α^β is the supremum of the ordinal product α^γα over all γ < β.

[edit] 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 captured by the Ackermann function.

[edit] Exponentiation 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, Microsoft Excel, TeX (and its derivatives), and most computer algebra systems
• x ** y: Ada, Bash, Fortran, FoxPro, Perl, Python, Ruby, SAS, ABAP
• x * y: APL
• Power(x, y): Microsoft Excel, Pascal
• pow(x, y): C, C++, PHP
• Math.pow(x, y): Java, JavaScript, Modula-3
• Math.Pow(x, y): C#
• (expt x y): Common Lisp, Scheme

In Bash, C, C++, C#, Java, JavaScript and Python, the symbol ^ represents bitwise XOR. In Pascal, it represents indirection.

[edit] History of the notation

The term power was used by Euclid for the square of a line. Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel. Samuel Jeake introduced the term indices in 1696.^[10] In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), surfolide (fifth), zenzicube (sixth), second surfolide (seventh) and Zenzizenzizenzic (eighth).

[edit] Table of powers

Table of kⁿ, with k on the left and n at the top.

[edit] References

1. ^ 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.
2. ^ 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 edition, MIT Press. ISBN 0262032937.  Online resource
   • Paul Cull, Mary Flahive, and Robby Robson (2005). Difference Equations: From Rabbits to Chaos, Undergraduate Texts in Mathematics, Springer. ISBN 0387232346.  Defined on page 351, available on Google books.
   • "Principal root of unity", MathWorld.
3. ^ Benson, Donald C. The Moment of Proof : Mathematical Epiphanies. New York Oxford University Press (UK), 1999. ISBN 9780195117219
4. ^ Examples of this include Edwards and Penny (1994). Calculus, 4th ed,, Prentice-Hall, p. 466, and Keedy, Bittinger, and Smith (1982). Algebra Two. Addison-Wesley, p. 32.
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 edition, Addison Wesley Longman Publishing Co, 162. ISBN 0-201-14236-8. 
6. ^ L. J. Paige (March 1954). "A note on indeterminate forms". American Mathematical Monthly 61 (3): 189–190. 
7. ^ Along the x-axis the limit is 1, along the y-axis the limit is 0, and any intermediate limit a can be obtained using the curve y = log(a)/log(x). However, if y is an analytic function of x, or if there exists a positive constant, a, such that y < ax, then the limit is 1.
8. ^ "... Let's start at x=0. Here x^x is undefined." Mark D. Meyerson "The X^{x Spindle." Mathematics Magazine, v. 69 n. 3, Jun 1996, pp. 198-206.}
9. ^ For example, see John Benito (April 2003). "Rationale for International Standard — Programming Languages — C". Revision 5.10.
10. ^ O'Connor, John J., and Edmund F. Robertson. "Etymology of some common mathematical terms". MacTutor History of Mathematics archive.

[edit] See also

Wikimedia Commons has media related to:

Exponential function

• Exponential growth and Exponential decay
• List of exponential topics
• Logarithm
• Modular exponentiation
• Nth root
• Unicode subscripts and superscripts

[edit] External links

• sci.math FAQ: What is 0⁰?
• Introducing 0th power on PlanetMath
• Laws of Exponents with derivation and examples
• 1058 Powers of Two