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Wikipedia

Exponentiation

"Exponent" redirects here. For other uses, see Exponent (disambiguation).

Exponentiation is a mathematical operation, written asbn, involving two numbers, the baseb and the exponent or powern, and pronounced as "b raised to the power ofn". Whenn is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is,bn is the product of multiplyingn bases:

Graphs ofy = bx for various bases b: base 10, base e, base 2, base1/2. Each curve passes through the point(0, 1) because any nonzero number raised to the power of 0 is 1. Atx = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
b n = b × × b n times . {\displaystyle b^{n}=\underbrace {b\times \dots \times b} _{n{\text{ times}}}.}

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", "the nth power of b", "b to the nth power", or most briefly as "b to the nth".

One hasb1 = b, and, for any positive integersm andn, one hasbnbm = bn+m. To extend this property to non-positive integer exponents,b0 is defined to be1, andbn (withn a positive integer andb not zero) is defined as1/bn. In particular,b−1 is equal to1/b, the reciprocal ofb.

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.

Contents

The term power (Latin: potentia, potestas, dignitas) 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. In The Sand Reckoner, Archimedes discovered and proved the law of exponents,10a ⋅ 10b = 10a+b, necessary to manipulate powers of10.[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"—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.

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

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. Samuel Jeake introduced the term indices in 1696. 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.

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 Isaac Newton) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, asax + bxx + cx3 + d.

Another historical synonym,[clarification needed] involution, is now rare 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."

The expressionb2 = bb is called "the square of b" or "b squared", because the area of a square with side-lengthb isb2.

Similarly, the expressionb3 = bbb is called "the cube of b" or "b cubed", because the volume of a cube with side-lengthb isb3.

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 base3 appears5 times in the multiplication, because the exponent is5. 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, so35 can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiationbn 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 as357 (which means3(57) and not(35)7), is called a tower of powers, or simply a tower.

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

b 1 = b {\displaystyle b^{1}=b}

and the recurrence is

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

The associativity of multiplication implies that for any positive integersm andn,

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

and

( b m ) n = b m n . {\displaystyle (b^{m})^{n}=b^{mn}.}

Zero exponent

By definition, any nonzero number raised to the0 power is1:

b 0 = 1. {\displaystyle b^{0}=1.}

This definition is the only possible that allows extending the formula

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

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

Intuitionally, b 0 {\displaystyle b^{0}} may be interpreted as the empty product of copies ofb. So, the equality b 0 = 1 {\displaystyle b^{0}=1} is a special case of the general convention for the empty product.

The case of00 is more complicated. In contexts where only integer powers are considered, the value1 is generally assigned to 0 0 , {\displaystyle 0^{0},} 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 integern and nonzerob:

b n = 1 b n . {\displaystyle b^{-n}={\frac {1}{b^{n}}}.}

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

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

The same definition applies to invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted1 (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible elementx is standardly denoted x 1 . {\displaystyle x^{-1}.}

Identities and properties

"Laws of Indices" redirects here. For the horse, see Laws of Indices (horse).

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

b m + n = b m b n ( b m ) n = b m n ( b c ) n = b n c n {\displaystyle {\begin{aligned}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{aligned}}}

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, whereas2(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 (or left-associative). That is,

b p q = b ( p q ) , {\displaystyle b^{p^{q}}=b^{\left(p^{q}\right)},}

which, in general, is different from

( b p ) q = b p q . {\displaystyle \left(b^{p}\right)^{q}=b^{pq}.}

Powers of a sum

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

( a + b ) n = i = 0 n ( n i ) a i b n i = i = 0 n n ! i ! ( n i ) ! a i b n i . {\displaystyle (a+b)^{n}=\sum _{i=0}^{n}{\binom {n}{i}}a^{i}b^{n-i}=\sum _{i=0}^{n}{\frac {n!}{i!(n-i)!}}a^{i}b^{n-i}.}

However, this formula is true only if the summands commute (i.e. thatab = ba), which is implied if they belong to a structure that is commutative. Otherwise, ifa andb 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 integersn andm, the value ofnm is the number of functions from a set ofm elements to a set ofn elements (see cardinal exponentiation). Such functions can be represented asm-tuples from ann-element set (or asm-letter words from ann-letter alphabet). Some examples for particular values ofm andn are given in the following table:

nm Thenm possiblem-tuples of elements from the set{1, ..., n}
05 = 0 none
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

Main article: Power of 10

In the base ten (decimal) number system, integer powers of10 are written as the digit1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example,103 =1000 and10−4 =0.0001.

Exponentiation with base10 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 as2.99792458×108 m/s and then approximated as2.998×108 m/s.

SI prefixes based on powers of10 are also used to describe small or large quantities. For example, the prefix kilo means103 =1000, so a kilometre is1000 m.

Powers of two

Main article: Power of two

The first negative powers of2 are commonly used, and have special names, e.g.: half and quarter.

Powers of2 appear in set theory, since a set withn members has a power set, the set of all of its subsets, which has2n members.

Integer powers of2 are important in computer science. The positive integer powers2n give the number of possible values for ann-bit integer binary number; for example, a byte may take28 = 256 different values. The binary number system expresses any number as a sum of powers of2, and denotes it as a sequence of0 and1, separated by a binary point, where1 indicates a power of2 that appears in the sum; the exponent is determined by the place of this1: the nonnegative exponents are the rank of the1 on the left of the point (starting from0), 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.

The first power of a number is the number itself: n 1 = n . {\displaystyle n^{1}=n.}

Powers of zero

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

If the exponentn is negative (n < 0), thenth power of zero0n is undefined, because it must equal 1 / 0 n {\displaystyle 1/0^{-n}} withn > 0, and this would be 1 / 0 {\displaystyle 1/0} according to above.

The expression00 is either defined as 1, or it is left undefined (see Zero to the power of zero).

Powers of negative one

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

Ifn 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 numberi, 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 → ∞ asn → ∞ whenb > 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 asn → ∞ when|b| < 1

Any power of one is always one:

bn = 1 for alln ifb = 1

Powers of–1 alternate between1 and–1 asn alternates between even and odd, and thus do not tend to any limit asn grows.

Ifb < –1,bn, alternates between larger and larger positive and negative numbers asn alternates between even and odd, and thus does not tend to any limit asn grows.

If the exponentiated number varies while tending to1 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 asn → ∞

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}
Power functions for n = 2 , 4 , 6 {\displaystyle n=2,4,6}

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

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

For c < 0 {\displaystyle c<0} , 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 2187 6561 19683 59049
4 16 64 256 1024 4096 16384 65536 262144 1048576
5 25 125 625 3125 15625 78125 390625 1953125 9765625
6 36 216 1296 7776 46656 279936 1679616 10077696 60466176
7 49 343 2401 16807 117649 823543 5764801 40353607 282475249
8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824
9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401
10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000
From top to bottom: x1/8, x1/4, x1/2, x1, x2, x4, x8.

Ifx is a nonnegative real number, andn is a positive integer, x 1 n {\displaystyle x^{\frac {1}{n}}} or x n {\displaystyle {\sqrt[{n}]{x}}} denotes the unique positive real nth root ofx, that is, the unique positive real numbery such that y n = x . {\displaystyle y^{n}=x.}

Ifx is a positive real number, and p q {\displaystyle {\frac {p}{q}}} is a rational number, withp andq ≠ 0 integers, then x p q {\textstyle x^{\frac {p}{q}}} is defined as

x p q = ( x p ) 1 q = ( x 1 q ) p . {\displaystyle x^{\frac {p}{q}}=\left(x^{p}\right)^{\frac {1}{q}}=(x^{\frac {1}{q}})^{p}.}

The equality on the right may be derived by setting y = x 1 q , {\displaystyle y=x^{\frac {1}{q}},} and writing ( x 1 q ) p = y p = ( ( y p ) q ) 1 q = ( ( y q ) p ) 1 q = ( x p ) 1 q . {\displaystyle (x^{\frac {1}{q}})^{p}=y^{p}=\left((y^{p})^{q}\right)^{\frac {1}{q}}=\left((y^{q})^{p}\right)^{\frac {1}{q}}=(x^{p})^{\frac {1}{q}}.}

Ifr is a positive rational number, 0 r = 0 , {\displaystyle 0^{r}=0,} by definition.

All these definitions are required for extending the identity ( x r ) s = x r s {\displaystyle (x^{r})^{s}=x^{rs}} 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 realnth root, which is negative ifn is odd, and no real root ifn is even. In the latter case, whichever complexnth root one chooses for x 1 n , {\displaystyle x^{\frac {1}{n}},} the identity ( x a ) b = x a b {\displaystyle (x^{a})^{b}=x^{ab}} cannot be satisfied. For example,

( ( 1 ) 2 ) 1 2 = 1 1 2 = 1 ( 1 ) 2 1 2 = ( 1 ) 1 = 1. {\displaystyle \left((-1)^{2}\right)^{\frac {1}{2}}=1^{\frac {1}{2}}=1\neq (-1)^{2\cdot {\frac {1}{2}}}=(-1)^{1}=-1.}

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

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

( b r ) s = b r s {\displaystyle \left(b^{r}\right)^{s}=b^{rs}}

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 ofe1/n ise0 = 1 whenn 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 numberb with an arbitrary real exponentx can be defined by continuity with the rule

b x = lim r ( Q ) x b r ( b R + , x R ) , {\displaystyle b^{x}=\lim _{r(\in \mathbb {Q} )\to x}b^{r}\quad (b\in \mathbb {R} ^{+},\,x\in \mathbb {R} ),}

where the limit is taken over rational values ofr only. This limit exists for every positiveb and every realx.

For example, ifx =π, the non-terminating decimal representationπ = 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 b π : {\displaystyle b^{\pi }:}

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

So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted b π . {\displaystyle b^{\pi }.}

This defines b x {\displaystyle b^{x}} for every positiveb and realx as a continuous function ofb andx. See also Well-defined expression.

The exponential function

Main article: Exponential function

The exponential function is often defined as x e x , {\displaystyle x\mapsto e^{x},} where e 2.718 {\displaystyle e\approx 2.718} is Euler's number. For avoiding circular reasoning, this definition cannot be used here. So, a definition of the exponential function, denoted exp ( x ) , {\displaystyle \exp(x),} 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

exp ( x ) = e x . {\displaystyle \exp(x)=e^{x}.}

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

exp ( x ) = lim n ( 1 + x n ) n . {\displaystyle \exp(x)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}.}

One has exp ( 0 ) = 1 , {\displaystyle \exp(0)=1,} and the exponential identity exp ( x + y ) = exp ( x ) exp ( y ) {\displaystyle \exp(x+y)=\exp(x)\exp(y)} holds as well, since

exp ( x ) exp ( y ) = lim n ( 1 + x n ) n ( 1 + y n ) n = lim n ( 1 + x + y n + x y n 2 ) n , {\displaystyle \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},}

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

Euler's number can be defined as e = exp ( 1 ) {\displaystyle e=\exp(1)} . It follows from the preceding equations that exp ( x ) = e x {\displaystyle \exp(x)=e^{x}} whenx is an integer (this results from the repeated-multiplication definition of the exponentiation). Ifx is real, exp ( x ) = e x {\displaystyle \exp(x)=e^{x}} results from the definitions given in preceding sections, by using the exponential identity ifx is rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for every complex value ofx, and therefore it can be used to extend the definition of exp ( z ) {\displaystyle \exp(z)} , and thus e z , {\displaystyle e^{z},} from the real numbers to any complex argumentz. 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 ofex as the exponential function allows definingbx for every positive real numbersb, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithmln(x) is the inverse of the exponential functionex means that one has

b = exp ( ln b ) = e ln b {\displaystyle b=\exp(\ln b)=e^{\ln b}}

for everyb > 0. For preserving the identity ( e x ) y = e x y , {\displaystyle (e^{x})^{y}=e^{xy},} one must have

b x = ( e ln b ) x = e x ln b {\displaystyle b^{x}=\left(e^{\ln b}\right)^{x}=e^{x\ln b}}

So, e x ln b {\displaystyle e^{x\ln b}} can be used as an alternative definition ofbx for any positive realb. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.

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

b z = e ( z ln b ) , {\displaystyle b^{z}=e^{(z\ln b)},}

where ln b {\displaystyle \ln b} denotes the natural logarithm ofb.

This satisfies the identity

b z + t = b z b t , {\displaystyle b^{z+t}=b^{z}b^{t},}

In general, ( b z ) t {\textstyle \left(b^{z}\right)^{t}} is not defined, sincebz 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,

( b z ) t b z t , {\displaystyle \left(b^{z}\right)^{t}\neq b^{zt},}

unlessz is real orw is integer.

Euler's formula,

e i y = cos y + i sin y , {\displaystyle e^{iy}=\cos y+i\sin y,}

allows expressing the polar form of b z {\displaystyle b^{z}} in terms of the real and imaginary parts ofz, namely

b x + i y = b x ( cos ( y ln b ) + i sin ( y ln b ) ) , {\displaystyle b^{x+iy}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)),}

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

b x + i y = b x b i y = b x e i y ln b = b x ( cos ( y ln b ) + i sin ( y ln b ) ) . {\displaystyle b^{x+iy}=b^{x}b^{iy}=b^{x}e^{iy\ln b}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)).}

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 ofnth roots, that is, of exponents 1 / n , {\displaystyle 1/n,} wheren is a positive integer. Although the general theory of exponentiation with non-integer exponents applies tonth 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 numberz may be written in polar form as

z = ρ e i θ = r ( c o s θ + i sin θ ) , {\displaystyle z=\rho e^{i\theta }=r(cos\theta +i\sin \theta ),}

where ρ {\displaystyle \rho } is the absolute value ofz, and θ {\displaystyle \theta } is its argument. The argument is defined up to an integer multiple of2π; this means that, if θ {\displaystyle \theta } is the argument of a complex number, then θ + 2 k π {\displaystyle \theta +2k\pi } 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 annth root of a complex number can be obtained by taking thenth root of the absolute value and dividing its argument byn:

( ρ e i θ ) 1 n = ρ n e i θ n . {\displaystyle \left(\rho e^{i\theta }\right)^{\frac {1}{n}}={\sqrt[{n}]{\rho }}\,e^{\frac {i\theta }{n}}.}

If 2 i π {\displaystyle 2i\pi } is added to θ , {\displaystyle \theta ,} the complex number in not changed, but this adds 2 i π / n {\displaystyle 2i\pi /n} to the argument of thenth root, and provides a newnth root. This can be donen times, and provides thennth roots of the complex number.

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

Roots of unity

Main article: Root of unity
The three third roots of 1

Thenth roots of unity are then complex numbers such thatwn = 1, wheren 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).

Thennth roots of unity are then first powers of ω = e 2 π i n {\displaystyle \omega =e^{\frac {2\pi i}{n}}} , that is 1 = ω 0 = ω n , ω = ω 1 , ω 2 , ω n 1 . {\displaystyle 1=\omega ^{0}=\omega ^{n},\omega =\omega ^{1},\omega ^{2},\omega ^{n-1}.} Thenth roots of unity that have this generating property are called primitiventh roots of unity; they have the form ω k = e 2 k π i n , {\displaystyle \omega ^{k}=e^{\frac {2k\pi i}{n}},} withk coprime withn. The unique primitive square root of unity is 1 ; {\displaystyle -1;} the primitive fourth roots of unity are i {\displaystyle i} and i . {\displaystyle -i.}

Thenth roots of unity allow expressing allnth roots of a complex numberz as then products of a givennth roots ofz with anth root of unity.

Geometrically, thenth roots of unity lie on the unit circle of the complex plane at the vertices of a regularn-gon with one vertex on the real number 1.

As the number e 2 k π i n {\displaystyle e^{\frac {2k\pi i}{n}}} is the primitiventh root of unity with the smallest positive argument, it is called the principal primitiventh root of unity, sometimes shortened as principalnth root of unity, although this terminology can be confused with the principal value of 1 1 / n {\displaystyle 1^{1/n}} 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 z w z^{w} . So, either a principal value is defined, which is not continuous for the values ofz that are real and nonpositive, or z w z^{w} is defined as a multivalued function.

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

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

where log z {\displaystyle \log z} is the variant of the complex logarithm that is used, which is, a function or a multivalued function such that

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

for everyz in its domain of definition.

Principal value

The principal value of the complex logarithm is the unique function, commonly denoted log , {\displaystyle \log ,} such that, for every nonzero complex numberz,

e log z = z , {\displaystyle e^{\log z}=z,}

and the imaginary part ofz satisfies

π < I m π . {\displaystyle -\pi <\mathrm {Im} \leq \pi .}

The principal value of the complex logarithm is not defined for z = 0 , {\displaystyle z=0,} it is discontinuous at negative real values ofz, and it is holomorphic (that is, complex differentiable) elsewhere. Ifz is real and positive, the principal value of the complex logarithm is the natural logarithm: log z = ln z . {\displaystyle \log z=\ln z.}

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

The function ( z , w ) z w {\displaystyle (z,w)\to z^{w}} is holomorphic except in the neibourhood of the points wherez is real and nonpositive.

Ifz is real and positive, the principal value of z w {\displaystyle z^{w}} equals its usual value defined above. If w = 1 / n , {\displaystyle w=1/n,} wheren 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 log z {\displaystyle \log z} and z w {\displaystyle z^{w}} at the negative real values ofz. In this case, it is useful to consider these functions as multivalued functions.

If log z {\displaystyle \log z} denotes one of the values of the multivalued logarithm (typically its principal value), the other values are 2 i k π + log z , {\displaystyle 2ik\pi +\log z,} wherek is any integer. Similarly, if z w {\displaystyle z^{w}} is one value of the exponentiation, then the other values are given by

e w ( 2 i k π + log z ) = z w e 2 i k π w , {\displaystyle e^{w(2ik\pi +\log z)}=z^{w}e^{2ik\pi w},}

wherek is any integer.

Different values ofk give different values of z w {\displaystyle z^{w}} unlessw is a rational number, that is, there is an integerd such thatdw is an integer. This results from the periodicity of the exponential function, more specifically, that e a = e b {\displaystyle e^{a}=e^{b}} if and only if a b {\displaystyle a-b} is an integer multiple of 2 π i . {\displaystyle 2\pi i.}

If w = m n {\displaystyle w={\frac {m}{n}}} is a rational number withm andn coprime integers with n > 0 , {\displaystyle n>0,} then z w {\displaystyle z^{w}} has exactlyn values. In the case m = 1 , {\displaystyle m=1,} these values are the same as those described in §nth roots of a complex number. Ifw is an integer, there is only one value that agrees with that of § Integer exponents.

The multivalued exponentiation is holomorphic for z 0 , {\displaystyle z\neq 0,} in the sense that its graph consists of several sheets that define each a holomorphic function in the neighborhood of every point. Ifz varies continuously along a circle around0, then, after a turn, the value of z w {\displaystyle z^{w}} has changed of sheet.

Computation

The canonical form x + i y {\displaystyle x+iy} of z w {\displaystyle z^{w}} can be computed from the canonical form ofz andw. Although this can be described by a single formula, it is clearer to split the computation in several steps.

  • Polar form ofz. If z = a + i b {\displaystyle z=a+ib} is the canonical form ofz (a andb being real), then its polar form is
    z = ρ e i θ = ρ ( cos θ + i sin θ ) , {\displaystyle z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),}
    where ρ = a 2 + b 2 {\displaystyle \rho ={\sqrt {a^{2}+b^{2}}}} and θ = atan2 ( a , b ) {\displaystyle \theta =\operatorname {atan2} (a,b)} (see atan2 for the definition of this function).
  • Logarithm ofz. The principal value of this logarithm is log z = ln ρ + i θ , {\displaystyle \log z=\ln \rho +i\theta ,} where ln {\displaystyle \ln } denotes the natural logarithm. The other values of the logarithm are obtained by adding 2 i k π {\displaystyle 2ik\pi } for any integerk.
  • Canonical form of w log z . {\displaystyle w\log z.} If w = c + d i {\displaystyle w=c+di} withc andd real, the values of w log z {\displaystyle w\log z} are
    w log z = ( c ln ρ d θ 2 d k π ) + i ( d ln ρ + c θ + 2 c k π ) , {\displaystyle w\log z=(c\ln \rho -d\theta -2dk\pi )+i(d\ln \rho +c\theta +2ck\pi ),}
    the principal value corresponding to k = 0. {\displaystyle k=0.}
  • Final result. Using the identities e x + y e x = e y {\displaystyle e^{x+y}e^{x}=e^{y}} and e y ln x = x y , {\displaystyle e^{y\ln x}=x^{y},} one gets
    z w = ρ c e d ( θ + 2 k π ) ( cos ( d ln ρ + c θ + 2 c k π ) + i sin ( d ln ρ + c θ + 2 c k π ) ) , {\displaystyle 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),}
    with k = 0 {\displaystyle k=0} for the principal value.
Examples
  • i i {\displaystyle i^{i}}
    The polar form ofi is i = e i π / 2 , {\displaystyle i=e^{i\pi /2},} and the values of log i {\displaystyle \log i} are thus
    log i = i ( π 2 + 2 k π ) . {\displaystyle \log i=i\left({\frac {\pi }{2}}+2k\pi \right).}
    It follows that
    i i = e i log i = e π 2 e 2 k π . {\displaystyle i^{i}=e^{i\log i}=e^{-{\frac {\pi }{2}}}e^{-2k\pi }.}
    So, all values of i i {\displaystyle i^{i}} are real, the principal one being
    e π 2 0.2079. {\displaystyle e^{-{\frac {\pi }{2}}}\approx 0.2079.}
  • ( 2 ) 3 + 4 i {\displaystyle (-2)^{3+4i}}
    Similarly, the polar form of−2 is 2 = 2 e i π . {\displaystyle -2=2e^{i\pi }.} So, the above described method gives the values
    ( 2 ) 3 + 4 i = 2 3 e 4 ( π + 2 k π ) ( cos ( 4 ln 2 + 3 ( π + 2 k π ) ) + i sin ( 4 ln 2 + 3 ( π + 2 k π ) ) ) = 2 3 e 4 ( π + 2 k π ) ( cos ( 4 ln 2 ) + i sin ( 4 ln 2 ) ) . {\displaystyle {\begin{aligned}(-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{aligned}}}
    In this case, all the values have the same argument 4 ln 2 , {\displaystyle 4\ln 2,} and different absolute values.

In both examples, all values of z w {\displaystyle z^{w}} have the same argument. More generally, this is true if and only if the real part ofw 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 identitylog(bx) = x ⋅ log b holds wheneverb is a positive real number andx is a real number. But for the principal branch of the complex logarithm one has
    log ( ( i ) 2 ) = log ( 1 ) = i π 2 log ( i ) = 2 log ( e i π / 2 ) = 2 i π 2 = i π {\displaystyle \log((-i)^{2})=\log(-1)=i\pi \neq 2\log(-i)=2\log(e^{-i\pi /2})=2\,{\frac {-i\pi }{2}}=-i\pi }

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

    log w z z log w ( mod 2 π i ) {\displaystyle \log w^{z}\equiv z\log w{\pmod {2\pi i}}}

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

    { log w z } = { z Log w + z 2 π i n + 2 π i m m , n Z } { z log w } = { z Log w + z 2 π i n n Z } {\displaystyle {\begin{aligned}\left\{\log w^{z}\right\}&=\left\{z\cdot \operatorname {Log} w+z\cdot 2\pi in+2\pi im\mid m,n\in \mathbb {Z} \right\}\\\left\{z\log w\right\}&=\left\{z\operatorname {Log} w+z\cdot 2\pi in\mid n\in \mathbb {Z} \right\}\end{aligned}}}
  • The identities(bc)x = bxcx and(b/c)x = bx/cx are valid whenb andc are positive real numbers andx is a real number. But, for the principal values, one has
    ( 1 1 ) 1 2 = 1 ( 1 ) 1 2 ( 1 ) 1 2 = 1 {\displaystyle (-1\cdot -1)^{\frac {1}{2}}=1\not =(-1)^{\frac {1}{2}}(-1)^{\frac {1}{2}}=-1}

    and

    ( 1 1 ) 1 2 = ( 1 ) 1 2 = i 1 1 2 ( 1 ) 1 2 = 1 i = i {\displaystyle \left({\frac {1}{-1}}\right)^{\frac {1}{2}}=(-1)^{\frac {1}{2}}=i\not ={\frac {1^{\frac {1}{2}}}{(-1)^{\frac {1}{2}}}}={\frac {1}{i}}=-i}

    On the other hand, whenx 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 numbersx andy, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen: For any integern, we have:
    1. e 1 + 2 π i n = e 1 e 2 π i n = e 1 = e {\displaystyle e^{1+2\pi in}=e^{1}e^{2\pi in}=e\cdot 1=e}
    2. ( e 1 + 2 π i n ) 1 + 2 π i n = e {\displaystyle \left(e^{1+2\pi in}\right)^{1+2\pi in}=e\qquad } (taking the ( 1 + 2 π i n ) {\displaystyle (1+2\pi in)} -th power of both sides)
    3. e 1 + 4 π i n 4 π 2 n 2 = e {\displaystyle e^{1+4\pi in-4\pi ^{2}n^{2}}=e\qquad } (using ( e x ) y = e x y {\displaystyle \left(e^{x}\right)^{y}=e^{xy}} and expanding the exponent)
    4. e 1 e 4 π i n e 4 π 2 n 2 = e {\displaystyle e^{1}e^{4\pi in}e^{-4\pi ^{2}n^{2}}=e\qquad } (using e x + y = e x e y {\displaystyle e^{x+y}=e^{x}e^{y}} )
    5. e 4 π 2 n 2 = 1 {\displaystyle e^{-4\pi ^{2}n^{2}}=1\qquad } (dividing bye)
    but this is false when the integern is nonzero. The error is the following: by definition, e y {\displaystyle e^{y}} is a notation for exp ( y ) , {\displaystyle \exp(y),} a true function, and x y {\displaystyle x^{y}} is a notation for exp ( y log x ) , {\displaystyle \exp(y\log x),} which is a multi-valued function. Thus the notation is ambiguous whenx = e. Here, before expanding the exponent, the second line should be
    exp ( ( 1 + 2 π i n ) log exp ( 1 + 2 π i n ) ) = exp ( 1 + 2 π i n ) . {\displaystyle \exp \left((1+2\pi in)\log \exp(1+2\pi in)\right)=\exp(1+2\pi in).}
    Therefore, when expanding the exponent, one has implicitly supposed that log exp z = z {\displaystyle \log \exp z=z} for complex values ofz, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity(ex)y = exy must be replaced by the identity
    ( e x ) y = e y log e x , {\displaystyle \left(e^{x}\right)^{y}=e^{y\log e^{x}},}
    which is a true identity between multivalued functions.

Ifb is a positive real algebraic number, andx is a rational number, thenbx is an algebraic number. This results from the theory of algebraic extensions. This remains true ifb is any algebraic number, in which case, all values ofbx (as a multivalued function) are algebraic. Ifx is irrational (that is, not rational), and bothb andx are algebraic, Gelfond–Schneider theorem asserts that all values ofbx are transcendental (that is, not algebraic), except ifb equals0 or1.

In other words, ifx is irrational and b { 0 , 1 } , {\displaystyle b\not \in \{0,1\},} then at least one ofb,x andbx is transcendental.

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication. The definition of x 0 {\displaystyle x^{0}} 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 1 is a monoid. In such a monoid, exponentiation of an elementx is defined inductively by

  • x 0 = 1 , {\displaystyle x^{0}=1,}
  • x n + 1 = x x n {\displaystyle x^{n+1}=xx^{n}} for every nonnegative integern.

Ifn is a negative integer, x n {\displaystyle x^{n}} is defined only ifx has a multiplicative inverse. In this case, the inverse ofx is denoted x 1 , {\displaystyle x^{-1},} and x n {\displaystyle x^{n}} is defined as ( x 1 ) n . {\displaystyle \left(x^{-1}\right)^{-n}.}

Exponentiation with integer exponents obeys the following laws, forx andy in the algebraic structure, andm andn integers:

x 0 = 1 x m + n = x m x n ( x m ) n = x m n ( x y ) n = x n y n if x y = y x , and, in particular, if the multiplication is commutative. {\displaystyle {\begin{aligned}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{aligned}}}

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, iff is a real function whose valued can be multiplied, f n {\displaystyle f^{n}} denotes the exponentiation with respect of multiplication, and f n {\displaystyle f^{\circ n}} may denote exponentiation with respect of function composition. That is,

( f n ) ( x ) = ( f ( x ) ) n = f ( x ) f ( x ) f ( x ) , {\displaystyle (f^{n})(x)=(f(x))^{n}=f(x)\,f(x)\cdots f(x),}

and

( f n ) ( x ) = f ( f ( f ( f ( x ) ) ) ) . {\displaystyle (f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots )).}

Commonly, ( f n ) ( x ) {\displaystyle (f^{n})(x)} is denoted f ( x ) n , {\displaystyle f(x)^{n},} while ( f n ) ( x ) {\displaystyle (f^{\circ n})(x)} is denoted f n ( x ) . {\displaystyle f^{n}(x).}

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, ifG is a group, x n {\displaystyle x^{n}} is defined for every x G {\displaystyle x\in G} and every integern.

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 elementx is the cyclic group generated byx. If all the powers ofx are distinct, the group is isomorphic to the additive group Z {\displaystyle \mathbb {Z} } of the integers. Otherwise, the cyclic group is finite (it has a finite number of elements), and its number of elements is the order ofx. If the order ofx isn, then x n = x 0 = 1 , {\displaystyle x^{n}=x^{0}=1,} and the cyclic group generated byx consists of then first powers ofx (starting indifferently from the exponent0 or1).

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 ( g h ) k = g h k {\displaystyle (g^{h})^{k}=g^{hk}} and ( g h ) k = g k h k . {\displaystyle (gh)^{k}=g^{k}h^{k}.}

In a ring

In a ring, it may occurs that some nonzero elements satisfy x n = 0 {\displaystyle x^{n}=0} for some integern. 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 x 0 {\displaystyle x\neq 0} implies x n 0 {\displaystyle x^{n}\neq 0} for every positive integern), 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 idealI in a commutative ringR, the set of the elements ofR that have a power inI is an ideal, called the radical ofI. The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring k [ x 1 , , x n ] {\displaystyle k[x_{1},\ldots ,x_{n}]} over a fieldk, 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 A 0 {\displaystyle A^{0}} is defined to be the identity matrix, and if A is invertible, then A n = ( A 1 ) n {\displaystyle A^{-n}=\left(A^{-1}\right)^{n}} .

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. This is the standard interpretation of a Markov chain, for example. Then A 2 x {\displaystyle A^{2}x} is the state of the system after two time steps, and so forth: A n x {\displaystyle A^{n}x} is the state of the system after n time steps. The matrix power A n {\displaystyle A^{n}} 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, d / d x {\displaystyle d/dx} , which is a linear operator acting on functions f ( x ) {\displaystyle f(x)} to give a new function ( d / d x ) f ( x ) = f ( x ) {\displaystyle (d/dx)f(x)=f'(x)} . The n-th power of the differentiation operator is the n-th derivative:

( d d x ) n f ( x ) = d n d x n f ( x ) = f ( n ) ( x ) . {\displaystyle \left({\frac {d}{dx}}\right)^{n}f(x)={\frac {d^{n}}{dx^{n}}}f(x)=f^{(n)}(x).}

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

Main article: Finite field

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 of0. 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 q = p k , {\displaystyle q=p^{k},} wherep is a prime number, andk is a positive integer. For every suchq, there are fields withq elements. The fields withq elements are all isomorphic, which allows, in general, working as if there were only one field withq elements, denoted F q . {\displaystyle \mathbb {F} _{q}.}

One has

x q = x {\displaystyle x^{q}=x}

for every x F q . {\displaystyle x\in \mathbb {F} _{q}.}

A primitive element in F q {\displaystyle \mathbb {F} _{q}} is an elementg such the set of theq − 1 first powers ofg (that is, { g 1 = g , g 2 , , g p 1 = g 0 = 1 } {\displaystyle \{g^{1}=g,g^{2},\ldots ,g^{p-1}=g^{0}=1\}} ) equals the set of the nonzero elements of F q . {\displaystyle \mathbb {F} _{q}.} There are φ ( p 1 ) {\displaystyle \varphi (p-1)} primitive elements in F q , {\displaystyle \mathbb {F} _{q},} where φ {\displaystyle \varphi } is Euler's totient function.

In F q , {\displaystyle \mathbb {F} _{q},} the Freshman's dream identity

( x + y ) p = x p + y p {\displaystyle (x+y)^{p}=x^{p}+y^{p}}

is true for the exponentp. As x p = x {\displaystyle x^{p}=x} in F q , {\displaystyle \mathbb {F} _{q},} It follows that the map

F : F q F q x x p {\displaystyle {\begin{aligned}F\colon {}&\mathbb {F} _{q}\to \mathbb {F} _{q}\\&x\mapsto x^{p}\end{aligned}}}

is linear over F q , {\displaystyle \mathbb {F} _{q},} and is a field automorphism, called the Frobenius automorphism. If q = p k , {\displaystyle q=p^{k},} the field F q {\displaystyle \mathbb {F} _{q}} hask automorphisms, which are thek first powers (under composition) ofF. In other words, the Galois group of F q {\displaystyle \mathbb {F} _{q}} is cyclic of orderk, 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, ifg is a primitive element in F q , {\displaystyle \mathbb {F} _{q},} then g e {\displaystyle g^{e}} can be efficiently computed with exponentiation by squaring for anye, even ifq is large, while there is no known algorithm allowing retrievinge from g e {\displaystyle g^{e}} ifq is sufficiently large.

The Cartesian product of two setsS andT is the set of the ordered pairs ( x , y ) {\displaystyle (x,y)} such that x S {\displaystyle x\in S} and y T . {\displaystyle y\in T.} This operation is not properly commutative nor associative, but has these properties up to canonical isomorphisms, that allow identifying, for example, ( x , ( y , z ) ) , {\displaystyle (x,(y,z)),} ( ( x , y ) , z ) , {\displaystyle ((x,y),z),} and ( x , y , z ) . {\displaystyle (x,y,z).}

This allows defining thenth power S n {\displaystyle S^{n}} of a setS as the set of alln-tuples ( x 1 , , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} of elements ofS.

WhenS is endowed with some structure, it is frequent that S n {\displaystyle S^{n}} 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 R n {\displaystyle \mathbb {R} ^{n}} (where R {\displaystyle \mathbb {R} } denotes the real numbers) denotes the Cartesian product ofn copies of R , {\displaystyle \mathbb {R} ,} as well as their direct product as vector space, topological spaces, rings, etc.

Sets as exponents

An-tuple ( x 1 , , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} of elements ofS can be considered as a function from { 1 , , n } . {\displaystyle \{1,\ldots ,n\}.} This generalizes to the following notation.

Given two setsS andT, the set of all functions fromT toS is denoted S T {\displaystyle S^{T}} This exponential notation is justified by the following canonical isomorphisms (for the first one, see Currying):

( S T ) U S T × U , {\displaystyle (S^{T})^{U}\cong S^{T\times U},}
S T U S T × S U , {\displaystyle S^{T\sqcup U}\cong S^{T}\times S^{U},}

where × {\displaystyle \times } denotes the Cartesian product, and {\displaystyle \sqcup } 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, R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} denotes the vector space of the infinite sequences of real numbers, and R ( N ) {\displaystyle \mathbb {R} ^{(\mathbb {N} )}} 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 equals1, 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 { 0 , 1 } . {\displaystyle \{0,1\}.} So, 2 S {\displaystyle 2^{S}} denotes the power set ofS, that is the set of the functions fromS to { 0 , 1 } , {\displaystyle \{0,1\},} which can be identified with the set of the subsets ofS, by mapping each function to the inverse image of1.

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

In category theory

In the category of sets, the morphisms between setsX andY are the functions fromX toY. It results that the set of the functions fromX toY that is denoted Y X {\displaystyle Y^{X}} in the preceding section can also be denoted hom ( X , Y ) . {\displaystyle \hom(X,Y).} The isomorphism ( S T ) U S T × U {\displaystyle (S^{T})^{U}\cong S^{T\times U}} can be rewritten

hom ( U , S T ) hom ( T × U , S ) . {\displaystyle \hom(U,S^{T})\cong \hom(T\times U,S).}

This means the functor "exponentiation to the powerT" is a right adjoint to the functor "direct product withT".

This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor X X T {\displaystyle X\to X^{T}} is, if it exists, a right adjoint to the functor Y T × Y . {\displaystyle Y\to T\times Y.} A category is called a Cartesian closed category, if direct products exist, and the functor Y X × Y {\displaystyle Y\to X\times Y} has a right adjoint for everyT.

Main articles: Tetration and Hyperoperation

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, and7625597484987 (= 327 = 333 = 33) respectively.

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 functionxy has no limit at the point(0, 0). One may consider at what points this function does have a limit.

More precisely, consider the functionf(x, y) = xy defined onD = {(x, y) ∈ R2 : x > 0}. ThenD can be viewed as a subset ofR2 (that is, the set of all pairs(x, y) withx,y belonging to the extended real number lineR = [−∞, +∞], endowed with the product topology), which will contain the points at which the functionf has a limit.

In fact,f has a limit at all accumulation points ofD, except for(0, 0),(+∞, 0),(1, +∞) and(1, −∞). Accordingly, this allows one to define the powersxy by continuity whenever0 ≤ x ≤ +∞,−∞ ≤ y ≤ +∞, except for 00, (+∞)0, 1+∞ and 1−∞, which remain indeterminate forms.

Under this definition by continuity, we obtain:

  • x+∞ = +∞ andx−∞ = 0, when1 < x ≤ +∞.
  • x+∞ = 0 andx−∞ = +∞, when0 ≤ x < 1.
  • 0y = 0 and(+∞)y = +∞, when0 < y ≤ +∞.
  • 0y = +∞ and(+∞)y = 0, when−∞ ≤ y < 0.

These powers are obtained by taking limits ofxy for positive values ofx. This method does not permit a definition ofxy whenx < 0, since pairs(x, y) withx < 0 are not accumulation points ofD.

On the other hand, whenn is an integer, the powerxn is already meaningful for all values ofx, including negative ones. This may make the definition0n = +∞ obtained above for negativen problematic whenn is odd, since in this casexn → +∞ asx tends to0 through positive values, but not negative ones.

Computing bn using iterated multiplication requiresn − 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:

100 = 2 2 + 2 5 + 2 6 = 2 2 ( 1 + 2 3 ( 1 + 2 ) ) {\displaystyle 100=2^{2}+2^{5}+2^{6}=2^{2}(1+2^{3}(1+2))} .

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 = 250 =1125899906842624
(250)2 = 2100 =1267650600228229401496703205376

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

In general, the number of multiplication operations required to computebn can be reduced to n + log 2 n 1 , {\displaystyle \sharp n+\lfloor \log _{2}n\rfloor -1,} by using exponentiation by squaring, where n {\displaystyle \sharp n} denotes the number of1 in the binary representation ofn. 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) forbn 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.

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 g f , {\displaystyle g\circ f,} and defined as

( g f ) ( x ) = g ( f ( x ) )<

Exponentiation
Exponentiation Language Watch Edit 160 160 Redirected from Zeroth power Exponent redirects here For other uses see Exponent disambiguation Exponentiation is a mathematical operation written as bn involving two numbers the base b and the exponent or power n and pronounced as b raised to the power of n 1 When n is a positive integer exponentiation corresponds to repeated multiplication of the base that is bn is the product of multiplying n bases 1 Graphs of y bx 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 b n b b n times displaystyle b n underbrace b times dots times b n text times 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 the nth power of b b to the nth power 2 or most briefly as b to the nth One has b1 b and for any positive integers m and n one has bn bm bn m To extend this property to non positive integer exponents b0 is defined to be 1 and b n 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 Contents 1 History of the notation 2 Terminology 3 Integer exponents 3 1 Positive exponents 3 2 Zero exponent 3 3 Negative exponents 3 4 Identities and properties 3 5 Powers of a sum 3 6 Combinatorial interpretation 3 7 Particular bases 3 7 1 Powers of ten 3 7 2 Powers of two 3 7 3 Powers of one 3 7 4 Powers of zero 3 7 5 Powers of negative one 3 8 Large exponents 3 9 Power functions 3 10 Table of powers of decimal digits 4 Rational exponents 5 Real exponents 5 1 Limits of rational exponents 5 2 The exponential function 5 3 Powers via logarithms 6 Complex exponents with a positive real base 7 Non integer powers of complex numbers 7 1 n th roots of a complex number 7 1 1 Roots of unity 7 2 Complex exponentiation 7 2 1 Principal value 7 2 2 Multivalued function 7 2 3 Computation 7 2 3 1 Examples 7 2 4 Failure of power and logarithm identities 8 Irrationality and transcendence 9 Integer powers in algebra 9 1 In a group 9 2 In a ring 9 3 Matrices and linear operators 9 4 Finite fields 10 Powers of sets 10 1 Sets as exponents 10 2 In category theory 11 Repeated exponentiation 12 Limits of powers 13 Efficient computation with integer exponents 14 Iterated functions 15 In programming languages 16 See also 17 Notes 18 ReferencesHistory of the notation EditThe term power Latin potentia potestas dignitas is a mistranslation 3 4 of the ancient Greek dynamis dunamis here amplification 3 used by the Greek mathematician Euclid for the square of a line 5 following Hippocrates of Chios 6 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 Musa al Khwarizmi used the terms م ال mal 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 7 and ك ع ب ة kaʿbah cube for a cube which later Islamic mathematicians represented in mathematical notation as the letters mim m and kaf k respectively by the 15th century as seen in the work of Abu al Hasan ibn Ali al Qalasadi 8 In the late 16th century Jost Burgi used Roman numerals for exponents 9 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 10 11 Samuel Jeake introduced the term indices in 1696 5 In the 16th century Robert Recorde used the terms square cube zenzizenzic fourth power sursolid fifth zenzicube sixth second sursolid seventh and zenzizenzizenzic eighth 7 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 Rene Descartes in his text titled La Geometrie there the notation is introduced in Book I 12 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 13 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 14 Terminology EditThe expression b2 b b 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 b b b 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 EditThe exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations Positive exponents Edit The definition of the exponentiation as an iterated multiplication can be formalized by using induction 15 and this definition can be used as soon one has an associative multiplication The base case is b 1 b displaystyle b 1 b and the recurrence is b n 1 b n b displaystyle b n 1 b n cdot b The associativity of multiplication implies that for any positive integers m and n b m n b m b n displaystyle b m n b m cdot b n and b m n b m n displaystyle b m n b mn Zero exponent Edit By definition any nonzero number raised to the 0 power is 1 16 1 b 0 1 displaystyle b 0 1 This definition is the only possible that allows extending the formula b m n b m b n displaystyle b m n b m cdot b n to zero exponents It may be used in every algebraic structure with a multiplication that has an identity Intuitionally b 0 displaystyle b 0 may be interpreted as the empty product of copies of b So the equality b 0 1 displaystyle b 0 1 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 1 is generally assigned to 0 0 displaystyle 0 0 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 Edit Exponentiation with negative exponents is defined by the following identity which holds for any integer n and nonzero b b n 1 b n displaystyle b n frac 1 b n 1 Raising 0 to a negative exponent is undefined but in some circumstances it may be interpreted as infinity displaystyle infty This definition of exponentiation with negative exponents is the only one that allows extending the identity b m n b m b n displaystyle b m n b m cdot b n to negative exponents consider the case m n displaystyle m n 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 x 1 displaystyle x 1 Identities and properties Edit Laws of Indices redirects here For the horse see Laws of Indices horse The following identities often called exponent rules hold for all integer exponents provided that the base is non zero 1 b m n b m b n b m n b m n b c n b n c n displaystyle begin aligned b m n amp b m cdot b n left b m right n amp b m cdot n b cdot c n amp b n cdot c n end aligned 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 17 18 19 20 or left associative That is b p q b p q displaystyle b p q b left p q right which in general is different from b p q b p q displaystyle left b p right q b pq Powers of a sum Edit The powers of a sum can normally be computed from the powers of the summands by the binomial formula a b n i 0 n n i a i b n i i 0 n n i n i a i b n i displaystyle a b n sum i 0 n binom n i a i b n i sum i 0 n frac n i n i a i b n i 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 Edit See also Exponentiation over sets 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 nm The nm possible m tuples of elements from the set 1 n 05 0 none14 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 Edit Powers of ten Edit See also Scientific notation Main article Power of 10 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 299792 458 m s the speed of light in vacuum in metres per second can be written as 2 997924 58 108 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 Edit Main article Power 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 Edit The powers of one are all one 1n 1 The first power of a number is the number itself n 1 n displaystyle n 1 n Powers of zero Edit If the exponent n is positive n gt 0 the n th power of zero is zero 0n 0 If the exponent n is negative n lt 0 the n th power of zero 0n is undefined because it must equal 1 0 n displaystyle 1 0 n with n gt 0 and this would be 1 0 displaystyle 1 0 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 Edit 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 Edit 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 gt 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 lt 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 lt 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 n e 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 Edit Power functions for n 1 3 5 displaystyle n 1 3 5 Power functions for n 2 4 6 displaystyle n 2 4 6 Real functions of the form f x c x n displaystyle f x cx n where c 0 displaystyle c neq 0 are sometimes called power functions citation needed When n displaystyle n is an integer and n 1 displaystyle n geq 1 two primary families exist for n displaystyle n even and for n displaystyle n odd In general for c gt 0 displaystyle c gt 0 when n displaystyle n is even f x c x n displaystyle f x cx n will tend towards positive infinity with increasing x displaystyle x and also towards positive infinity with decreasing x displaystyle x All graphs from the family of even power functions have the general shape of y c x 2 displaystyle y cx 2 flattening more in the middle as n displaystyle n increases 21 Functions with this kind of symmetry f x f x displaystyle f x f x are called even functions When n displaystyle n is odd f x displaystyle f x s asymptotic behavior reverses from positive x displaystyle x to negative x displaystyle x For c gt 0 displaystyle c gt 0 f x c x n displaystyle f x cx n will also tend towards positive infinity with increasing x displaystyle x but towards negative infinity with decreasing x displaystyle x All graphs from the family of odd power functions have the general shape of y c x 3 displaystyle y cx 3 flattening more in the middle as n displaystyle n increases and losing all flatness there in the straight line for n 1 displaystyle n 1 Functions with this kind of symmetry f x f x displaystyle f x f x are called odd functions For c lt 0 displaystyle c lt 0 the opposite asymptotic behavior is true in each case 21 Table of powers of decimal digits Edit n n2 n3 n4 n5 n6 n7 n8 n9 n101 1 1 1 1 1 1 1 1 12 4 8 16 32 64 128 256 512 10243 9 27 81 243 729 2187 6561 19683 590494 16 64 256 1024 4096 16384 65536 262144 1048 5765 25 125 625 3125 15625 78125 390625 1953 125 9765 6256 36 216 1296 7776 46656 279936 1679 616 10077 696 60466 1767 49 343 2401 16807 117649 823543 5764 801 40353 607 282475 2498 64 512 4096 32768 262144 2097 152 16777 216 134217 728 1073 741 8249 81 729 6561 59049 531441 4782 969 43046 721 387420 489 3486 784 40110 100 1000 10000 100000 1000 000 10000 000 100000 000 1000 000 000 10000 000 000Rational exponents Edit From top to bottom x1 8 x1 4 x1 2 x1 x2 x4 x8 If x is a nonnegative real number and n is a positive integer x 1 n displaystyle x frac 1 n or x n displaystyle sqrt n x denotes the unique positive real n th root of x that is the unique positive real number y such that y n x displaystyle y n x If x is a positive real number and p q displaystyle frac p q is a rational number with p and q 0 integers then x p q textstyle x frac p q is defined as x p q x p 1 q x 1 q p displaystyle x frac p q left x p right frac 1 q x frac 1 q p The equality on the right may be derived by setting y x 1 q displaystyle y x frac 1 q and writing x 1 q p y p y p q 1 q y q p 1 q x p 1 q displaystyle x frac 1 q p y p left y p q right frac 1 q left y q p right frac 1 q x p frac 1 q If r is a positive rational number 0 r 0 displaystyle 0 r 0 by definition All these definitions are required for extending the identity x r s x r s displaystyle x r s x rs 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 n th root which is negative if n is odd and no real root if n is even In the latter case whichever complex n th root one chooses for x 1 n displaystyle x frac 1 n the identity x a b x a b displaystyle x a b x ab cannot be satisfied For example 1 2 1 2 1 1 2 1 1 2 1 2 1 1 1 displaystyle left 1 2 right frac 1 2 1 frac 1 2 1 neq 1 2 cdot frac 1 2 1 1 1 See Real exponents and Powers of complex numbers for details on the way these problems may be handled Real exponents EditFor 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 b r s b r s displaystyle left b r right s b rs 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 Edit The limit of e1 n is e0 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 22 b x lim r Q x b r b R x R displaystyle b x lim r in mathbb Q to x b r quad b in mathbb R x in mathbb R 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 p the non terminating decimal representation p 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 b p displaystyle b pi b 3 b 4 b 3 1 b 3 2 b 3 14 b 3 15 b 3 141 b 3 142 b 3 1415 b 3 1416 b 3 14159 b 3 14160 displaystyle 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 So the upper bounds and the lower bounds of the intervals form two sequences that have the same limit denoted b p displaystyle b pi This defines b x displaystyle b x for every positive b and real x as a continuous function of b and x See also Well defined expression The exponential function Edit Main article Exponential function The exponential function is often defined as x e x displaystyle x mapsto e x where e 2 718 displaystyle e approx 2 718 is Euler s number For avoiding circular reasoning this definition cannot be used here So a definition of the exponential function denoted exp x displaystyle exp x 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 exp x e x displaystyle exp x e x There are many equivalent ways to define the exponential function one of them being exp x lim n 1 x n n displaystyle exp x lim n rightarrow infty left 1 frac x n right n One has exp 0 1 displaystyle exp 0 1 and the exponential identity exp x y exp x exp y displaystyle exp x y exp x exp y holds as well since exp x exp y lim n 1 x n n 1 y n n lim n 1 x y n x y n 2 n displaystyle 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 and the second order term x y n 2 displaystyle frac xy n 2 does not affect the limit yielding exp x exp y exp x y displaystyle exp x exp y exp x y Euler s number can be defined as e exp 1 displaystyle e exp 1 It follows from the preceding equations that exp x e x displaystyle exp x e x when x is an integer this results from the repeated multiplication definition of the exponentiation If x is real exp x e x displaystyle exp x e x 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 limit that defines the exponential function converges for every complex value of x and therefore it can be used to extend the definition of exp z displaystyle exp z and thus e z displaystyle e z from the real numbers to 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 Edit 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 b exp ln b e ln b displaystyle b exp ln b e ln b for every b gt 0 For preserving the identity e x y e x y displaystyle e x y e xy one must have b x e ln b x e x ln b displaystyle b x left e ln b right x e x ln b So e x ln b displaystyle e x ln b 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 EditIf 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 b z e z ln b displaystyle b z e z ln b where ln b displaystyle ln b denotes the natural logarithm of b This satisfies the identity b z t b z b t displaystyle b z t b z b t In general b z t textstyle left b z right t 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 b z t b z t displaystyle left b z right t neq b zt unless z is real or w is integer Euler s formula e i y cos y i sin y displaystyle e iy cos y i sin y allows expressing the polar form of b z displaystyle b z in terms of the real and imaginary parts of z namely b x i y b x cos y ln b i sin y ln b displaystyle b x iy b x cos y ln b i sin y ln b where the absolute value of the trigonometric factor is one This results from b x i y b x b i y b x e i y ln b b x cos y ln b i sin y ln b displaystyle b x iy b x b iy b x e iy ln b b x cos y ln b i sin y ln b Non integer powers of complex numbers EditIn 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 n th roots that is of exponents 1 n displaystyle 1 n where n is a positive integer Although the general theory of exponentiation with non integer exponents applies to n th roots this case deserves to be considered first since it does not need to use complex logarithms and is therefore easier to understand n th roots of a complex number Edit Every nonzero complex number z may be written in polar form as z r e i 8 r c o s 8 i sin 8 displaystyle z rho e i theta r cos theta i sin theta where r displaystyle rho is the absolute value of z and 8 displaystyle theta is its argument The argument is defined up to an integer multiple of 2p this means that if 8 displaystyle theta is the argument of a complex number then 8 2 k p displaystyle theta 2k pi 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 n th root of a complex number can be obtained by taking the n th root of the absolute value and dividing its argument by n r e i 8 1 n r n e i 8 n displaystyle left rho e i theta right frac 1 n sqrt n rho e frac i theta n If 2 i p displaystyle 2i pi is added to 8 displaystyle theta the complex number in not changed but this adds 2 i p n displaystyle 2i pi n to the argument of the n th root and provides a new n th root This can be done n times and provides the n n th roots of the complex number It is usual to choose one of the n n th root as the principal root The common choice is to choose the n th root for which p lt 8 p displaystyle pi lt theta leq pi that is the n th root that has the largest real part and if they are two the one with positive imaginary part This makes the principal n th root a continuous function in the whole complex plane except for negative real values of the radicand This function equals the usual n th root for positive real radicands For negative real radicands and odd exponents the principal n th root is not real although the usual n th root is real Analytic continuation shows that the principal n th root is the unique complex differentiable function that extends the usual n th 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 2 p displaystyle 2 pi the complex number comes back to its initial position and its n th roots are permuted circularly they are multiplied by e 2 i p n e 2i pi n This shows that it is not possible to define a n th root function that is not continuous in the whole complex plane Roots of unity Edit Main article Root of unity The three third roots of 1 The n th 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 n n th roots of unity are the n first powers of w e 2 p i n displaystyle omega e frac 2 pi i n that is 1 w 0 w n w w 1 w 2 w n 1 displaystyle 1 omega 0 omega n omega omega 1 omega 2 omega n 1 The n th roots of unity that have this generating property are called primitive n th roots of unity they have the form w k e 2 k p i n displaystyle omega k e frac 2k pi i n with k coprime with n The unique primitive square root of unity is 1 displaystyle 1 the primitive fourth roots of unity are i displaystyle i and i displaystyle i The n th roots of unity allow expressing all n th roots of a complex number z as the n products of a given n th roots of z with a n th root of unity Geometrically the n th 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 e 2 k p i n displaystyle e frac 2k pi i n is the primitive n th root of unity with the smallest positive argument it is called the principal primitive n th root of unity sometimes shortened as principal n th root of unity although this terminology can be confused with the principal value of 1 1 n displaystyle 1 1 n which is 1 23 24 25 Complex exponentiation Edit 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 z w z w So either a principal value is defined which is not continuous for the values of z that are real and nonpositive or z w z w is defined as a multivalued function In all cases the complex logarithm is used to define complex exponentiation as z w e w log z displaystyle z w e w log z where log z displaystyle log z is the variant of the complex logarithm that is used which is a function or a multivalued function such that e log z z displaystyle e log z z for every z in its domain of definition Principal value Edit The principal value of the complex logarithm is the unique function commonly denoted log displaystyle log such that for every nonzero complex number z e log z z displaystyle e log z z and the imaginary part of z satisfies p lt I m p displaystyle pi lt mathrm Im leq pi The principal value of the complex logarithm is not defined for z 0 displaystyle z 0 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 log z ln z displaystyle log z ln z The principal value of z w displaystyle z w is defined as z w e w log z displaystyle z w e w log z where log z displaystyle log z is the principal value of the logarithm The function z w z w displaystyle z w to z w 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 z w displaystyle z w equals its usual value defined above If w 1 n displaystyle w 1 n where n is an integer this principal value is the same as the one defined above Multivalued function Edit In some contexts there is a problem with the discontinuity of the principal values of log z displaystyle log z and z w displaystyle z w at the negative real values of z In this case it is useful to consider these functions as multivalued functions If log z displaystyle log z denotes one of the values of the multivalued logarithm typically its principal value the other values are 2 i k p log z displaystyle 2ik pi log z where k is any integer Similarly if z w displaystyle z w is one value of the exponentiation then the other values are given by e w 2 i k p log z z w e 2 i k p w displaystyle e w 2ik pi log z z w e 2ik pi w where k is any integer Different values of k give different values of z w displaystyle z w 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 e a e b displaystyle e a e b if and only if a b displaystyle a b is an integer multiple of 2 p i displaystyle 2 pi i If w m n displaystyle w frac m n is a rational number with m and n coprime integers with n gt 0 displaystyle n gt 0 then z w displaystyle z w has exactly n values In the case m 1 displaystyle m 1 these values are the same as those described in n th 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 z 0 displaystyle z neq 0 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 z w displaystyle z w has changed of sheet Computation Edit The canonical form x i y displaystyle x iy of z w displaystyle z w 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 z a i b displaystyle z a ib is the canonical form of z a and b being real then its polar form is z r e i 8 r cos 8 i sin 8 displaystyle z rho e i theta rho cos theta i sin theta where r a 2 b 2 displaystyle rho sqrt a 2 b 2 and 8 atan2 a b displaystyle theta operatorname atan2 a b see atan2 for the definition of this function Logarithm of z The principal value of this logarithm is log z ln r i 8 displaystyle log z ln rho i theta where ln displaystyle ln denotes the natural logarithm The other values of the logarithm are obtained by adding 2 i k p displaystyle 2ik pi for any integer k Canonical form of w log z displaystyle w log z If w c d i displaystyle w c di with c and d real the values of w log z displaystyle w log z are w log z c ln r d 8 2 d k p i d ln r c 8 2 c k p displaystyle w log z c ln rho d theta 2dk pi i d ln rho c theta 2ck pi the principal value corresponding to k 0 displaystyle k 0 Final result Using the identities e x y e x e y displaystyle e x y e x e y and e y ln x x y displaystyle e y ln x x y one gets z w r c e d 8 2 k p cos d ln r c 8 2 c k p i sin d ln r c 8 2 c k p displaystyle 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 with k 0 displaystyle k 0 for the principal value Examples Edit i i displaystyle i i The polar form of i is i e i p 2 displaystyle i e i pi 2 and the values of log i displaystyle log i are thus log i i p 2 2 k p displaystyle log i i left frac pi 2 2k pi right It follows that i i e i log i e p 2 e 2 k p displaystyle i i e i log i e frac pi 2 e 2k pi So all values of i i displaystyle i i are real the principal one being e p 2 0 2079 displaystyle e frac pi 2 approx 0 2079 2 3 4 i displaystyle 2 3 4i Similarly the polar form of 2 is 2 2 e i p displaystyle 2 2e i pi So the above described method gives the values 2 3 4 i 2 3 e 4 p 2 k p cos 4 ln 2 3 p 2 k p i sin 4 ln 2 3 p 2 k p 2 3 e 4 p 2 k p cos 4 ln 2 i sin 4 ln 2 displaystyle begin aligned 2 3 4i amp 2 3 e 4 pi 2k pi cos 4 ln 2 3 pi 2k pi i sin 4 ln 2 3 pi 2k pi amp 2 3 e 4 pi 2k pi cos 4 ln 2 i sin 4 ln 2 end aligned In this case all the values have the same argument 4 ln 2 displaystyle 4 ln 2 and different absolute values In both examples all values of z w displaystyle z w 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 Edit 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 log i 2 log 1 i p 2 log i 2 log e i p 2 2 i p 2 i p displaystyle log i 2 log 1 i pi neq 2 log i 2 log e i pi 2 2 frac i pi 2 i pi Regardless of which branch of the logarithm is used a similar failure of the identity will exist The best that can be said if only using this result is that log w z z log w mod 2 p i displaystyle log w z equiv z log w pmod 2 pi i This identity does not hold even when considering log as a multivalued function The possible values of log wz contain those of z log w as a 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 log w z z Log w z 2 p i n 2 p i m m n Z z log w z Log w z 2 p i n n Z displaystyle begin aligned left log w z right amp left z cdot operatorname Log w z cdot 2 pi in 2 pi im mid m n in mathbb Z right left z log w right amp left z operatorname Log w z cdot 2 pi in mid n in mathbb Z right end aligned 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 1 1 1 2 1 1 1 2 1 1 2 1 displaystyle 1 cdot 1 frac 1 2 1 not 1 frac 1 2 1 frac 1 2 1 and 1 1 1 2 1 1 2 i 1 1 2 1 1 2 1 i i displaystyle left frac 1 1 right frac 1 2 1 frac 1 2 i not frac 1 frac 1 2 1 frac 1 2 frac 1 i i On the other hand when x is an integer the identities are valid for all nonzero complex numbers If exponentiation is considered as a multivalued function then the possible values of 1 1 1 2 are 1 1 The identity holds but saying 1 1 1 1 2 is wrong The identity ex y exy holds for real numbers x and y but assuming its truth for complex numbers leads to the following paradox discovered in 1827 by Clausen 26 For any integer n we have e 1 2 p i n e 1 e 2 p i n e 1 e displaystyle e 1 2 pi in e 1 e 2 pi in e cdot 1 e e 1 2 p i n 1 2 p i n e displaystyle left e 1 2 pi in right 1 2 pi in e qquad taking the 1 2 p i n displaystyle 1 2 pi in th power of both sides e 1 4 p i n 4 p 2 n 2 e displaystyle e 1 4 pi in 4 pi 2 n 2 e qquad using e x y e x y displaystyle left e x right y e xy and expanding the exponent e 1 e 4 p i n e 4 p 2 n 2 e displaystyle e 1 e 4 pi in e 4 pi 2 n 2 e qquad using e x y e x e y displaystyle e x y e x e y e 4 p 2 n 2 1 displaystyle e 4 pi 2 n 2 1 qquad dividing by e but this is false when the integer n is nonzero The error is the following by definition e y displaystyle e y is a notation for exp y displaystyle exp y a true function and x y displaystyle x y is a notation for exp y log x displaystyle exp y log x which is a multi valued function Thus the notation is ambiguous when x e Here before expanding the exponent the second line should be exp 1 2 p i n log exp 1 2 p i n exp 1 2 p i n displaystyle exp left 1 2 pi in log exp 1 2 pi in right exp 1 2 pi in Therefore when expanding the exponent one has implicitly supposed that log exp z z displaystyle log exp z z 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 e x y e y log e x displaystyle left e x right y e y log e x which is a true identity between multivalued functions Irrationality and transcendence EditMain article Gelfond Schneider theorem 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 b 0 1 displaystyle b not in 0 1 then at least one of b x and bx is transcendental Integer powers in algebra EditThe 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 x 0 displaystyle x 0 requires further the existence of a multiplicative identity 27 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 x 0 1 displaystyle x 0 1 x n 1 x x n displaystyle x n 1 xx n for every nonnegative integer n If n is a negative integer x n displaystyle x n is defined only if x has a multiplicative inverse 28 In this case the inverse of x is denoted x 1 displaystyle x 1 and x n displaystyle x n is defined as x 1 n displaystyle left x 1 right n Exponentiation with integer exponents obeys the following laws for x and y in the algebraic structure and m and n integers x 0 1 x m n x m x n x m n x m n x y n x n y n if x y y x and in particular if the multiplication is commutative displaystyle begin aligned x 0 amp 1 x m n amp x m x n x m n amp x mn xy n amp x n y n quad text if xy yx text and in particular if the multiplication is commutative end aligned 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 f n displaystyle f n denotes the exponentiation with respect of multiplication and f n displaystyle f circ n may denote exponentiation with respect of function composition That is f n x f x n f x f x f x displaystyle f n x f x n f x f x cdots f x and f n x f f f f x displaystyle f circ n x f f cdots f f x cdots Commonly f n x displaystyle f n x is denoted f x n displaystyle f x n while f n x displaystyle f circ n x is denoted f n x displaystyle f n x In a group Edit 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 x n displaystyle x n is defined for every x G displaystyle x in G 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 Z displaystyle mathbb Z 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 x n x 0 1 displaystyle x n x 0 1 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 g h k g h k displaystyle g h k g hk and g h k g k h k displaystyle gh k g k h k In a ring Edit In a ring it may occurs that some nonzero elements satisfy x n 0 displaystyle x n 0 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 x 0 displaystyle x neq 0 implies x n 0 displaystyle x n neq 0 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 k x 1 x n displaystyle k x 1 ldots x n 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 Edit If A is a square matrix then the product of A with itself n times is called the matrix power Also A 0 displaystyle A 0 is defined to be the identity matrix 29 and if A is invertible then A n A 1 n displaystyle A n left A 1 right n 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 30 This is the standard interpretation of a Markov chain for example Then A 2 x displaystyle A 2 x is the state of the system after two time steps and so forth A n x displaystyle A n x is the state of the system after n time steps The matrix power A n displaystyle A n 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 d d x displaystyle d dx which is a linear operator acting on functions f x displaystyle f x to give a new function d d x f x f x displaystyle d dx f x f x The n th power of the differentiation operator is the n th derivative d d x n f x d n d x n f x f n x displaystyle left frac d dx right n f x frac d n dx n f x f n x 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 31 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 Schrodinger 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 Edit Main article Finite field 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 q p k displaystyle q p k 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 F q displaystyle mathbb F q One has x q x displaystyle x q x for every x F q displaystyle x in mathbb F q A primitive element in F q displaystyle mathbb F q is an element g such the set of the q 1 first powers of g that is g 1 g g 2 g p 1 g 0 1 displaystyle g 1 g g 2 ldots g p 1 g 0 1 equals the set of the nonzero elements of F q displaystyle mathbb F q There are f p 1 displaystyle varphi p 1 primitive elements in F q displaystyle mathbb F q where f displaystyle varphi is Euler s totient function In F q displaystyle mathbb F q the Freshman s dream identity x y p x p y p displaystyle x y p x p y p is true for the exponent p As x p x displaystyle x p x in F q displaystyle mathbb F q It follows that the map F F q F q x x p displaystyle begin aligned F colon amp mathbb F q to mathbb F q amp x mapsto x p end aligned is linear over F q displaystyle mathbb F q and is a field automorphism called the Frobenius automorphism If q p k displaystyle q p k the field F q displaystyle mathbb F q has k automorphisms which are the k first powers under composition of F In other words the Galois group of F q displaystyle mathbb F q 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 F q displaystyle mathbb F q then g e displaystyle g e 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 g e displaystyle g e if q is sufficiently large Powers of sets EditThe Cartesian product of two sets S and T is the set of the ordered pairs x y displaystyle x y such that x S displaystyle x in S and y T displaystyle y in T This operation is not properly commutative nor associative but has these properties up to canonical isomorphisms that allow identifying for example x y z displaystyle x y z x y z displaystyle x y z and x y z displaystyle x y z This allows defining the n th power S n displaystyle S n of a set S as the set of all n tuples x 1 x n displaystyle x 1 ldots x n of elements of S When S is endowed with some structure it is frequent that S n displaystyle S n 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 R n displaystyle mathbb R n where R displaystyle mathbb R denotes the real numbers denotes the Cartesian product of n copies of R displaystyle mathbb R as well as their direct product as vector space topological spaces rings etc Sets as exponents Edit A n tuple x 1 x n displaystyle x 1 ldots x n of elements of S can be considered as a function from 1 n displaystyle 1 ldots n This generalizes to the following notation Given two sets S and T the set of all functions from T to S is denoted S T displaystyle S T This exponential notation is justified by the following canonical isomorphisms for the first one see Currying S T U S T U displaystyle S T U cong S T times U S T U S T S U displaystyle S T sqcup U cong S T times S U where displaystyle times denotes the Cartesian product and displaystyle sqcup 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 R N displaystyle mathbb R mathbb N denotes the vector space of the infinite sequences of real numbers and R N displaystyle mathbb R mathbb N 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 0 1 displaystyle 0 1 So 2 S displaystyle 2 S denotes the power set of S that is the set of the functions from S to 0 1 displaystyle 0 1 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 Edit Main article Cartesian closed category 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 Y X displaystyle Y X in the preceding section can also be denoted hom X Y displaystyle hom X Y The isomorphism S T U S T U displaystyle S T U cong S T times U can be rewritten hom U S T hom T U S displaystyle hom U S T cong hom T times U S 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 X X T displaystyle X to X T is if it exists a right adjoint to the functor Y T Y displaystyle Y to T times Y A category is called a Cartesian closed category if direct products exist and the functor Y X Y displaystyle Y to X times Y has a right adjoint for every T Repeated exponentiation EditMain articles Tetration and Hyperoperation 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 7625 597 484 987 327 333 33 respectively Limits of powers EditZero 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 gt 0 Then D can be viewed as a subset of R 2 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 32 Accordingly this allows one to define the powers xy by continuity whenever 0 x y except for 00 0 1 and 1 which remain indeterminate forms Under this definition by continuity we obtain x and x 0 when 1 lt x x 0 and x when 0 x lt 1 0y 0 and y when 0 lt y 0y and y 0 when y lt 0 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 lt 0 since pairs x y with x lt 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 EditComputing 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 100 2 2 2 5 2 6 2 2 1 2 3 1 2 displaystyle 100 2 2 2 5 2 6 2 2 1 2 3 1 2 Then compute the following terms in order reading Horner s rule from right to left 22 42 22 23 8 23 2 26 64 26 2 212 4096 212 2 224 16777 2162 224 225 33554 432 225 2 250 1125 899 906 842 624 250 2 2100 1267 650 600 228 229 401 496 703 205 376 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 n log 2 n 1 displaystyle sharp n lfloor log 2 n rfloor 1 by using exponentiation by squaring where n displaystyle sharp n 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 33 However in practical computations exponentiation by squaring is efficient enough and much more easy to implement Iterated functions EditFunction 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 g f displaystyle g circ f and defined as g f x g f x msty, wikipedia, wiki, book,

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