Exponentiation

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Graphs of y = b for various bases b:
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base 10,
base e,
base 2,
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1/2. Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself. Expo02.svg
Graphs of y = b for various bases b:    base 10,   base e,   base 2,  base 1/2. Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.

Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or powern, and pronounced as "b raised to the power of n". [1] [2] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases: [2]

Contents

The exponent is usually shown as a superscript to the right of the base. In that case, bn is called "b raised to the nth power", "b raised to the power of n", [1] "the nth power of b", "b to the nth power", [3] or most briefly as "b to the nth".

One has b1 = b, and, for any positive integers m and n, one has bnbm = bn+m. To extend this property to non-positive integer exponents, b0 is defined to be 1, and bn (with n a positive integer and b not zero) is defined as 1/bn. In particular, b−1 is equal to 1/b, the reciprocal of b.

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

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

History of the notation

The term power (Latin : potentia, potestas, dignitas) is a mistranslation [4] [5] of the ancient Greek δύναμις (dúnamis, here: "amplification" [4] ) used by the Greek mathematician Euclid for the square of a line, [6] following Hippocrates of Chios. [7] In The Sand Reckoner , Archimedes discovered and proved the law of exponents, 10a ⋅ 10b = 10a+b, necessary to manipulate powers of 10.[ citation needed ] In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms مَال (māl, "possessions", "property") for a square—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property" [8] —and كَعْبَة ( kaʿbah , "cube") for a cube, which later Islamic mathematicians represented in mathematical notation as the letters mīm (m) and kāf (k), respectively, by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī. [9]

In the late 16th century, Jost Bürgi used Roman numerals for exponents. [10]

Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel in the 16th century. The word exponent was coined in 1544 by Michael Stifel. [11] [12] Samuel Jeake introduced the term indices in 1696. [6] In the 16th century, Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth). [8] Biquadrate has been used to refer to the fourth power as well.

Early in the 17th century, the first form of our modern exponential notation was introduced by René Descartes in his text titled La Géométrie ; there, the notation is introduced in Book I. [13]

Some mathematicians (such as 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,[ clarification needed ]involution, is now rare [14] and should not be confused with its more common meaning.

In 1748, Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing:

"consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant." [15]

Terminology

The expression b2 = bb is called "the square of b" or "b squared", because the area of a square with side-length b is b2.

Similarly, the expression b3 = bbb is called "the cube of b" or "b cubed", because the volume of a cube with side-length b is b3.

When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, 35 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power.

The word "raised" is usually omitted, and sometimes "power" as well, so 35 can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiation bn can be expressed as "b to the power of n", "b to the nth power", "b to the nth", or most briefly as "b to the n".

A formula with nested exponentiation, such as 357 (which means 3(57) and not (35)7), is called a tower of powers, or simply a tower.

Integer exponents

The exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations.

Positive exponents

The definition of the exponentiation as an iterated multiplication can be formalized by using induction, [16] and this definition can be used as soon one has an associative multiplication:

The base case is

and the recurrence is

The associativity of multiplication implies that for any positive integers m and n,

and

Zero exponent

By definition, any nonzero number raised to the 0 power is 1: [17] [2]

This definition is the only possible that allows extending the formula

to zero exponents. It may used in every algebraic structure with a multiplication that has an identity.

Intuitionally, may be interpreted as the empty product of copies of b. So, the equality is a special case of the general convention for the empty product.

The case of 00 is more complicated. In contexts where only integer powers are considered, the value 0 is generally assigned to but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context. For more details, see Zero to the power of zero.

Negative exponents

Exponentiation with negative exponents is defined by the following identity, which holds for any integer n and nonzero b:

[2]

Raising 0 to a negative exponent is undefined, but in some circumstances, it may be interpreted as infinity ().

This definition of exponentiation with negative exponents is the only one that allows extending the identity to negative exponents (consider the case ).

The same definition applies to invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted 1 (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element x is standardly denoted

Identities and properties

The following identities, often called exponent rules, hold for all integer exponents, provided that the base is non-zero: [2]

Unlike addition and multiplication, exponentiation is not commutative. For example, 23 = 8 ≠ 32 = 9. Also unlike addition and multiplication, exponentiation is not associative. For example, (23)2 = 82 = 64, whereas 2(32) = 29 = 512. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up [18] [19] [20] [21] (or left-associative). That is,

which, in general, is different from

Powers of a sum

The powers of a sum can normally be computed from the powers of the summands by the binomial formula

However, this formula is true only if the summands commute (i.e. that ab = ba), which is implied if they belong to a structure that is commutative. Otherwise, if a and b are, say, square matrices of the same size, this formula cannot be used. It follows that in computer algebra, many algorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose computer algebra systems use a different notation (sometimes ^^ instead of ^) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

Combinatorial interpretation

For nonnegative integers n and m, the value of nm is the number of functions from a set of m elements to a set of n elements (see cardinal exponentiation). Such functions can be represented as m-tuples from an n-element set (or as m-letter words from an n-letter alphabet). Some examples for particular values of m and n are given in the following table:

nmThe nm possible m-tuples of elements from the set {1, ..., n}
05 = 0none
14 = 1(1, 1, 1, 1)
23 = 8(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)
32 = 9(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)
41 = 4(1), (2), (3), (4)
50 = 1()

Particular bases

Powers of ten

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

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

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

Powers of two

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

Powers of 2 appear in set theory, since a set with n members has a power set, the set of all of its subsets, which has 2n members.

Integer powers of 2 are important in computer science. The positive integer powers 2n give the number of possible values for an n-bit integer binary number; for example, a byte may take 28 = 256 different values. The binary number system expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a binary point, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and the negative exponents are determined by the rank on the right of the point.

Powers of one

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

Powers of zero

If the exponent n is positive (n > 0), the nth power of zero is zero: 0n = 0.

If the exponent n is negative (n < 0), the nth power of zero 0n is undefined, because it must equal with n > 0, and this would be according to above.

The expression 00 is either defined as 1, or it is left undefined (see Zero to the power of zero ).

Powers of negative one

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

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

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

Large exponents

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

bn → ∞ as n → ∞ when b > 1

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

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

bn → 0 as n → ∞ when |b| < 1

Any power of one is always one:

bn = 1 for all n if b = 1

Powers of –1 alternate between 1 and –1 as n alternates between even and odd, and thus do not tend to any limit as n grows.

If b < –1, bn, alternates between larger and larger positive and negative numbers as n alternates between even and odd, and thus does not tend to any limit as n grows.

If the exponentiated number varies while 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 exponential function below.

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

Power functions

Power functions for
n
=
1
,
3
,
5
{\displaystyle n=1,3,5} Potenssi 1 3 5.svg
Power functions for
Power functions for
n
=
2
,
4
,
6
{\displaystyle n=2,4,6} Potenssi 2 4 6.svg
Power functions for

Real functions of the form , where , are sometimes called power functions.[ citation needed ] When is an integer and , two primary families exist: for even, and for odd. In general for , when is even will tend towards positive infinity with increasing , and also towards positive infinity with decreasing . All graphs from the family of even power functions have the general shape of , flattening more in the middle as increases. [22] Functions with this kind of symmetry () are called even functions.

When is odd, 's asymptotic behavior reverses from positive to negative . For , will also tend towards positive infinity with increasing , but towards negative infinity with decreasing . All graphs from the family of odd power functions have the general shape of , flattening more in the middle as increases and losing all flatness there in the straight line for . Functions with this kind of symmetry () are called odd functions.

For , the opposite asymptotic behavior is true in each case. [22]

List of whole-number powers

nn2n3n4n5n6n7n8n9n10
2481632641282565121024
392781243729218765611968359049
416642561024409616384655362621441048576
5251256253125156257812539062519531259765625
636216129677764665627993616796161007769660466176
749343240116807117649823543576480140353607282475249
8645124096327682621442097152167772161342177281073741824
9817296561590495314414782969430467213874204893486784401
10100100010000100000100000010000000100000000100000000010000000000

Rational exponents

From top to bottom: x , x , x , x , x , x , x . Mplwp roots 01.svg
From top to bottom: x , x , x , x , x , x , x .

If x is a nonnegative real number, and n is a positive integer, or denotes the unique positive real nth root of x, that is, the unique positive real number y such that

If x is a positive real number, and is a rational number, with p and q ≠ 0 integers, then is defined as

The equality on the right may be derived by setting and writing

If r is a positive rational number, by definition.

All these definitions are required for extending the identity to rational exponents.

On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real nth root, which is negative if n is odd, and no real root if n is even. In the latter case, whichever complex nth root one chooses for the identity cannot be satisfied. For example,

See § Real exponents and § Powers of complex numbers for details on the way these problems may be handled.

Real exponents

For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (§ Limits of rational exponents, below), or in terms of the logarithm of the base and the exponential function (§ Powers via logarithms, below). The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to complex exponents.

On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values (see § Real exponents with negative bases). One may choose one of these values, called the principal value, but there is no choice of the principal value for which the identity

is true; see § Failure of power and logarithm identities. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.

Limits of rational exponents

The limit of e is e = 1 when n tends to the infinity. Continuity of the Exponential at 0.svg
The limit of e is e = 1 when n tends to the infinity.

Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by continuity with the rule [23]

where the limit is taken over rational values of r only. This limit exists for every positive b and every real x.

For example, if x = π, the non-terminating decimal representation π = 3.14159... and the monotonicity of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain

So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted

This defines for every positive b and real x as a continuous function of b and x. See also Well-defined expression.

The exponential function

The exponential function is often defined as where is Euler's number. For avoiding circular reasoning, this definition cannot be used here. So, a definition of the exponential function, denoted and of Euler's number are given, which are independent from exponentiation. Then a proof is sketched that, if one uses the definition of exponentiation given in preceding sections, one has

There are many equivalent ways to define the exponential function, one of them being to define it as the inverse function of the natural logarithm. Precisely, the natural logarithm is the antiderivative of that takes the value 0 for x = 1:

This defines the logarithm as an increasing function from the positive reals to the real numbers. It inverse function, the exponential function, is thus an increasing function from the real numbers to the positive reals, which is commonly denoted exp. One has

and the exponential identity

for every x and y.

Euler's number can be defined as . It follows from the preceding equations that when x is an integer (this results from the repeated-multiplication definition of the exponentiation). If x is real, results from the definitions given in preceding sections, by using the exponential identity if x is rational, and the continuity of the exponential function otherwise.

The exponential function satisfies the equation

As this series converges for every complex value of x this equation allows defining the exponential function, and thus for any complex argument z. This extended exponential function still satifies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.

Powers via logarithms

The definition of ex as the exponential function allows defining bx for every positive real numbers b, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function ex means that one has

for every b > 0. For preserving the identity one must have

So, can be used as an alternative definition of bx for any positive real b. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.

Complex exponents with a positive real base

If b is a positive real number, exponentiation with base b and complex exponent is defined by mean of the exponential function with complex argument (see the end of § The exponential function, above) as

where denotes the natural logarithm of b.

This satisfies the identity

In general, is not defined, since bz is not a real number. If a meaning is given to the exponentiation of a complex number (see § Powers of complex numbers, below), one has, in general,

unless z is real or w is integer.

Euler's formula,

allows expressing the polar form of in terms of the real and imaginary parts of z, namely

where the absolute value of the trigonometric factor is one. This results from

Non-integer powers of complex numbers

In the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only. For other bases, difficulties appear already with the apparently simple case of nth roots, that is, of exponents where n is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to nth roots, this case deserves to be considered first, since it does not need to use complex logarithms, and is therefore easier to understand.

nth roots of a complex number

Every nonzero complex number z may be witten in polar form as

where is the absolute value of z, and is its argument. The argument is defined up to an integer multiple of 2π; this means that, if is the argument of a complex number, then is also an argument of the same complex number.

The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an nth root of a complex number can be obtained by taking the nth root of the absolute value and dividing its argument by n:

If is added to the complex number in not changed, but this adds to the argument of the nth root, and provides a new nth root. This can be done n times, and provides the nnth roots of the complex number.

It is usual to choose one of the nnth root as the principal root. The common choice is to choose the nth root for which that is, the nth root that has the largest real part, and, if they are two, the one with positive imaginary part. This makes the principal nth root a continuous function in the whole complex plane, except for negative real values of the radicand. This function equals the usual nth root for positive real radicands. For negative real radicands, and odd exponents, the principal nth root is not real, although the usual nth root is real. Analytic continuation shows that the principal nth root is the unique complex differentiable function that extends the usual nth root to the complex plane without the nonpositive real numbers.

If the complex number is moved around zero by increasing its argument, after an increment of the complex number comes back to its initial position, and its nth roots are permuted circularly (they are multiplied by ). This shows that it is not possible to define a nth root function that is not continuous in the whole complex plane.

Roots of unity

The three third roots of 1 One3Root.svg
The three third roots of 1

The nth roots of unity are the n complex numbers such that wn = 1, where n is a positive integer. They arise in various areas of mathematics, such as in discrete Fourier transform or algebraic solutions of algebraic equations (Lagrange resolvent).

The nnth roots of unity are the n first powers of , that is The nth roots of unity that have this generating property are called primitive nth roots of unity; they have the form with k coprime with n. The unique primitive square root of unity is the primitive fourth roots of unity are and

The nth roots of unity allow expressing all nth roots of a complex number z as the n products of a given nth roots of z with a 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.

As the number is the primitive nth root of unity with the smallest positive argument, it is called the principal primitive nth root of unity, sometimes shortened as principal nth root of unity, although this terminology can be confused with the principal value of which is 1. [24] [25] [26]

Complex exponentiation

Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, there are infinitely many possible values for . So, either a principal value is defined, which is not continuous for the values of z that are real and nonpositive, or is defined as a multivalued function.

In all cases, the complex logarithm is used to define complex exponentiation as

where is the variant of the complex logarithm that is used, which is, a function or a multivalued function such that

for every z in its domain of definition.

Principal value

The principal value of the complex logarithm is the unique function, commonly denoted such that, for every nonzero complex number z,

and the imaginary part of z satisfies

The principal value of the complex logarithm is not defined for it is discontinuous at negative real values of z, and it is holomorphic (that is, complex differentiable) elsewhere. If z is real and positive, the principal value of the complex logarithm is the natural logarithm:

The principal value of is defined as where is the principal value of the logarithm.

The function is holomorphic except in the neibourhood of the points where z is real and nonpositive.

If z is real and positive, the principal value of equals its usual value defined above. If where n is an integer, this principal value is the same as the one defined above.

Multivalued function

In some contexts, there is a problem with the discontinuity of the principal values of and at the negative real values of z. In this case, it is useful to consider these functions as multivalued functions.

If denotes one of the values of the multivalued logarithm (typically its principal value), the other values are where k is any integer. Similarly, if is one value of the exponentiation, then the other values are given by

where k is any integer.

Different values of k give different values of unless w is a rational number, that is, there is an integer d such that dw is an integer. This results from the periodicity of the exponential function, more specifically, that if and only if is an integer multiple of

If is a rational number with m and n coprime integers with then has exactly n values. In the case these values are the same as those described in § nth roots of a complex number. If w is an integer, there is only one value that agrees with that of § Integer exponents.

The multivalued exponentiation is holomorphic for in the sense that its graph consists of several sheets that define each a holomorphic function in the neighborhood of every point. If z varies continuously along a circle around 0, then, after a turn, the value of has changed of sheet.

Computation

The canonical form of can be computed from the canonical form of z and w. Although this can be described by a single formula, it is clearer to split the computation in several steps.

  • Polar form of z. If is the canonical form of z (a and b being real), then its polar form is
    where and (see atan2 for the definition of this function).
  • Logarithm of z. The principal value of this logarithm is where denotes the natural logarithm. The other values of the logarithm are obtained by adding for any integer k.
  • Canonical form of If with c and d real, the values of are
    the principal value corresponding to
  • Final result. Using the identities and one gets
    with for the principal value.
Examples

  • The polar form of i is and the values of are thus
    It follows that
    So, all values of are real, the principal one being

  • Similarly, the polar form of −2 is So, the above described method gives the values
    In this case, all the values have the same argument and different absolute values.

In both examples, all values of have the same argument. More generally, this is true if and only if the real part of w is an integer.

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

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

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

  • The identities (bc)x = bxcx and (b/c)x = bx/cx are valid when b and c are positive real numbers and x is a real number. But, for the principal values, one has

    and

    On the other hand, when x is an integer, the identities are valid for all nonzero complex numbers.

    If exponentiation is considered as a multivalued function then the possible values of (−1 ⋅ −1)1/2 are {1, −1}. The identity holds, but saying {1} = {(−1 ⋅ −1)1/2} is wrong.
  • The identity (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: [27] For any integer n, we have:
    1. (taking the -th power of both sides)
    2. (using and expanding the exponent)
    3. (using )
    4. (dividing by e)

    but this is false when the integer n is nonzero.

    The error is the following: by definition, is a notation for a true function, and is a notation for which is a multi-valued function. Thus the notation is ambiguous when x = e. Here, before expanding the exponent, the second line should be

    Therefore, when expanding the exponent, one has implicitly supposed that for complex values of z, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity (ex)y = exy must be replaced by the identity

    which is a true identity between multivalued functions.

Irrationality and transcendence

If b is a positive real algebraic number, and x is a rational number, then bx is an algebraic number. This results from the theory of algebraic extensions. This remains true if b is any algebraic number, in which case, all values of bx (as a multivalued function) are algebraic. If x is irrational (that is, not rational), and both b and x are algebraic, Gelfond–Schneider theorem asserts that all values of bx are transcendental (that is, not algebraic), except if b equals 0 or 1.

In other words, if x is irrational and then at least one of b, x and bx is transcendental.

Integer powers in algebra

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication. [nb 1] The definition of requires further the existence of a multiplicative identity. [28]

An algebraic structure consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a monoid. In such a monoid, exponentiation of an element x is defined inductively by

If n is a negative integer, is defined only if x has a multiplicative inverse. [29] In this case, the inverse of x is denoted and is defined as

Exponentiation with integer exponents obeys the following laws, for x and y in the algebraic structure, and m and n integers:

These definitions are widely used in many areas of mathematics, notably for groups, rings, fields, square matrices (which form a ring). They apply also to functions from a set to itself, which form a monoid under function composition. This includes, as specific instances, geometric transformations, and endomorphisms of any mathematical structure.

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if f is a real function whose valued can be multiplied, denotes the exponentiation with respect of multiplication, and may denote exponentiation with respect of function composition. That is,

and

Commonly, is denoted while is denoted

In a group

A multiplicative group is a set with as associative operation denoted as multiplication, that has an identity element, and such that every element has an inverse.

So, if G is a group, is defined for every and every integer n.

The set of all powers of an element of a group form a subgroup. A group (or subgroup) that consists of all powers of a specific element x is the cyclic group generated by x. If all the powers of x are distinct, the group is isomorphic to the additive group of the integers. Otherwise, the cyclic group is finite (it has a finite number of elements), and its number of elements is the order of x. If the order of x is n, then and the cyclic group generated by x consists of the n first powers of x (starting indifferently from the exponent 0 or 1).

Order of elements play a fundamental role in group theory. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the order of the group). The possible orders of group elements are important in the study of the structure of a group (see Sylow theorems), and in the classification of finite simple groups.

Superscript notation is also used for conjugation; that is, gh = h−1gh, where g and h are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely and

In a ring

In a ring, it may occurs that some nonzero elements satisfy for some integer n. Such an element is said nilpotent. In a commutative ring, the nilpotent elements form an ideal, called the nilradical of the ring.

If the nilradical is reduced to the zero ideal (that is, if implies for every positive integer n), the commutative ring is said reduced. Reduced rings important in algebraic geometry, since the coordinate ring of an affine algebraic set is always a reduced ring.

More generally, given an ideal I in a commutative ring R, the set of the elements of R that have a power in I is an ideal, called the radical of I. The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring over a field k, an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of Hilbert's Nullstellensatz).

Matrices and linear operators

If A is a square matrix, then the product of A with itself n times is called the matrix power. Also is defined to be the identity matrix, [30] and if A is invertible, then .

Matrix powers appear often in the context of discrete dynamical systems, where the matrix A expresses a transition from a state vector x of some system to the next state Ax of the system. [31] This is the standard interpretation of a Markov chain, for example. Then is the state of the system after two time steps, and so forth: is the state of the system after n time steps. The matrix power is the transition matrix between the state now and the state at a time n steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors.

Apart from matrices, more general linear operators can also be exponentiated. An example is the derivative operator of calculus, , which is a linear operator acting on functions to give a new function . The n-th power of the differentiation operator is the n-th derivative:

These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups. [32] Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the heat equation, Schrödinger equation, wave equation, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.

Finite fields

A field is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is associative and every nonzero element has a multiplicative inverse. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of 0. Common examples are the complex numbers and their subfields, the rational numbers and the real numbers, which have been considered earlier in this article, and are all infinite.

A finite field is a field with a finite number of elements. This number of elements is either a prime number or a prime power; that is, it has the form where p is a prime number, and k is a positive integer. For every such q, there are fields with q elements. The fields with q elements are all isomorphic, which allows, in general, working as if there were only one field with q elements, denoted

One has

for every

A primitive element in is an element g such the set of the q − 1 first powers of g (that is, ) equals the set of the nonzero elements of There are primitive elements in where is Euler's totient function.

In the Freshman's dream identity

is true for the exponent p. As in It follows that the map

is linear over and is a field automorphism, called the Frobenius automorphism. If the field has k automorphisms, which are the k first powers (under composition) of F. In other words, the Galois group of is cyclic of order k, generated by the Frobenius automorphism.

The Diffie–Hellman key exchange is an application of exponentiation in finite fields that is widely used for secure communications. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the discrete logarithm, is computationally expensive. More precisely, if g is a primitive element in then can be efficiently computed with exponentiation by squaring for any e, even if q is large, while there is no known algorithm allowing retrieving e from if q is sufficiently large.

Powers of sets

The Cartesian product of two sets S and T is the set of the ordered pairs such that and This operation is not properly commutative nor associative, but has these properties up to canonical isomorphisms, that allow identifying, for example, and

This allows defining the nth power of a set S as the set of all n-tuples of elements of S.

When S is endowed with some structure, it is frequent that is naturally endowed with a similar structure. In this case, the term "direct product" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example (where denotes the real numbers) denotes the Cartesian product of n copies of as well as their direct product as vector space, topological spaces, rings, etc.

Sets as exponents

A n-tuple of elements of S can be considered as a function from This generalizes to the following notation.

Given two sets S and T, the set of all functions from T to S is denoted This exponential notation is justified by the following canonical isomorphisms (for the first one, see Currying):

where denotes the Cartesian product, and the disjoint union.

One can use sets as exponents for other operations on sets, typically for direct sums of abelian groups, vector spaces, or modules. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, denotes the vector space of the infinite sequences of real numbers, and the vector space of those sequences that have a finite number of nonzero elements. The latter has a basis consisting of the sequences with exactly one nonzero element that equals 1, while the Hamel bases of the former cannot be explicitly described (because there existence involves Zorn's lemma).

In this context, 2 can represents the set So, denotes the power set of S, that is the set of the functions from S to which can be identified with the set of the subsets of S, by mapping each function to the inverse image of 1.

This fits in with the exponentiation of cardinal numbers, in the sense that |ST| = |S||T|, where |X| is the cardinality of X.

In category theory

In the category of sets, the morphisms between sets X and Y are the functions from X to Y. It results that the set of the functions from X to Y that is denoted in the preceding section can also be denoted The isomorphism can be rewritten

This means the functor "exponentiation to the power T" is a right adjoint to the functor "direct product with T".

This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor is, if it exists, a right adjoint to the functor A category is called a Cartesian closed category, if direct products exist, and the functor has a right adjoint for every T.

Repeated exponentiation

Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation. This sequence of operations is expressed by the Ackermann function and Knuth's up-arrow notation. Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at (3, 3), the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and 7625597484987 (= 327 = 333 = 33) respectively.

Limits of powers

Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 00. The limits in these examples exist, but have different values, showing that the two-variable function xy has no limit at the point (0, 0). One may consider at what points this function does have a limit.

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

In fact, f has a limit at all accumulation points of D, except for (0, 0), (+∞, 0), (1, +∞) and (1, −∞). [33] 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:

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

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

Efficient computation with integer exponents

Computing bn using iterated multiplication requires n − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2100, apply Horner's rule to the exponent 100 written in binary:

.

Then compute the following terms in order, reading Horner's rule from right to left.

22 = 4
2 (22) = 23 = 8
(23)2 = 26 = 64
(26)2 = 212 = 4096
(212)2 = 224 = 16777216
2 (224) = 225 = 33554432
(225) 2 2 = 250 = 1125899906842624
(225)2 = 2100 = 1267650600228229401496703205376

This series of steps only requires 8 multiplications instead of 99.

In general, the number of multiplication operations required to compute bn can be reduced to by using exponentiation by squaring, where denotes the number of 1 in the binary representation of n. For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal 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. [34] However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.

Iterated functions

Function composition is a binary operation that is defined on functions such that the codomain of the function written on the right is included in the domain of the function written on the left. It is denoted and defined as

for every x in the domain of f.

If the domain of a function f equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the nth power of the function under composition, commonly called the nth iterate of the function. Thus denotes generally the nth iterate of f; for example, means [35]

When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the pointwise multiplication, which induces another exponentiation. When using functional notation, the two kinds of exponentiation are generaly distinguished by placing the exponent of the functional iteration before the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication after the parentheses. Thus and When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example and For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the trigonometric functions. So, and mean both and not which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors. [36] [37] [38]

In this context, the exponent denotes always the inverse function, if it exists. So For the multiplicative inverse fractions are generally used as in

In programming languages

Programming languages generally express exponentiation either as an infix operator or as a (prefix) function, as they are linear notations which do not support superscripts:

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

For certain exponents there are special ways to compute xy much faster than through generic exponentiation. These cases include small positive and negative integers (prefer x · x over x2; prefer 1/x over x−1) and roots (prefer sqrt(x) over x0.5, prefer cbrt(x) over x1/3).

Not all programming languages adhere to the same association convention for exponentiation: while the Wolfram language, Google Search and others use right-association (i.e. a^b^c is evaluated as a^(b^c)), many computer programs such as Microsoft Office Excel and Matlab associate to the left (i.e. a^b^c is evaluated as (a^b)^c).

See also

Notes

  1. More generally, power associativity is sufficient for the definition.

Related Research Articles

Binomial coefficient Positive integers that occur as coefficients in the binomial theorem

In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers nk ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and is given by the formula

Complex number Element of a number system in which –1 has a square root

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no real number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

Exponential function Class of specific mathematical functions

In mathematics, the exponential function is the function where the base e = 2.71828... is Euler's number and the argument x occurs as an exponent. More generally, an exponential function is a function of the form where the base b is a positive real number.

In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation. These can be of quite general use, for example in modular arithmetic or powering of matrices. For semigroups for which additive notation is commonly used, like elliptic curves used in cryptography, this method is also referred to as double-and-add.

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n:

Gamma function Extension of the factorial function

In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For any positive integer n,

Logarithm Inverse of the exponential function, which maps products to sums

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g. since 1000 = 10 × 10 × 10 = 103, the "logarithm base 10" of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logbx, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

Natural logarithm Logarithm to the base of the mathematical constant e

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

Square root Number whose square is a given number

In mathematics, a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 42 = (−4)2 = 16. Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by where the symbol is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by because 32 = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9.

In calculus, the power rule is used to differentiate functions of the form , whenever is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series as it relates a power series with a function's derivatives.

Root of unity Number that has an integer power equal to 1

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

Harmonic number Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

In mathematics, an nth root of a number x is a number r which, when raised to the power n, yields x:

Digamma function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

Tetration Repeated or iterated exponentiation

In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.

In mathematics, the Mahler measureof a polynomial with complex coefficients is defined as

Stable distribution Distribution of variables which satisfies a stability property under linear combinations

In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.

Complex logarithm Logarithm of a complex number

In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:

The Bailey–Borwein–Plouffe formula is a formula for π. It was discovered in 1995 by Simon Plouffe and is named after the authors of the article in which it was published, David H. Bailey, Peter Borwein, and Plouffe. Before that, it had been published by Plouffe on his own site. The formula is

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