Английская Википедия:Exponentiation

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Файл:Expo02.svg

Graphs of Шаблон:Math for various bases Шаблон:Math: Шаблон:Legend-line Шаблон:Legend-line Шаблон:Legend-line Шаблон:Legend-line Each curve passes through the point Шаблон:Math because any nonzero number raised to the power of Шаблон:Math is Шаблон:Math. At Шаблон:Math, the value of Шаблон:Math equals the base because any number raised to the power of Шаблон:Math is the number itself.

In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power. Exponentiation is written as Шаблон:Math, where Шаблон:Mvar is the base and Шаблон:Mvar is the power; this is pronounced as "Шаблон:Mvar (raised) to the (power of) Шаблон:Mvar". When Шаблон:Mvar is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, Шаблон:Math is the product of multiplying Шаблон:Mvar bases: <math display="block">b^n = \underbrace{b \times b \times \dots \times b \times b}_{n \text{ times}}.</math>

The exponent is usually shown as a superscript to the right of the base. In that case, Шаблон:Math is called "b raised to the nth power", "b (raised) to the power of n", "the nth power of b", "b to the nth power", or most briefly as "b to the n(th)".

Starting from the basic fact stated above that, for any positive integer <math>n</math>, <math>b^n</math> is <math>n</math> occurrences of <math>b</math> all multiplied by each other, several other properties of exponentiation directly follow. In particular:

<math display="block"> \begin{align} b^{n+m} & = \underbrace{b \times \dots \times b}_{n+m \text{ times}} \\[1ex] & = \underbrace{b \times \dots \times b}_{n \text{ times}} \times \underbrace{b \times \dots \times b}_{m \text{ times}} \\[1ex] & = b^n \times b^m \end{align} </math>

In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that <math>b^0</math> must be equal to 1 for any <math>b \neq 0</math>, as follows. For any <math>n</math>, <math>b^0 \times b^n = b^{0+n} = b^n</math>. Dividing both sides by <math>b^n</math> gives <math>b^0 = b^n / b^n = 1</math>.

The fact that <math>b^1 = b</math> can similarly be derived from the same rule. For example, <math> (b^1)^3 = b^1 \times b^1 \times b^1 = b^{1+1+1} = b^3 </math>. Taking the cube root of both sides gives <math>b^1 = b</math>.

The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what <math>b^{-1}</math> should mean. In order to respect the "exponents add" rule, it must be the case that <math>b^{-1} \times b^1 = b^{-1+1} = b^0 = 1 </math>. Dividing both sides by <math>b^{1}</math> gives <math>b^{-1} = 1 / b^1</math>, which can be more simply written as <math>b^{-1} = 1 / b</math>, using the result from above that <math>b^1 = b</math>. By a similar argument, <math>b^{-n} = 1 / b^n</math>.

The properties of fractional exponents also follow from the same rule. For example, suppose we consider <math>\sqrt{b}</math> and ask if there is some suitable exponent, which we may call <math>r</math>, such that <math> b^r = \sqrt{b}</math>. From the definition of the square root, we have that <math> \sqrt{b} \times \sqrt{b} = b </math>. Therefore, the exponent <math>r</math> must be such that <math> b^r \times b^r = b </math>. Using the fact that multiplying makes exponents add gives <math> b^{r+r} = b </math>. The <math> b </math> on the right-hand side can also be written as <math> b^1 </math>, giving <math> b^{r+r} = b^1 </math>. Equating the exponents on both sides, we have <math> r+r = 1 </math>. Therefore, <math> r = \frac{1}{2} </math>, so <math>\sqrt{b} = b^{1/2} </math>.

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.

Etymology

The term exponent originates from the Latin exponentem, the present participle of exponere, meaning "to put forth". The term power (Шаблон:Lang-la) is a mistranslation of the ancient Greek δύναμις (dúnamis, here: "amplification") used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios.

History

Antiquity

The Sand Reckoner

Шаблон:Main In The Sand Reckoner, Archimedes proved the law of exponents, Шаблон:Math, necessary to manipulate powers of Шаблон:Math. He then used powers of Шаблон:Math to estimate the number of grains of sand that can be contained in the universe.

Islamic Golden Age

Māl and kaʿbah ("square" and "cube")

In the 9th century, the Persian mathematician Al-Khwarizmi 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"—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 Abu'l-Hasan ibn Ali al-Qalasadi.

15th–18th century

Introducing exponents

Nicolas Chuquet used a form of exponential notation in the 15th century, for example Шаблон:Math to represent Шаблон:Math. This was later used by Henricus Grammateus and Michael Stifel in the 16th century. In the late 16th century, Jost Bürgi would use Roman numerals for exponents in a way similar to that of Chuquet, for example Шаблон:Overset for Шаблон:Math.

"Exponent"; "square" and "cube"

The word exponent was coined in 1544 by Michael Stifel. In the 16th century, Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth). Biquadrate has been used to refer to the fourth power as well.

Modern exponential notation

In 1636, James Hume used in essence modern notation, when in L'algèbre de Vietè he wrote Шаблон:Math for Шаблон:Math. 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.

Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as Шаблон:Math.

"Indices"

Samuel Jeake introduced the term indices in 1696. The term involution was used synonymously with the term indices, but had declined in usage and should not be confused with its more common meaning.

Variable exponents, non-integer exponents

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

Terminology

The expression Шаблон:Math is called "the square of Шаблон:Math" or "Шаблон:Math squared", because the area of a square with side-length Шаблон:Math is Шаблон:Math. (It is true that it could also be called "Шаблон:Math to the second power", but "the square of Шаблон:Math" and "Шаблон:Math squared" are so ingrained by tradition and convenience that "Шаблон:Math to the second power" tends to sound unusual or clumsy.)

Similarly, the expression Шаблон:Math is called "the cube of Шаблон:Math" or "Шаблон:Math cubed", because the volume of a cube with side-length Шаблон:Math is Шаблон:Math.

When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example, Шаблон:Math. The base Шаблон:Math appears Шаблон:Math times in the multiplication, because the exponent is Шаблон:Math. Here, Шаблон:Math 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 Шаблон:Math can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiation Шаблон:Math 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".

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, and this definition can be used as soon one has an associative multiplication:

The base case is

<math>b^1 = b</math>

and the recurrence is

<math>b^{n+1} = b^n \cdot b.</math>

The associativity of multiplication implies that for any positive integers Шаблон:Mvar and Шаблон:Mvar,

<math>b^{m+n} = b^m \cdot b^n,</math>

and

<math>(b^m)^n=b^{mn}.</math>

Zero exponent

By definition, any nonzero number raised to the Шаблон:Math power is Шаблон:Math:

<math>b^0=1.</math>

This is the only possible definition of the zero exponent that allows applying the formula

<math>b^{m+n}=b^m\cdot b^n</math>

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

Intuitionally, <math>b^0</math> may be interpreted as the empty product of copies of Шаблон:Mvar. So, the equality <math>b^0=1</math> is a special case of the general convention for the empty product.

The case of Шаблон:Math is more complicated. In contexts where only integer powers are considered, the value Шаблон:Math is generally assigned to Шаблон:Math but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context. Шаблон:Crossreference

Negative exponents

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

<math>b^{-n} = \frac{1}{b^n}</math>.

Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (<math>\infty</math>).Шаблон:Citation needed

This definition of exponentiation with negative exponents is the only one that allows extending the identity <math>b^{m+n}=b^m\cdot b^n</math> to negative exponents (consider the case <math>m=-n</math>).

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 Шаблон:Math (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element Шаблон:Mvar is standardly denoted <math>x^{-1}.</math>

Identities and properties

Шаблон:Redirect The following identities, often called Шаблон:Vanchor, hold for all integer exponents, provided that the base is non-zero:

<math>\begin{align}

          b^{m + n} &= b^m \cdot b^n \\
 \left(b^m\right)^n &= b^{m \cdot n} \\
      (b \cdot c)^n &= b^n \cdot c^n

\end{align}</math>

Unlike addition and multiplication, exponentiation is not commutative. For example, Шаблон:Math. Also unlike addition and multiplication, exponentiation is not associative. For example, Шаблон:Math, whereas Шаблон:Math. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up (or left-associative). That is,

<math>b^{p^q} = b^{\left(p^q\right)},</math>

which, in general, is different from

<math>\left(b^p\right)^q = b^{p q} .</math>

Powers of a sum

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

<math>(a+b)^n=\sum_{i=0}^n \binom{n}{i}a^ib^{n-i}=\sum_{i=0}^n \frac{n!}{i!(n-i)!}a^ib^{n-i}.</math>

However, this formula is true only if the summands commute (i.e. that Шаблон:Math), which is implied if they belong to a structure that is commutative. Otherwise, if Шаблон:Mvar and Шаблон:Mvar 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 Шаблон:Math instead of Шаблон:Math) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

Combinatorial interpretation

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

Шаблон:Math	The Шаблон:Math possible Шаблон:Mvar-tuples of elements from the set Шаблон:Math
0Шаблон:Sup = 0	Шаблон:CNone
1Шаблон:Sup = 1	(1, 1, 1, 1)
2Шаблон:Sup = 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)
3Шаблон:Sup = 9	(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)
4Шаблон:Sup = 4	(1), (2), (3), (4)
5Шаблон:Sup = 1	()

Particular bases

Powers of ten

Шаблон:See also Шаблон:Main In the base ten (decimal) number system, integer powers of Шаблон:Math are written as the digit Шаблон:Math followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, Шаблон:Math and Шаблон:Math.

Exponentiation with base Шаблон:Math is used in scientific notation to denote large or small numbers. For instance, Шаблон:Val (the speed of light in vacuum, in metres per second) can be written as Шаблон:Val and then approximated as Шаблон:Val.

SI prefixes based on powers of Шаблон:Math are also used to describe small or large quantities. For example, the prefix kilo means Шаблон:Math, so a kilometre is Шаблон:Val.

Шаблон:AnchorPowers of two

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

Powers of Шаблон:Math appear in set theory, since a set with Шаблон:Math members has a power set, the set of all of its subsets, which has Шаблон:Math members.

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

Powers of one

Every power of one equals: Шаблон:Math. This is true even if Шаблон:Mvar is negative.

The first power of a number is the number itself: Шаблон:Math.

Powers of zero

If the exponent Шаблон:Mvar is positive (Шаблон:Math), the Шаблон:Mvarth power of zero is zero: Шаблон:Math.

If the exponent Шаблон:Mvar is negative (Шаблон:Math), the Шаблон:Mvarth power of zero Шаблон:Math is undefined, because it must equal <math>1/0^{-n}</math> with Шаблон:Math, and this would be <math>1/0</math> according to above.

The expression [[zero to the power of zero|Шаблон:Math]] is either defined as Шаблон:Math, or it is left undefined.

Powers of negative one

If Шаблон:Math is an even integer, then Шаблон:Math. This is because a negative number multiplied by another negative number cancels the sign, and thus gives a positive number.

If Шаблон:Math is an odd integer, then Шаблон:Math. This is because there will be a remaining Шаблон:Math after removing Шаблон:Math pairs.

Because of this, powers of Шаблон:Math are useful for expressing alternating sequences. For a similar discussion of powers of the complex number Шаблон:Math, see Шаблон:Slink.

Large exponents

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

Шаблон:Math as Шаблон:Math when Шаблон:Math

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:

Шаблон:Math as Шаблон:Math when Шаблон:Math

Any power of one is always one:

Шаблон:Math for all Шаблон:Math if Шаблон:Math

Powers of Шаблон:Math alternate between Шаблон:Math and Шаблон:Math as Шаблон:Math alternates between even and odd, and thus do not tend to any limit as Шаблон:Math grows.

If Шаблон:Math, Шаблон:Math alternates between larger and larger positive and negative numbers as Шаблон:Math alternates between even and odd, and thus does not tend to any limit as Шаблон:Math grows.

If the exponentiated number varies while tending to Шаблон:Math as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is

Шаблон:Math as Шаблон:Math

See Шаблон:Slink below.

Other limits, in particular those of expressions that take on an indeterminate form, are described in Шаблон:Slink below.

Power functions

Файл:Potenssi 1 3 5.svg

Power functions for Шаблон:Math

Файл:Potenssi 2 4 6.svg

Power functions for Шаблон:Math

Real functions of the form <math>f(x) = cx^n</math>, where <math>c \ne 0</math>, are sometimes called power functions. When <math>n</math> is an integer and <math>n \ge 1</math>, two primary families exist: for <math>n</math> even, and for <math>n</math> odd. In general for <math>c > 0</math>, when <math>n</math> is even <math>f(x) = cx^n</math> will tend towards positive infinity with increasing <math>x</math>, and also towards positive infinity with decreasing <math>x</math>. All graphs from the family of even power functions have the general shape of <math>y=cx^2</math>, flattening more in the middle as <math>n</math> increases. Functions with this kind of symmetry {{{1}}} are called even functions.

When <math>n</math> is odd, <math>f(x)</math>'s asymptotic behavior reverses from positive <math>x</math> to negative <math>x</math>. For <math>c > 0</math>, <math>f(x) = cx^n</math> will also tend towards positive infinity with increasing <math>x</math>, but towards negative infinity with decreasing <math>x</math>. All graphs from the family of odd power functions have the general shape of <math>y=cx^3</math>, flattening more in the middle as <math>n</math> increases and losing all flatness there in the straight line for <math>n=1</math>. Functions with this kind of symmetry {{{1}}} are called odd functions.

For <math>c < 0</math>, the opposite asymptotic behavior is true in each case.

Table of powers of decimal digits

n	n2	n3	n4	n5	n6	n7	n8	n9	n10
1	1	1	1	1	1	1	1	1	1
2	4	8	16	32	64	128	256	512	1024
3	9	27	81	243	729	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val
4	16	64	256	1024	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val
5	25	125	625	3125	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val
6	36	216	1296	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val
7	49	343	2401	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val
8	64	512	4096	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val
9	81	729	6561	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val
10	100	1000	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val	Шаблон:Val

Rational exponents

Файл:Mplwp roots 01.svg

From top to bottom: Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math.

If Шаблон:Mvar is a nonnegative real number, and Шаблон:Mvar is a positive integer, <math>x^{1/n}</math> or <math>\sqrt[n]x</math> denotes the unique positive real [[nth root|Шаблон:Mvarth root]] of Шаблон:Mvar, that is, the unique positive real number Шаблон:Mvar such that <math>y^n=x.</math>

If Шаблон:Mvar is a positive real number, and <math>\frac pq</math> is a rational number, with Шаблон:Mvar and Шаблон:Mvar integers, then <math display="inline">x^{p/q}</math> is defined as

<math>x^\frac pq= \left(x^p\right)^\frac 1q=(x^\frac 1q)^p.</math>

The equality on the right may be derived by setting <math>y=x^\frac 1q,</math> and writing <math>(x^\frac 1q)^p=y^p=\left((y^p)^q\right)^\frac 1q=\left((y^q)^p\right)^\frac 1q=(x^p)^\frac 1q.</math>

If Шаблон:Mvar is a positive rational number, Шаблон:Math, by definition.

All these definitions are required for extending the identity <math>(x^r)^s = x^{rs}</math> 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 Шаблон:Mvarth root, which is negative, if Шаблон:Mvar is odd, and no real root if Шаблон:Mvar is even. In the latter case, whichever complex Шаблон:Mvarth root one chooses for <math>x^\frac 1n,</math> the identity <math>(x^a)^b=x^{ab}</math> cannot be satisfied. For example,

<math>\left((-1)^2\right)^\frac 12 = 1^\frac 12= 1\neq (-1)^{2\cdot\frac 12} =(-1)^1=-1.</math>

See Шаблон:Slink and Шаблон:Slink 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 (Шаблон:Slink, below), or in terms of the logarithm of the base and the exponential function (Шаблон:Slink, 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 Шаблон:Slink). One may choose one of these values, called the principal value, but there is no choice of the principal value for which the identity

<math>\left(b^r\right)^s = b^{r s}</math>

is true; see Шаблон:Slink. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.

Limits of rational exponents

Файл:Continuity of the Exponential at 0.svg

The limit of Шаблон:Math is Шаблон:Math when Шаблон:Mvar 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 Шаблон:Mvar with an arbitrary real exponent Шаблон:Mvar can be defined by continuity with the rule

<math> b^x = \lim_{r (\in \mathbb{Q}) \to x} b^r \quad (b \in \mathbb{R}^+,\, x \in \mathbb{R}),</math>

where the limit is taken over rational values of Шаблон:Mvar only. This limit exists for every positive Шаблон:Mvar and every real Шаблон:Mvar.

For example, if Шаблон:Math, the non-terminating decimal representation Шаблон:Math 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 <math>b^\pi:</math>

<math>\left[b^3, b^4\right], \left[b^{3.1}, b^{3.2}\right], \left[b^{3.14}, b^{3.15}\right], \left[b^{3.141}, b^{3.142}\right], \left[b^{3.1415}, b^{3.1416}\right], \left[b^{3.14159}, b^{3.14160}\right], \ldots</math>

So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted <math>b^\pi.</math>

This defines <math>b^x</math> for every positive Шаблон:Mvar and real Шаблон:Mvar as a continuous function of Шаблон:Mvar and Шаблон:Mvar. See also Well-defined expression.

Exponential function

Шаблон:Main The exponential function is often defined as <math>x\mapsto e^x,</math> where <math>e\approx 2.718</math> is Euler's number. To avoid circular reasoning, this definition cannot be used here. So, a definition of the exponential function, denoted <math>\exp(x),</math> and of Euler's number are given, which rely only on exponentiation with positive integer exponents. Then a proof is sketched that, if one uses the definition of exponentiation given in preceding sections, one has

<math>\exp(x)=e^x.</math>

There are many equivalent ways to define the exponential function, one of them being

<math>\exp(x) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n.</math>

One has <math>\exp(0)=1,</math> and the exponential identity <math>\exp(x+y)=\exp(x)\exp(y)</math> holds as well, since

<math>\exp(x)\exp(y) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n\left(1 + \frac{y}{n}\right)^n = \lim_{n\rightarrow\infty} \left(1 + \frac{x+y}{n} + \frac{xy}{n^2}\right)^n,</math>

and the second-order term <math>\frac{xy}{n^2}</math> does not affect the limit, yielding <math>\exp(x)\exp(y) = \exp(x+y)</math>.

Euler's number can be defined as <math>e=\exp(1)</math>. It follows from the preceding equations that <math>\exp(x)=e^x</math> when Шаблон:Mvar is an integer (this results from the repeated-multiplication definition of the exponentiation). If Шаблон:Mvar is real, <math>\exp(x)=e^x</math> results from the definitions given in preceding sections, by using the exponential identity if Шаблон:Mvar is rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for every complex value of Шаблон:Mvar, and therefore it can be used to extend the definition of <math>\exp(z)</math>, and thus <math>e^z,</math> from the real numbers to any complex argument Шаблон:Mvar. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.

Powers via logarithms

The definition of Шаблон:Math as the exponential function allows defining Шаблон:Math for every positive real numbers Шаблон:Mvar, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm Шаблон:Math is the inverse of the exponential function Шаблон:Math means that one has

<math>b = \exp(\ln b)=e^{\ln b}</math>

for every Шаблон:Math. For preserving the identity <math>(e^x)^y=e^{xy},</math> one must have

<math>b^x=\left(e^{\ln b} \right)^x = e^{x \ln b}</math>

So, <math>e^{x \ln b}</math> can be used as an alternative definition of Шаблон:Math for any positive real Шаблон:Mvar. 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 Шаблон:Mvar is a positive real number, exponentiation with base Шаблон:Mvar and complex exponent Шаблон:Mvar is defined by means of the exponential function with complex argument (see the end of Шаблон:Slink, above) as

<math>b^z = e^{(z\ln b)},</math>

where <math>\ln b</math> denotes the natural logarithm of Шаблон:Mvar.

This satisfies the identity

<math>b^{z+t} = b^z b^t,</math>

In general, <math DISPLAY=inline>\left(b^z\right)^t</math> is not defined, since Шаблон:Math is not a real number. If a meaning is given to the exponentiation of a complex number (see Шаблон:Slink, below), one has, in general,

<math>\left(b^z\right)^t \ne b^{zt},</math>

unless Шаблон:Mvar is real or Шаблон:Mvar is an integer.

Euler's formula,

<math>e^{iy} = \cos y + i \sin y,</math>

allows expressing the polar form of <math>b^z</math> in terms of the real and imaginary parts of Шаблон:Mvar, namely

<math>b^{x+iy}= b^x(\cos(y\ln b)+i\sin(y\ln b)),</math>

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

<math>b^{x+iy}=b^x b^{iy}=b^x e^{iy\ln b} =b^x(\cos(y\ln b)+i\sin(y\ln b)).</math>

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 Шаблон:Mvarth roots, that is, of exponents <math>1/n,</math> where Шаблон:Mvar is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to Шаблон:Mvarth roots, this case deserves to be considered first, since it does not need to use complex logarithms, and is therefore easier to understand.

Шаблон:Mvarth roots of a complex number

Every nonzero complex number Шаблон:Mvar may be written in polar form as

<math>z=\rho e^{i\theta}=\rho(\cos \theta +i \sin \theta),</math>

where <math>\rho</math> is the absolute value of Шаблон:Mvar, and <math>\theta</math> is its argument. The argument is defined up to an integer multiple of Шаблон:Math; this means that, if <math>\theta</math> is the argument of a complex number, then <math>\theta +2k\pi</math> is also an argument of the same complex number for every integer <math>k</math>.

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 Шаблон:Mvarth root of a complex number can be obtained by taking the Шаблон:Mvarth root of the absolute value and dividing its argument by Шаблон:Mvar:

<math>\left(\rho e^{i\theta}\right)^\frac 1n=\sqrt[n]\rho \,e^\frac{i\theta}n.</math>

If <math>2\pi</math> is added to <math>\theta</math>, the complex number is not changed, but this adds <math>2i\pi/n</math> to the argument of the Шаблон:Mvarth root, and provides a new Шаблон:Mvarth root. This can be done Шаблон:Mvar times, and provides the Шаблон:Mvar Шаблон:Mvarth roots of the complex number.

It is usual to choose one of the Шаблон:Mvar Шаблон:Mvarth root as the principal root. The common choice is to choose the Шаблон:Mvarth root for which <math>-\pi<\theta\le \pi,</math> that is, the Шаблон:Mvarth root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal Шаблон:Mvarth root a continuous function in the whole complex plane, except for negative real values of the radicand. This function equals the usual Шаблон:Mvarth root for positive real radicands. For negative real radicands, and odd exponents, the principal Шаблон:Mvarth root is not real, although the usual Шаблон:Mvarth root is real. Analytic continuation shows that the principal Шаблон:Mvarth root is the unique complex differentiable function that extends the usual Шаблон:Mvarth 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 <math>2\pi,</math> the complex number comes back to its initial position, and its Шаблон:Mvarth roots are permuted circularly (they are multiplied by <math DISPLAY=textstyle>e^{2i\pi/n}</math>). This shows that it is not possible to define a Шаблон:Mvarth root function that is continuous in the whole complex plane.

Roots of unity

Файл:One3Root.svg

The three third roots of Шаблон:Math

The Шаблон:Mvarth roots of unity are the Шаблон:Mvar complex numbers such that Шаблон:Math, where Шаблон:Mvar 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 Шаблон:Mvar Шаблон:Mvarth roots of unity are the Шаблон:Mvar first powers of <math>\omega =e^\frac{2\pi i}{n}</math>, that is <math>1=\omega^0=\omega^n, \omega=\omega^1, \omega^2, \omega^{n-1}.</math> The Шаблон:Mvarth roots of unity that have this generating property are called primitive Шаблон:Mvarth roots of unity; they have the form <math>\omega^k=e^\frac{2k\pi i}{n},</math> with Шаблон:Mvar coprime with Шаблон:Mvar. The unique primitive square root of unity is <math>-1;</math> the primitive fourth roots of unity are <math>i</math> and <math>-i.</math>

The Шаблон:Mvarth roots of unity allow expressing all Шаблон:Mvarth roots of a complex number Шаблон:Mvar as the Шаблон:Mvar products of a given Шаблон:Mvarth roots of Шаблон:Mvar with a Шаблон:Mvarth root of unity.

Geometrically, the Шаблон:Mvarth roots of unity lie on the unit circle of the complex plane at the vertices of a [[regular polygon|regular Шаблон:Mvar-gon]] with one vertex on the real number 1.

As the number <math>e^\frac{2k\pi i}{n}</math> is the primitive Шаблон:Mvarth root of unity with the smallest positive argument, it is called the principal primitive Шаблон:Mvarth root of unity, sometimes shortened as principal Шаблон:Mvarth root of unity, although this terminology can be confused with the principal value of <math>1^{1/n}</math>, which is 1.

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, infinitely many possible values for <math DISPLAY=textstyle>z^w</math>. So, either a principal value is defined, which is not continuous for the values of Шаблон:Mvar that are real and nonpositive, or <math DISPLAY=textstyle>z^w</math> is defined as a multivalued function.

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

<math>z^w=e^{w\log z},</math>

where <math>\log z</math> is the variant of the complex logarithm that is used, which is, a function or a multivalued function such that

<math>e^{\log z}=z</math>

for every Шаблон:Mvar in its domain of definition.

Principal value

The principal value of the complex logarithm is the unique continuous function, commonly denoted <math>\log,</math> such that, for every nonzero complex number Шаблон:Mvar,

<math>e^{\log z}=z,</math>

and the argument of Шаблон:Mvar satisfies

<math>-\pi <\operatorname{Arg}z \le \pi.</math>

The principal value of the complex logarithm is not defined for <math>z=0,</math> it is discontinuous at negative real values of Шаблон:Mvar, and it is holomorphic (that is, complex differentiable) elsewhere. If Шаблон:Mvar is real and positive, the principal value of the complex logarithm is the natural logarithm: <math>\log z=\ln z.</math>

The principal value of <math>z^w</math> is defined as <math>z^w=e^{w\log z},</math> where <math>\log z</math> is the principal value of the logarithm.

The function <math>(z,w)\to z^w</math> is holomorphic except in the neighbourhood of the points where Шаблон:Mvar is real and nonpositive.

If Шаблон:Mvar is real and positive, the principal value of <math>z^w</math> equals its usual value defined above. If <math>w=1/n,</math> where Шаблон:Mvar 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 <math>\log z</math> and <math>z^w</math> at the negative real values of Шаблон:Mvar. In this case, it is useful to consider these functions as multivalued functions.

If <math>\log z</math> denotes one of the values of the multivalued logarithm (typically its principal value), the other values are <math>2ik\pi +\log z,</math> where Шаблон:Mvar is any integer. Similarly, if <math>z^w</math> is one value of the exponentiation, then the other values are given by

<math>e^{w(2ik\pi +\log z)} = z^we^{2ik\pi w},</math>

where Шаблон:Mvar is any integer.

Different values of Шаблон:Mvar give different values of <math>z^w</math> unless Шаблон:Mvar is a rational number, that is, there is an integer Шаблон:Mvar such that Шаблон:Mvar is an integer. This results from the periodicity of the exponential function, more specifically, that <math>e^a=e^b</math> if and only if <math>a-b</math> is an integer multiple of <math>2\pi i.</math>

If <math>w=\frac mn</math> is a rational number with Шаблон:Mvar and Шаблон:Mvar coprime integers with <math>n>0,</math> then <math>z^w</math> has exactly Шаблон:Mvar values. In the case <math>m=1,</math> these values are the same as those described in [[#nth roots of a complex number|§ Шаблон:Mvarth roots of a complex number]]. If Шаблон:Mvar is an integer, there is only one value that agrees with that of Шаблон:Slink.

The multivalued exponentiation is holomorphic for <math>z\ne 0,</math> in the sense that its graph consists of several sheets that define each a holomorphic function in the neighborhood of every point. If Шаблон:Mvar varies continuously along a circle around Шаблон:Math, then, after a turn, the value of <math>z^w</math> has changed of sheet.

Computation

The canonical form <math>x+iy</math> of <math>z^w</math> can be computed from the canonical form of Шаблон:Mvar and Шаблон:Mvar. Although this can be described by a single formula, it is clearer to split the computation in several steps.

• Polar form of Шаблон:Mvar. If <math>z=a+ib</math> is the canonical form of Шаблон:Mvar (Шаблон:Mvar and Шаблон:Mvar being real), then its polar form is <math DISPLAY=block>z=\rho e^{i\theta}= \rho (\cos\theta + i \sin\theta),</math> where <math>\rho=\sqrt{a^2+b^2}</math> and <math>\theta=\operatorname{atan2}(a,b)</math> (see atan2 for the definition of this function).
• Logarithm of Шаблон:Mvar. The principal value of this logarithm is <math>\log z=\ln \rho+i\theta,</math> where <math>\ln</math> denotes the natural logarithm. The other values of the logarithm are obtained by adding <math>2ik\pi</math> for any integer Шаблон:Mvar.
• Canonical form of <math>w\log z.</math> If <math>w=c+di</math> with Шаблон:Mvar and Шаблон:Mvar real, the values of <math>w\log z</math> are <math DISPLAY=block>w\log z = (c\ln \rho - d\theta-2dk\pi) +i (d\ln \rho + c\theta+2ck\pi),</math> the principal value corresponding to <math>k=0.</math>
• Final result. Using the identities <math>e^{x+y}=e^xe^y</math> and <math>e^{y\ln x} = x^y,</math> one gets <math DISPLAY=block>z^w=\rho^c e^{-d(\theta+2k\pi)} \left(\cos (d\ln \rho + c\theta+2ck\pi) +i\sin(d\ln \rho + c\theta+2ck\pi)\right),</math> with <math>k=0</math> for the principal value.

Examples

• <math>i^i</math>
  The polar form of Шаблон:Mvar is <math>i=e^{i\pi/2},</math> and the values of <math>\log i</math> are thus <math DISPLAY=block>\log i=i\left(\frac \pi 2 +2k\pi\right).</math> It follows that <math DISPLAY=block>i^i=e^{i\log i}=e^{-\frac \pi 2} e^{-2k\pi}.</math>So, all values of <math>i^i</math> are real, the principal one being <math DISPLAY=block> e^{-\frac \pi 2} \approx 0.2079.</math>
• <math>(-2)^{3+4i}</math>
  Similarly, the polar form of Шаблон:Math is <math>-2 = 2e^{i \pi}.</math> So, the above described method gives the values <math DISPLAY=block>\begin{align}

(-2)^{3 + 4i} &= 2^3 e^{-4(\pi+2k\pi)} (\cos(4\ln 2 + 3(\pi +2k\pi)) +i\sin(4\ln 2 + 3(\pi+2k\pi)))\\ &=-2^3 e^{-4(\pi+2k\pi)}(\cos(4\ln 2) +i\sin(4\ln 2)). \end{align}</math>In this case, all the values have the same argument <math>4\ln 2,</math> and different absolute values.

In both examples, all values of <math>z^w</math> have the same argument. More generally, this is true if and only if the real part of Шаблон:Mvar 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:

Шаблон:Bulleted list{(-1)^\frac{1}{2}} = \frac{1}{i} = -i</math>

On the other hand, when Шаблон:Mvar 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 Шаблон:Math are Шаблон:Math. The identity holds, but saying Шаблон:Math is incorrect. | The identity Шаблон:Math holds for real numbers Шаблон:Mvar and Шаблон:Mvar, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:

For any integer Шаблон:Mvar, we have:

1. <math>e^{1 + 2 \pi i n} = e^1 e^{2 \pi i n} = e \cdot 1 = e</math>
2. <math>\left(e^{1 + 2\pi i n}\right)^{1 + 2 \pi i n} = e\qquad</math> (taking the <math>(1 + 2 \pi i n)</math>-th power of both sides)
3. <math>e^{1 + 4 \pi i n - 4 \pi^2 n^2} = e\qquad</math> (using <math>\left(e^x\right)^y = e^{xy}</math> and expanding the exponent)
4. <math>e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e\qquad</math> (using <math>e^{x+y} = e^x e^y</math>)
5. <math>e^{-4 \pi^2 n^2} = 1\qquad</math> (dividing by Шаблон:Mvar)

but this is false when the integer Шаблон:Mvar is nonzero.

The error is the following: by definition, <math>e^y</math> is a notation for <math>\exp(y),</math> a true function, and <math>x^y</math> is a notation for <math>\exp(y\log x),</math> which is a multi-valued function. Thus the notation is ambiguous when Шаблон:Math. Here, before expanding the exponent, the second line should be <math display="block">\exp\left((1 + 2\pi i n)\log \exp(1 + 2\pi i n)\right) = \exp(1 + 2\pi i n).</math>

Therefore, when expanding the exponent, one has implicitly supposed that <math>\log \exp z =z</math> for complex values of Шаблон:Mvar, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity Шаблон:Math must be replaced by the identity <math display="block">\left(e^x\right)^y = e^{y\log e^x},</math> which is a true identity between multivalued functions. }}

Irrationality and transcendence

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

In other words, if Шаблон:Mvar is irrational and <math>b\not\in \{0,1\},</math> then at least one of Шаблон:Mvar, Шаблон:Mvar and Шаблон:Math 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. The definition of Шаблон:Math requires further the existence of a multiplicative identity.

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

• <math>x^0 = 1,</math>
• <math>x^{n+1} = x x^n</math> for every nonnegative integer Шаблон:Mvar.

If Шаблон:Mvar is a negative integer, <math>x^n</math> is defined only if Шаблон:Mvar has a multiplicative inverse. In this case, the inverse of Шаблон:Mvar is denoted Шаблон:Math, and Шаблон:Math is defined as <math>\left(x^{-1}\right)^{-n}.</math>

Exponentiation with integer exponents obeys the following laws, for Шаблон:Mvar and Шаблон:Mvar in the algebraic structure, and Шаблон:Mvar and Шаблон:Mvar integers:

<math>\begin{align}

x^0&=1\\ x^{m+n}&=x^m x^n\\ (x^m)^n&=x^{mn}\\ (xy)^n&=x^n y^n \quad \text{if} xy=yx, \text{and, in particular, if the multiplication is commutative.} \end{align}</math>

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 Шаблон:Mvar is a real function whose valued can be multiplied, <math>f^n</math> denotes the exponentiation with respect of multiplication, and <math>f^{\circ n}</math> may denote exponentiation with respect of function composition. That is,

<math>(f^n)(x)=(f(x))^n=f(x) \,f(x) \cdots f(x),</math>

and

<math>(f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots)).</math>

Commonly, <math>(f^n)(x)</math> is denoted <math>f(x)^n,</math> while <math>(f^{\circ n})(x)</math> is denoted <math>f^n(x).</math>

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 Шаблон:Mvar is a group, <math>x^n</math> is defined for every <math>x\in G</math> and every integer Шаблон:Mvar.

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 Шаблон:Mvar is the cyclic group generated by Шаблон:Mvar. If all the powers of Шаблон:Mvar are distinct, the group is isomorphic to the additive group <math>\Z</math> 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 Шаблон:Mvar. If the order of Шаблон:Mvar is Шаблон:Mvar, then <math>x^n=x^0=1,</math> and the cyclic group generated by Шаблон:Mvar consists of the Шаблон:Mvar first powers of Шаблон:Mvar (starting indifferently from the exponent Шаблон:Math or Шаблон:Math).

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, Шаблон:Math, where Шаблон:Math and Шаблон:Math 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 <math>(g^h)^k=g^{hk}</math> and <math>(gh)^k=g^kh^k.</math>

In a ring

In a ring, it may occur that some nonzero elements satisfy <math>x^n=0</math> for some integer Шаблон:Mvar. Such an element is said to be 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 <math>x\neq 0</math> implies <math>x^n\neq 0</math> for every positive integer Шаблон:Mvar), 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 Шаблон:Mvar in a commutative ring Шаблон:Mvar, the set of the elements of Шаблон:Mvar that have a power in Шаблон:Mvar is an ideal, called the radical of Шаблон:Mvar. The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring <math>k[x_1, \ldots, x_n]</math> over a field Шаблон:Mvar, 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 Шаблон:Math is a square matrix, then the product of Шаблон:Math with itself Шаблон:Math times is called the matrix power. Also <math>A^0</math> is defined to be the identity matrix, and if Шаблон:Math is invertible, then <math>A^{-n} = \left(A^{-1}\right)^n</math>.

Matrix powers appear often in the context of discrete dynamical systems, where the matrix Шаблон:Math expresses a transition from a state vector Шаблон:Math of some system to the next state Шаблон:Math of the system. This is the standard interpretation of a Markov chain, for example. Then <math>A^2x</math> is the state of the system after two time steps, and so forth: <math>A^nx</math> is the state of the system after Шаблон:Math time steps. The matrix power <math>A^n</math> is the transition matrix between the state now and the state at a time Шаблон:Math 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, <math>d/dx</math>, which is a linear operator acting on functions <math>f(x)</math> to give a new function <math>(d/dx)f(x) = f'(x)</math>. The Шаблон:Mathth power of the differentiation operator is the Шаблон:Mathth derivative:

<math>\left(\frac{d}{dx}\right)^nf(x) = \frac{d^n}{dx^n}f(x) = f^{(n)}(x).</math>

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. 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

Шаблон:Main 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 Шаблон:Math. Common examples are the field of complex numbers, the real numbers and the rational numbers, considered earlier in this article, which 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 <math>q=p^k,</math> where Шаблон:Mvar is a prime number, and Шаблон:Mvar is a positive integer. For every such Шаблон:Mvar, there are fields with Шаблон:Mvar elements. The fields with Шаблон:Mvar elements are all isomorphic, which allows, in general, working as if there were only one field with Шаблон:Mvar elements, denoted <math>\mathbb F_q.</math>

One has

<math>x^q=x</math>

for every <math>x\in \mathbb F_q.</math>

A primitive element in <math>\mathbb F_q</math> is an element Шаблон:Mvar such that the set of the Шаблон:Math first powers of Шаблон:Mvar (that is, <math>\{g^1=g, g^2, \ldots, g^{p-1}=g^0=1\}</math>) equals the set of the nonzero elements of <math>\mathbb F_q.</math> There are <math>\varphi (p-1)</math> primitive elements in <math>\mathbb F_q,</math> where <math>\varphi</math> is Euler's totient function.

In <math>\mathbb F_q,</math> the freshman's dream identity

<math>(x+y)^p = x^p+y^p</math>

is true for the exponent Шаблон:Mvar. As <math>x^p=x</math> in <math>\mathbb F_q,</math> It follows that the map

<math>\begin{align}

F\colon{} & \mathbb F_q \to \mathbb F_q\\ & x\mapsto x^p \end{align}</math> is linear over <math>\mathbb F_q,</math> and is a field automorphism, called the Frobenius automorphism. If <math>q=p^k,</math> the field <math>\mathbb F_q</math> has Шаблон:Mvar automorphisms, which are the Шаблон:Mvar first powers (under composition) of Шаблон:Mvar. In other words, the Galois group of <math>\mathbb F_q</math> is cyclic of order Шаблон:Mvar, 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 Шаблон:Mvar is a primitive element in <math>\mathbb F_q,</math> then <math>g^e</math> can be efficiently computed with exponentiation by squaring for any Шаблон:Mvar, even if Шаблон:Mvar is large, while there is no known computationally practical algorithm that allows retrieving Шаблон:Mvar from <math>g^e</math> if Шаблон:Mvar is sufficiently large.

Powers of sets Шаблон:Anchor

The Cartesian product of two sets Шаблон:Mvar and Шаблон:Mvar is the set of the ordered pairs <math>(x,y)</math> such that <math>x\in S</math> and <math>y\in T.</math> This operation is not properly commutative nor associative, but has these properties up to canonical isomorphisms, that allow identifying, for example, <math>(x,(y,z)),</math> <math>((x,y),z),</math> and <math>(x,y,z).</math>

This allows defining the Шаблон:Mvarth power <math>S^n</math> of a set Шаблон:Mvar as the set of all Шаблон:Mvar-tuples <math>(x_1, \ldots, x_n)</math> of elements of Шаблон:Mvar.

When Шаблон:Mvar is endowed with some structure, it is frequent that <math>S^n</math> 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 <math>\R^n</math> (where <math>\R</math> denotes the real numbers) denotes the Cartesian product of Шаблон:Mvar copies of <math>\R,</math> as well as their direct product as vector space, topological spaces, rings, etc.

Sets as exponents

Шаблон:See also A Шаблон:Mvar-tuple <math>(x_1, \ldots, x_n)</math> of elements of Шаблон:Mvar can be considered as a function from <math>\{1,\ldots, n\}.</math> This generalizes to the following notation.

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

<math>(S^T)^U\cong S^{T\times U},</math>

<math>S^{T\sqcup U}\cong S^T\times S^U,</math>

where <math>\times</math> denotes the Cartesian product, and <math>\sqcup</math> 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, <math>\R^\N</math> denotes the vector space of the infinite sequences of real numbers, and <math>\R^{(\N)}</math> 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 Шаблон:Math, while the Hamel bases of the former cannot be explicitly described (because their existence involves Zorn's lemma).

In this context, Шаблон:Math can represents the set <math>\{0,1\}.</math> So, <math>2^S</math> denotes the power set of Шаблон:Mvar, that is the set of the functions from Шаблон:Mvar to <math>\{0,1\},</math> which can be identified with the set of the subsets of Шаблон:Mvar, by mapping each function to the inverse image of Шаблон:Math.

This fits in with the exponentiation of cardinal numbers, in the sense that Шаблон:Math, where Шаблон:Math is the cardinality of Шаблон:Math.

In category theory

Шаблон:Main In the category of sets, the morphisms between sets Шаблон:Mvar and Шаблон:Mvar are the functions from Шаблон:Mvar to Шаблон:Mvar. It results that the set of the functions from Шаблон:Mvar to Шаблон:Mvar that is denoted <math>Y^X</math> in the preceding section can also be denoted <math>\hom(X,Y).</math> The isomorphism <math>(S^T)^U\cong S^{T\times U}</math> can be rewritten

<math>\hom(U,S^T)\cong \hom(T\times U,S).</math>

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

This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor <math>X\to X^T</math> is, if it exists, a right adjoint to the functor <math>Y\to T\times Y.</math> A category is called a Cartesian closed category, if direct products exist, and the functor <math>Y\to X\times Y</math> has a right adjoint for every Шаблон:Mvar.

Repeated exponentiation

Шаблон:Main 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 Шаблон:Math, the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and Шаблон:Val (Шаблон:Math) 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 Шаблон:Math has no limit at the point Шаблон:Math. One may consider at what points this function does have a limit.

More precisely, consider the function <math>f(x,y) = x^y</math> defined on <math> D = \{(x, y) \in \mathbf{R}^2 : x > 0 \}</math>. Then Шаблон:Math can be viewed as a subset of Шаблон:Math (that is, the set of all pairs Шаблон:Math with Шаблон:Math, Шаблон:Math belonging to the extended real number line Шаблон:Math, endowed with the product topology), which will contain the points at which the function Шаблон:Math has a limit.

In fact, Шаблон:Math has a limit at all accumulation points of Шаблон:Math, except for Шаблон:Math, Шаблон:Math, Шаблон:Math and Шаблон:Math. Accordingly, this allows one to define the powers Шаблон:Math by continuity whenever Шаблон:Math, Шаблон:Math, except for Шаблон:Math, Шаблон:Math, Шаблон:Math and Шаблон:Math, which remain indeterminate forms.

Under this definition by continuity, we obtain:

• Шаблон:Math and Шаблон:Math, when Шаблон:Math.
• Шаблон:Math and Шаблон:Math, when Шаблон:Math.
• Шаблон:Math and Шаблон:Math, when Шаблон:Math.
• Шаблон:Math and Шаблон:Math, when Шаблон:Math.

These powers are obtained by taking limits of Шаблон:Math for positive values of Шаблон:Math. This method does not permit a definition of Шаблон:Math when Шаблон:Math, since pairs Шаблон:Math with Шаблон:Math are not accumulation points of Шаблон:Math.

On the other hand, when Шаблон:Math is an integer, the power Шаблон:Math is already meaningful for all values of Шаблон:Math, including negative ones. This may make the definition Шаблон:Math obtained above for negative Шаблон:Math problematic when Шаблон:Math is odd, since in this case Шаблон:Math as Шаблон:Math tends to Шаблон:Math through positive values, but not negative ones.

Efficient computation with integer exponents

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

<math>100 = 2^2 +2^5 + 2^6 = 2^2(1+2^3(1+2))</math>.

Then compute the following terms in order, reading Horner's rule from right to left. Шаблон:Static row numbers

22 = 4

2 (22) = 23 = 8

(23)2 = 26 = 64

(26)2 = 212 = Шаблон:Val

(212)2 = 224 = Шаблон:Val

2 (224) = 225 = Шаблон:Val

(225)2 = 250 = Шаблон:Val

(250)2 = 2100 = Шаблон:Val

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

In general, the number of multiplication operations required to compute Шаблон:Math can be reduced to <math>\sharp n +\lfloor \log_{2} n\rfloor -1,</math> by using exponentiation by squaring, where <math>\sharp n</math> denotes the number of Шаблон:Maths in the binary representation of Шаблон:Mvar. 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 Шаблон:Math is a difficult problem, for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available. However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.

Iterated functions

Шаблон:See also 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 <math>g\circ f,</math> and defined as

<math>(g\circ f)(x)=g(f(x))</math>

for every Шаблон:Mvar in the domain of Шаблон:Mvar.

If the domain of a function Шаблон:Mvar equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the Шаблон:Mvarth power of the function under composition, commonly called the Шаблон:Mvarth iterate of the function. Thus <math>f^n</math> denotes generally the Шаблон:Mvarth iterate of Шаблон:Mvar; for example, <math>f^3(x)</math> means <math>f(f(f(x))).</math>

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 generally 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 <math>f^2(x)= f(f(x)),</math> and <math>f(x)^2= f(x)\cdot f(x).</math> When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example <math>f^{\circ 3}=f\circ f \circ f,</math> and <math>f^3=f\cdot f\cdot f.</math> For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the trigonometric functions. So, <math>\sin^2 x</math> and <math>\sin^2(x)</math> both mean <math>\sin(x)\cdot\sin(x)</math> and not <math>\sin(\sin(x)),</math> which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.

In this context, the exponent <math>-1</math> denotes always the inverse function, if it exists. So <math>\sin^{-1}x=\sin^{-1}(x) = \arcsin x.</math> For the multiplicative inverse fractions are generally used as in <math>1/\sin(x)=\frac 1{\sin x}.</math>

In programming languages

Programming languages generally express exponentiation either as an infix operator or as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is the caret (^). The original version of ASCII included an uparrow symbol (↑), intended for exponentiation, but this was replaced by the caret in 1967, so the caret became usual in programming languages. The notations include:

• x ^ y: AWK, BASIC, J, MATLAB, Wolfram Language (Mathematica), R, Microsoft Excel, Analytica, TeX (and its derivatives), TI-BASIC, bc (for integer exponents), Haskell (for nonnegative integer exponents), Lua, and most computer algebra systems.
• x ** y. The Fortran character set did not include lowercase characters or punctuation symbols other than +-*/()&=.,' and so used ** for exponentiation (the initial version used a xx b instead.). Many other languages followed suit: Ada, Z shell, KornShell, Bash, COBOL, CoffeeScript, Fortran, FoxPro, Gnuplot, Groovy, JavaScript, OCaml, F#, Perl, PHP, PL/I, Python, Rexx, Ruby, SAS, Seed7, Tcl, ABAP, Mercury, Haskell (for floating-point exponents), Turing, and VHDL.
• x ↑ y: Algol Reference language, Commodore BASIC, TRS-80 Level II/III BASIC.
• x ^^ y: Haskell (for fractional base, integer exponents), D.
• x⋆y: APL.

In most programming languages with an infix exponentiation operator, it is right-associative, that is, a^b^c is interpreted as a^(b^c). This is because (a^b)^c is equal to a^(b*c) and thus not as useful. In some languages, it is left-associative, notably in Algol, Matlab, and the Microsoft Excel formula language.

Other programming languages use functional notation:

• (expt x y): Common Lisp.
• pown x y: F# (for integer base, integer exponent).

Still others only provide exponentiation as part of standard libraries:

• pow(x, y): C, C++ (in math library).
• Math.Pow(x, y): C#.
• math:pow(X, Y): Erlang.
• Math.pow(x, y): Java.
• [Math]::Pow(x, y): PowerShell.

In some statically typed languages that prioritize type safety such as Rust, exponentiation is performed via a multitude of methods:

• x.pow(y) for x and y as integers
• x.powf(y) for x and y as floating point numbers
• x.powi(y) for x as a float and y as an integer

See also

• Double exponential function
• Exponential decay
• Exponential field
• Exponential growth
• Hyperoperation
• List of exponential topics
• Modular exponentiation
• Scientific notation
• Tetration
• Unicode subscripts and superscripts
• x^y = y^x
• Zero to the power of zero