b n = b × ⋯ × b ⏟ n . displaystyle b^ n =underbrace btimes cdots times b _ n . The exponent is usually shown as a superscript to the right of the
base. In that case, bn is called "b raised to the n-th power", "b
raised to the power of n", or "the n-th power of b".
When n is a positive integer and b is not zero, b−n is naturally
defined as 1/bn, preserving the property bn × bm = bn + m. With
exponent −1, b−1 is equal to 1/b, and is the reciprocal of b.
The definition of exponentiation can be extended to allow any real or
complex exponent.
Contents 1 History of the notation
2 Terminology
3
3.1 Positive exponents 3.2 Zero exponent 3.3 Negative exponents 3.4 Combinatorial interpretation 3.5 Identities and properties 3.6 Particular bases 3.6.1 Powers of ten 3.6.2 Powers of two 3.6.3 Powers of one 3.6.4 Powers of zero 3.6.5 Powers of minus one 3.7 Large exponents 3.8 Power functions 3.9 List of whole-number powers 4 Rational exponents 5 Real exponents 5.1 Limits of rational exponents 5.2 The exponential function 5.3 Powers via logarithms 5.4 Real exponents with negative bases 5.5 Irrational exponents 6 Complex exponents with positive real bases 6.1 Imaginary exponents with base e 6.2 Trigonometric functions 6.3 Complex exponents with base e 6.4 Complex exponents with positive real bases 7 Powers of complex numbers 7.1 Complex exponents with complex bases 7.2 Complex roots of unity 7.3 Roots of arbitrary complex numbers 7.4 Computing complex powers 7.5 Failure of power and logarithm identities 8 Generalizations 8.1 Monoids 8.2 Matrices and linear operators 8.3 Finite fields 8.4 In abstract algebra 8.5 Over sets 8.6 In category theory 8.7 Of cardinal and ordinal numbers 9 Repeated exponentiation 10 Limits of powers 11 Efficient computation with integer exponents 12 Exponential notation for function names 13 In programming languages 14 See also 15 References 16 External links History of the notation[edit]
The term power was used by the Greek mathematician
b 1 = b displaystyle b^ 1 =b and the recurrence relation b n + 1 = b n ⋅ b . displaystyle b^ n+1 =b^ n cdot b. From the associativity of multiplication, it follows that for any positive integers m and n, b m + n = b m ⋅ b n . displaystyle b^ m+n =b^ m cdot b^ n . Zero exponent[edit] Any nonzero number raised to the 0 power is 1[11]: b 0 = 1. displaystyle b^ 0 =1. One interpretation of such a power is as an empty product. The case of 00 is discussed at Zero to the power of zero. Negative exponents[edit] The following identity holds for an arbitrary integer n and nonzero b: b − n = 1 / b n . displaystyle b^ -n =1/b^ n . Raising 0 to a negative exponent is undefined, but in some circumstances, it may be interpreted as infinity (∞). The identity above may be derived through a definition aimed at extending the range of exponents to negative integers. For non-zero b and positive n, the recurrence relation above can be rewritten as b n = b n + 1 / b , n ≥ 1. displaystyle b^ n =b^ n+1 /b,quad ngeq 1. By defining this relation as valid for all integer n and nonzero b, it follows that b 0 = b 1 / b = 1 , b − 1 = b 0 / b = 1 / b , displaystyle begin aligned b^ 0 &=b^ 1 /b=1,\b^ -1 &=b^ 0 /b=1/b,end aligned and more generally for any nonzero b and any nonnegative integer n, b − n = 1 / b n . displaystyle b^ -n =1/b^ n . This is then readily shown to be true for every integer n.
Combinatorial interpretation[edit]
See also:
nm The nm possible m-tuples of elements from the set 1,...,n 0 5 = 0 displaystyle 0^ 5 =0 none 1 4 = 1 displaystyle 1^ 4 =1 ( 1 , 1 , 1 , 1 ) displaystyle (1,1,1,1) 2 3 = 8 displaystyle 2^ 3 =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 ) displaystyle (1,1,1),(1,1,2),(1,2,1),(1,2,2),(2,1,1),(2,1,2),(2,2,1),(2,2,2) 3 2 = 9 displaystyle 3^ 2 =9 ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 3 ) , ( 3 , 1 ) , ( 3 , 2 ) , ( 3 , 3 ) displaystyle (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3) 4 1 = 4 displaystyle 4^ 1 =4 ( 1 ) , ( 2 ) , ( 3 ) , ( 4 ) displaystyle (1),(2),(3),(4) 5 0 = 1 displaystyle 5^ 0 =1 ( ) displaystyle () Identities and properties[edit] The following identities 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 \(b^ m )^ n &=b^ mcdot n \(bcdot c)^ n &=b^ n cdot c^ n end aligned Unlike addition and multiplication:
b p q = b ( p q ) ≠ ( b p ) q = b ( p ⋅ q ) = b p ⋅ q . displaystyle b^ p^ q =b^ (p^ q ) neq (b^ p )^ q =b^ (pcdot q) =b^ pcdot q . While
Particular bases[edit]
Powers of ten[edit]
See also: Scientific notation
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,
7003100000000000000♠103 = 7003100000000000000♠1000 and
6996100000000000000♠10−4 = 6996100000000000000♠0.0001.
bn → ∞ as n → ∞ when b > 1 This can be read as "b to the power of n tends to +∞ as n tends to infinity when b is greater than one". Powers of a number with absolute value less than one tend to zero: bn → 0 as n → ∞ when b < 1 Any power of one is always one: bn = 1 for all n if b = 1 Powers of –1 alternate between 1 and –1 as n alternates between even and odd, and thus do not tend to any limit as n grows. If b < –1, bn, alternates between larger and larger positive and negative numbers as n alternates between even and odd, and thus does not tend to any limit as n grows. If the exponentiated number varies while tending to 1 as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is (1 + 1/n)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 with c ≠ 0 displaystyle cneq 0 are sometimes called power functions.[citation needed] When n displaystyle n is an integer and n ≥ 1 displaystyle ngeq 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.[13] 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.[14] List of whole-number powers[edit] n n2 n3 n4 n5 n6 n7 n8 n9 n10 2 4 8 16 32 64 128 256 512 1,024 3 9 27 81 243 729 2,187 6,561 19,683 59,049 4 16 64 256 1,024 4,096 16,384 65,536 262,144 1,048,576 5 25 125 625 3,125 15,625 78,125 390,625 1,953,125 9,765,625 6 36 216 1,296 7,776 46,656 279,936 1,679,616 10,077,696 60,466,176 7 49 343 2,401 16,807 117,649 823,543 5,764,801 40,353,607 282,475,249 8 64 512 4,096 32,768 262,144 2,097,152 16,777,216 134,217,728 1,073,741,824 9 81 729 6,561 59,049 531,441 4,782,969 43,046,721 387,420,489 3,486,784,401 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000 10,000,000,000 Rational exponents[edit] Main article: nth root From top to bottom: x1/8, x1/4, x1/2, x1, x2, x4, x8. An nth root of a number b is a number x such that xn = b. If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to xn = b. This solution is called the principal nth root of b. It is denoted n√b, where √ is the radical symbol; alternatively, the principal root may be written b1/n. For example: 41/2 = 2, 81/3 = 2. The fact that x = b 1 / n displaystyle x=b^ 1/n solves x n = b displaystyle x^ n =b follows from noting that x n = b 1 n × b 1 n × ⋯ × b 1 n ⏟ n = b ( 1 n + 1 n + ⋯ + 1 n ) = b n n = b 1 = b . displaystyle x^ n =underbrace b^ frac 1 n times b^ frac 1 n times cdots times b^ frac 1 n _ n =b^ left( frac 1 n + frac 1 n +cdots + frac 1 n right) =b^ frac n n =b^ 1 =b. If n is even, then xn = b has two real solutions if b is positive, which are the positive and negative nth roots, i.e., b1/n > 0 and -(b1/n) < 0. If b is negative, the equation has no solution in real numbers for even n. If n is odd, then xn = b has exactly one real solution. The solution b1/n is positive if b is positive and negative if b is negative. Taking a positive real number b to a rational exponent u/v, where u is an integer and v is a positive integer, and considering principal roots only, yields b u v = ( b u ) 1 v = b u v = ( b 1 v ) u = ( b v ) u . displaystyle b^ frac u v =left(b^ u right)^ frac 1 v = sqrt[ v ] b^ u =left(b^ frac 1 v right)^ u =left( sqrt[ v ] b right)^ u . Taking a negative real number b to a rational power u/v, where u/v is in lowest terms, yields a positive real result if u is even, and hence v is odd, because then bu is positive; and yields a negative real result, if u and v are both odd, because then bu is negative. The case of odd u and even v cannot be treated this way within the reals, since there is no real number x such that x2k = −1, the value of bu/v in this case must use the imaginary unit i, as described more fully in the section § Powers of complex numbers. Thus we have (−27)1/3 = −3 and (−27)2/3 = 9. The number 4 has two 3/2th powers, namely 8 and −8; however, by convention the notation 43/2 employs the principal root, and results in 8. For employing the v-th root the u/v-th power is also called the u/v-th root, and for even v the term principal root denotes also the positive result. This sign ambiguity needs to be taken care of when applying the power identities. For instance: − 27 = ( − 27 ) ( ( 2 / 3 ) ( 3 / 2 ) ) = ( ( − 27 ) 2 / 3 ) 3 / 2 = 9 3 / 2 = 27 displaystyle -27=(-27)^ ((2/3)(3/2)) =((-27)^ 2/3 )^ 3/2 =9^ 3/2 =27 is clearly wrong. The problem starts already in the first equality by introducing a standard notation for an inherently ambiguous situation –asking for an even root– and simply relying wrongly on only one, the conventional or principal interpretation. The same problem occurs also with an inappropriately introduced surd-notation, inherently enforcing a positive result: ( ( − 27 ) 2 / 3 ) 3 / 2 = ( ( − 27 ) 2 3 ) 3 = ( − 27 ) 2 ≠ − 27 displaystyle ((-27)^ 2/3 )^ 3/2 = sqrt left( sqrt[ 3 ] (-27)^ 2 right)^ 3 = sqrt (-27)^ 2 neq -27 instead of ( ( − 27 ) 2 / 3 ) 3 / 2 = − ( ( − 27 ) 2 3 ) 3 = − ( − 27 ) 2 = − 27. displaystyle ((-27)^ 2/3 )^ 3/2 =- sqrt left( sqrt[ 3 ] (-27)^ 2 right)^ 3 =- sqrt (-27)^ 2 =-27. In general the same sort of problems occur for complex numbers as described in the section § Failure of power and logarithm identities. Real exponents[edit] The identities and properties shown above for integer exponents are true for positive real numbers with non-integer exponents as well. However the identity ( b r ) s = b r ⋅ s displaystyle (b^ r )^ s =b^ rcdot s cannot be extended consistently to cases where b is a negative real
number (see § Real exponents with negative bases). The failure
of this identity is the basis for the problems with complex number
powers detailed under § Failure of power and logarithm
identities.
Because the exponential function is continuous we find lim n → ∞ e x n = e lim n → ∞ x n displaystyle lim _ nto infty e^ x_ n =e^ lim _ nto infty x_ n for convergent sequences (xn). This is shown here for xn = 1/n. 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[15] 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 (bin mathbb R ^ + ,,xin mathbb R ) where the limit as r gets close to x is taken only over rational values of r. This limit only exists for positive b. The (ε, δ)-definition of limit is used; this involves showing that for any desired accuracy of the result bx one can choose a sufficiently small interval around x so all the rational powers in the interval are within the desired accuracy. For example, if x = π, the nonterminating decimal representation π = 3.14159... can be used (based on strict monotonicity of the rational power) to obtain the intervals bounded by rational powers [ b 3 , b 4 ] displaystyle [b^ 3 ,b^ 4 ] , [ b 3.1 , b 3.2 ] displaystyle [b^ 3.1 ,b^ 3.2 ] , [ b 3.14 , b 3.15 ] displaystyle [b^ 3.14 ,b^ 3.15 ] , [ b 3.141 , b 3.142 ] displaystyle [b^ 3.141 ,b^ 3.142 ] , [ b 3.1415 , b 3.1416 ] displaystyle [b^ 3.1415 ,b^ 3.1416 ] , [ b 3.14159 , b 3.14160 ] displaystyle [b^ 3.14159 ,b^ 3.14160 ] , ... The bounded intervals converge to a unique real number, denoted by b π displaystyle b^ pi . This technique can be used to obtain the power of a positive real number b for any irrational exponent. The function fb(x) = bx is thus defined for any real number x. The exponential function[edit] Main article: Exponential function The important mathematical constant e, sometimes called Euler's number, is approximately equal to 2.718 and is the base of the natural logarithm. Although exponentiation of e could, in principle, be treated the same as exponentiation of any other real number, such exponentials turn out to have particularly elegant and useful properties. Among other things, these properties allow exponentials of e to be generalized in a natural way to other types of exponents, such as complex numbers or even matrices, while coinciding with the familiar meaning of exponentiation with rational exponents. As a consequence, the notation ex usually denotes a generalized exponentiation definition called the exponential function, exp(x), which can be defined in many equivalent ways, for example by: exp ( x ) = lim n → ∞ ( 1 + x n ) n displaystyle exp(x)=lim _ nrightarrow infty left(1+ frac x n right)^ n Among other properties, exp satisfies the exponential identity exp ( x + y ) = exp ( x ) ⋅ exp ( y ) . displaystyle exp(x+y)=exp(x)cdot exp(y). The exponential function is defined for all integer, fractional, real, and complex values of x. In fact, the matrix exponential is well-defined for square matrices (in which case this exponential identity only holds when x and y commute), and is useful for solving systems of linear differential equations. Since exp(1) is equal to e and exp(x) satisfies this exponential identity, it immediately follows that exp(x) coincides with the repeated-multiplication definition of ex for integer x, and it also follows that rational powers denote (positive) roots as usual, so exp(x) coincides with the ex definitions in the previous section for all real x by continuity. Powers via logarithms[edit] The natural logarithm ln(x) is the inverse of the exponential function ex. It is defined for b > 0, and satisfies b = e ln b displaystyle b=e^ ln b If bx is to preserve the logarithm and exponent rules, then one must have b x = ( e ln b ) x = e x ⋅ ln b displaystyle b^ x =(e^ ln b )^ x =e^ xcdot ln b for each real number x.
This can be used as an alternative definition of the real number power
bx and agrees with the definition given above using rational exponents
and continuity. The definition of exponentiation using logarithms is
more common in the context of complex numbers, as discussed below.
Real exponents with negative bases[edit]
Powers of a positive real number are always positive real numbers. The
solution of x2 = 4, however, can be either 2 or −2. The
principal value of 41/2 is 2, but −2 is also a valid square root. If
the definition of exponentiation of real numbers is extended to allow
negative results then the result is no longer well-behaved.
Neither the logarithm method nor the rational exponent method can be
used to define br as a real number for a negative real number b and an
arbitrary real number r. Indeed, er is positive for every real number
r, so ln(b) is not defined as a real number for b ≤ 0.
The rational exponent method cannot be used for negative values of b
because it relies on continuity. The function f(r) = br has a unique
continuous extension[15] from the rational numbers to the real numbers
for each b > 0. But when b < 0, the function f is not even
continuous on the set of rational numbers r for which it is defined.
For example, consider b = −1. The nth root of −1 is −1 for every
odd natural number n. So if n is an odd positive integer, (−1)(m/n)
= −1 if m is odd, and (−1)(m/n) = 1 if m is even. Thus the set of
rational numbers q for which (−1)q = 1 is dense in the rational
numbers, as is the set of q for which (−1)q = −1. This means that
the function (−1)q is not continuous at any rational number q where
it is defined.
On the other hand, arbitrary complex powers of negative numbers b can
be defined by choosing a complex logarithm of b.
Irrational exponents[edit]
Main article: Gelfond–Schneider theorem
If a is a positive algebraic number, and b is a rational number, it
has been shown above that ab is algebraic. This remains true even if
one accepts any algebraic number for a, with the only difference that
ab may take several values (see below), all algebraic. The
If a is an algebraic number different from 0 and 1, and b an irrational algebraic number, then all the values of ab are transcendental numbers (that is, not algebraic). Complex exponents with positive real bases[edit] Imaginary exponents with base e[edit] Main article: Exponential function The exponential function ez can be defined as the limit of (1 + z/N)N, as N approaches infinity, and thus eiπ is the limit of (1 + iπ/N)N. In this animation N takes values increasing from 1 to 100. The computation of (1 + iπ/N)N is displayed as the combined effect of N repeated multiplications in the complex plane, so that (1 + iπ/N)k, k = 0 ... N are the vertices of a polygonal path whose final, leftmost endpoint is the actual value of (1 + iπ/N)N. It can be seen that as N gets larger (1 + iπ/N)N approaches a limit of −1. Therefore, eiπ = −1, which is known as Euler's identity. A complex number is an expression of the form z = x + i y displaystyle z=x+iy , where x and y are real numbers, and i is the so-called imaginary unit, a number that satisfies the rule i 2 = − 1 displaystyle i^ 2 =-1 . A complex number can be visualized as a point in the (x,y) plane. The polar coordinates of a point in the (x,y) plane consist of a non-negative real number r and angle θ such that x = r cos θ and y = r sin θ. So x + i y = r ( cos θ + i sin θ ) . displaystyle x+iy=r(cos theta +isin theta ). The product of two complex numbers z1 = x1 + iy1, z2 = x2 + iy2 is obtained by expanding out the product of the binomials and simplifying using the rule i 2 = − 1 displaystyle i^ 2 =-1 : z 1 z 2 = ( x 1 + i y 1 ) ( x 2 + i y 2 ) = ( x 1 x 2 − y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ) . displaystyle z_ 1 z_ 2 =(x_ 1 +iy_ 1 )(x_ 2 +iy_ 2 )=(x_ 1 x_ 2 -y_ 1 y_ 2 )+i(x_ 1 y_ 2 +x_ 2 y_ 1 ). As a consequence of the angle sum formulas of trigonometry, if z1 and z2 have polar coordinates (r1, θ1), (r2, θ2), then their product z1z2 has polar coordinates equal to (r1r2, θ1 + θ2). Consider the right triangle in the complex plane which has 0, 1, 1 + ix/n as vertices. For large values of n, the triangle is almost a circular sector with a radius of 1 and a small central angle equal to x/n radians. 1 + ix/n may then be approximated by the number with polar coordinates (1, x/n). So, in the limit as n approaches infinity, (1 + ix/n)n approaches (1, x/n)n = (1n, nx/n) = (1, x), the point on the unit circle whose angle from the positive real axis is x radians. The Cartesian coordinates of this point are (cos x, sin x). So e ix = cos x + isin x; this is Euler's formula, connecting algebra to trigonometry by means of complex numbers. The solutions to the equation ez = 1 are the integer multiples of 2πi: z : e z = 1 = 2 k π i : k ∈ Z displaystyle z:e^ z =1 = 2kpi i:kin mathbb Z More generally, if ev = w, then every solution to ez = w can be obtained by adding an integer multiple of 2πi to v: z : e z = w = v + 2 k π i : k ∈ Z displaystyle z:e^ z =w = v+2kpi i:kin mathbb Z Thus the complex exponential function is a periodic function with
period 2πi.
More simply: eiπ = −1; ex + iy = ex(cos y + i sin y).
Trigonometric functions[edit]
Main article: Euler's formula
It follows from
cos ( z ) = e i z + e − i z 2 ; sin ( z ) = e i z − e − i z 2 i displaystyle cos(z)= frac e^ iz +e^ -iz 2 ;qquad sin(z)= frac e^ iz -e^ -iz 2i Before the invention of complex numbers, cosine and sine were defined geometrically. The above formula reduces the complicated formulas for trigonometric functions of a sum into the simple exponentiation formula e i ( x + y ) = e i x ⋅ e i y displaystyle e^ i(x+y) =e^ ix cdot e^ iy Using exponentiation with complex exponents may reduce problems in trigonometry to algebra. Complex exponents with base e[edit] The power z = ex + iy can be computed as ex ⋅ eiy. The real factor ex is the absolute value of z and the complex factor eiy identifies the direction of z. Complex exponents with positive real bases[edit] If b is a positive real number, and z is any complex number, the power bz is defined as ez ⋅ ln(b), where x = ln(b) is the unique real solution to the equation ex = b. So the same method working for real exponents also works for complex exponents. For example: 2 i = e i ln ( 2 ) = cos ( ln ( 2 ) ) + i sin ( ln ( 2 ) ) ≈ 0.76924 + 0.63896 i displaystyle 2^ i =e^ iln(2) =cos(ln(2))+isin(ln(2))approx 0.76924+0.63896i e i ≈ 0.54030 + 0.84147 i displaystyle e^ i approx 0.54030+0.84147i ( e 2 π ) i = 535.49176 … i = 1 displaystyle (e^ 2pi )^ i = 535.49176dots ^ i =1 The identity (bz)u=bzu is not generally valid for complex powers. The power bz is a complex number and any power of it has to follow the rules for powers of complex numbers below. A simple counterexample is given by: ( e 2 π i ) i = 1 i = 1 ≠ e − 2 π = e 2 π i ⋅ i displaystyle (e^ 2pi i )^ i =1^ i =1neq e^ -2pi =e^ 2pi icdot i The identity is, however, valid for arbitrary complex z displaystyle z when u displaystyle u is an integer.
Powers of complex numbers[edit]
w z = e z log w displaystyle w^ z =e^ zlog w because this agrees with the earlier definition in the case where w is a positive real number and the (real) principal value of log w is used. If z is an integer, then the value of wz is independent of the choice of log w, and it agrees with the earlier definition of exponentiation with an integer exponent. If z is a rational number m/n in lowest terms with z > 0, then the countably infinitely many choices of log w yield only n different values for wz; these values are the n complex solutions s to the equation sn = wm. If z is an irrational number, then the countably infinitely many choices of log w lead to infinitely many distinct values for wz. The computation of complex powers is facilitated by converting the base w to polar form, as described in detail below. A similar construction is employed in quaternions. Complex roots of unity[edit] Main article: Root of unity The three 3rd roots of 1 A complex number w such that wn = 1 for a positive integer n is an nth root of unity. Geometrically, the nth roots of unity lie on the unit circle of the complex plane at the vertices of a regular n-gon with one vertex on the real number 1. If wn = 1 but wk ≠ 1 for all natural numbers k such that 0 < k < n, then w is called a primitive nth root of unity. The negative unit −1 is the only primitive square root of unity. The imaginary unit i is one of the two primitive 4th roots of unity; the other one is −i. The number e2πi/n is the primitive nth root of unity with the smallest positive argument. (It is sometimes called the principal nth root of unity, although this terminology is not universal and should not be confused with the principal value of n√1, which is 1.[16]) The other nth roots of unity are given by ( e 2 n π i ) k = e 2 n π i k displaystyle left(e^ frac 2 n pi i right)^ k =e^ frac 2 n pi ik for 2 ≤ k ≤ n. Roots of arbitrary complex numbers[edit] Although there are infinitely many possible values for a general complex logarithm, there are only a finite number of values for the power wq in the important special case where q = 1/n and n is a positive integer. These are the nth roots of w; they are solutions of the equation zn = w. As with real roots, a second root is also called a square root and a third root is also called a cube root. It is usual in mathematics to define w1/n as the principal value of the root, which is, conventionally, the nth root whose argument has the smallest absolute value. When w is a positive real number, this is coherent with the usual convention of defining w1/n as the unique positive real nth root. On the other hand, when w is a negative real number, and n is an odd integer, the unique real nth root is not one of the two nth roots whose argument has the smallest absolute value. In this case, the meaning of w1/n may depend on the context, and some care may be needed for avoiding errors. The set of nth roots of a complex number w is obtained by multiplying the principal value w1/n by each of the nth roots of unity. For example, the fourth roots of 16 are 2, −2, 2i, and −2i, because the principal value of the fourth root of 16 is 2 and the fourth roots of unity are 1, −1, i, and −i. Computing complex powers[edit] It is often easier to compute complex powers by writing the number to be exponentiated in polar form. Every complex number z can be written in the polar form z = r e i θ = e log ( r ) + i θ displaystyle z=re^ itheta =e^ log(r)+itheta where r is a nonnegative real number and θ is the (real) argument of z. The polar form has a simple geometric interpretation: if a complex number u + iv is thought of as representing a point (u, v) in the complex plane using Cartesian coordinates, then (r, θ) is the same point in polar coordinates. That is, r is the "radius" r2 = u2 + v2 and θ is the "angle" θ = atan2(v, u). The polar angle θ is ambiguous since any integer multiple of 2π could be added to θ without changing the location of the point. Each choice of θ gives in general a different possible value of the power. A branch cut can be used to choose a specific value. The principal value (the most common branch cut), corresponds to θ chosen in the interval (−π, π]. For complex numbers with a positive real part and zero imaginary part using the principal value gives the same result as using the corresponding real number. In order to compute the complex power wz, write w in polar form: w = r e i θ displaystyle w=re^ itheta Then log ( w ) = log ( r ) + i θ displaystyle log(w)=log(r)+itheta and thus w z = e z log ( w ) = e z ( log ( r ) + i θ ) displaystyle w^ z =e^ zlog(w) =e^ z(log(r)+itheta ) If z is decomposed as c + di, then the formula for wz can be written more explicitly as ( r c e − d θ ) e i ( d log ( r ) + c θ ) = ( r c e − d θ ) [ cos ( d log ( r ) + c θ ) + i sin ( d log ( r ) + c θ ) ] displaystyle left(r^ c e^ -dtheta right)e^ i(dlog(r)+ctheta ) =left(r^ c e^ -dtheta right)left[cos(dlog(r)+ctheta )+isin(dlog(r)+ctheta )right] This final formula allows complex powers to be computed easily from decompositions of the base into polar form and the exponent into Cartesian form. It is shown here both in polar form and in Cartesian form (via Euler's identity). The following examples use the principal value, the branch cut which causes θ to be in the interval (−π, π]. To compute ii, write i in polar and Cartesian forms: i = 1 ⋅ e 1 2 i π i = 0 + 1 i displaystyle begin aligned i&=1cdot e^ frac 1 2 ipi \i&=0+1iend aligned Then the formula above, with r = 1, θ = π/2, c = 0, and d = 1, yields: i i = ( 1 0 e − 1 2 π ) e i [ 1 ⋅ log ( 1 ) + 0 ⋅ 1 2 π ] = e − 1 2 π ≈ 0.2079 displaystyle i^ i =left(1^ 0 e^ - frac 1 2 pi right)e^ ileft[1cdot log(1)+0cdot frac 1 2 pi right] =e^ - frac 1 2 pi approx 0.2079 Similarly, to find (−2)3 + 4i, compute the polar form of −2, − 2 = 2 e i π displaystyle -2=2e^ ipi and use the formula above to compute ( − 2 ) 3 + 4 i = ( 2 3 e − 4 π ) e i [ 4 log ( 2 ) + 3 π ] ≈ ( 2.602 − 1.006 i ) ⋅ 10 − 5 displaystyle (-2)^ 3+4i =left(2^ 3 e^ -4pi right)e^ i[4log(2)+3pi ] approx (2.602-1.006i)cdot 10^ -5 The value of a complex power depends on the branch used. For example, if the polar form i = 1e5πi/2 is used to compute ii, the power is found to be e−5π/2; the principal value of ii, computed above, is e−π/2. The set of all possible values for ii is given by:[17] i = 1 ⋅ e 1 2 i π + i 2 π k
k ∈ Z i i = e i ( 1 2 i π + i 2 π k ) = e − ( 1 2 π + 2 π k ) displaystyle begin aligned i&=1cdot e^ frac 1 2 ipi +i2pi k big kin mathbb Z \i^ i &=e^ ileft( frac 1 2 ipi +i2pi kright) \&=e^ -left( frac 1 2 pi +2pi kright) end aligned So there is an infinity of values which are possible candidates for the value of ii, one for each integer k. All of them have a zero imaginary part so one can say ii has an infinity of valid real values. Failure of power and logarithm identities[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 i π = log ( − 1 ) = log [ ( − i ) 2 ] ≠ 2 log ( − i ) = 2 ( − i π 2 ) = − i π displaystyle ipi =log(-1)=log left[(-i)^ 2 right]neq 2log(-i)=2left(- frac ipi 2 right)=-ipi 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 zcdot log(w) pmod 2pi 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 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 π i n + 2 π i m z ⋅ log ( w ) = z ⋅ Log ( w ) + z ⋅ 2 π i n displaystyle begin aligned left log(w^ z )right &=left zcdot operatorname Log (w)+zcdot 2pi in+2pi imright \left zcdot log(w)right &=left zcdot operatorname Log (w)+zcdot 2pi inright 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 a calculation using principal branches shows that 1 = ( − 1 ⋅ − 1 ) 1 2 ≠ ( − 1 ) 1 2 ( − 1 ) 1 2 = − 1 displaystyle 1=(-1cdot -1)^ frac 1 2 not =(-1)^ frac 1 2 (-1)^ frac 1 2 =-1 and i = ( − 1 ) 1 2 = ( 1 − 1 ) 1 2 ≠ 1 1 2 ( − 1 ) 1 2 = 1 i = − i displaystyle i=(-1)^ frac 1 2 =left( frac 1 -1 right)^ frac 1 2 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:[18] For any integer n, we have: e 1 + 2 π i n = e 1 e 2 π i n = e ⋅ 1 = e displaystyle e^ 1+2pi in =e^ 1 e^ 2pi in =ecdot 1=e ( e 1 + 2 π i n ) 1 + 2 π i n = e displaystyle left(e^ 1+2pi in right)^ 1+2pi in =eqquad (taking the ( 1 + 2 π i n ) displaystyle (1+2pi in) -th power of both sides) e 1 + 4 π i n − 4 π 2 n 2 = e displaystyle e^ 1+4pi in-4pi ^ 2 n^ 2 =eqquad (using ( e x ) y = e x y displaystyle (e^ x )^ y =e^ xy and expanding the exponent) e 1 e 4 π i n e − 4 π 2 n 2 = e displaystyle e^ 1 e^ 4pi in e^ -4pi ^ 2 n^ 2 =eqquad (using e x + y = e x e y displaystyle e^ x+y =e^ x e^ y ) e − 4 π 2 n 2 = 1 displaystyle e^ -4pi ^ 2 n^ 2 =1qquad (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(ylog 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 π i n ) log exp ( 1 + 2 π i n ) ) = exp ( 1 + 2 π i n ) . displaystyle exp left((1+2pi in)log exp(1+2pi in)right)=exp(1+2pi 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 (e^ x )^ y =e^ ylog e^ x , which is a true identity between multivalued functions. Generalizations[edit]
Monoids[edit]
x 0 = 1 displaystyle x^ 0 =1 for all x ∈ X displaystyle xin X x n + 1 = x n x displaystyle x^ n+1 =x^ n x for all x ∈ X displaystyle xin X and non-negative integers n If n is a negative integer then x n displaystyle x^ n is only defined[20] if x displaystyle x has an inverse in X. Monoids include many structures of importance in mathematics, including groups and rings (under multiplication), with more specific examples of the latter being matrix rings and fields. 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,[21] and if A is invertible, then A − n = ( A − 1 ) n displaystyle A^ -n =(A^ -1 )^ 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.[22] 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.[23] 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[edit] A field is an algebraic structure in which multiplication, addition, subtraction, and division are all well-defined and satisfy their familiar properties. The real numbers, for example, form a field, as do the complex numbers and rational numbers. Unlike these familiar examples of fields, which are all infinite sets, some fields have only finitely many elements. The simplest example is the field with two elements F 2 = 0 , 1 displaystyle F_ 2 = 0,1 with addition defined by 0 + 1 = 1 + 0 = 1 displaystyle 0+1=1+0=1 and 0 + 0 = 1 + 1 = 0 displaystyle 0+0=1+1=0 , and multiplication 0 ⋅ 0 = 1 ⋅ 0 = 0 ⋅ 1 = 0 displaystyle 0cdot 0=1cdot 0=0cdot 1=0 and 1 ⋅ 1 = 1 displaystyle 1cdot 1=1 .
p x = 0 displaystyle px=0 for all x in F; that is, x added to itself p times is zero. For example, in F 2 displaystyle F_ 2 , the prime number p = 2 has this property. This prime number is called the characteristic of the field. Suppose that F is a field of characteristic p, and consider the function f ( x ) = x p displaystyle f(x)=x^ p that raises each element of F to the power p. This is called the
( x + y ) p = x p + y p displaystyle (x+y)^ p =x^ p +y^ p . The
x 1 = x x n = x n − 1 x for n > 1 displaystyle begin aligned x^ 1 &=x\x^ n &=x^ n-1 xquad hbox for n>1end aligned One has the following properties ( x i x j ) x k = x i ( x j x k ) (power-associative property) x m + n = x m x n ( x m ) n = x m n displaystyle begin aligned (x^ i x^ j )x^ k &=x^ i (x^ j x^ k )quad text (power-associative property) \x^ m+n &=x^ m x^ n \(x^ m )^ n &=x^ mn end aligned If the operation has a two-sided identity element 1, then x0 is defined to be equal to 1 for any x. x 1 = 1 x = x (two-sided identity) x 0 = 1 displaystyle begin aligned x1&=1x=xquad text (two-sided identity) \x^ 0 &=1end aligned [citation needed] If the operation also has two-sided inverses and is associative, then the magma is a group. The inverse of x can be denoted by x−1 and follows all the usual rules for exponents. x x − 1 = x − 1 x = 1 (two-sided inverse) ( x y ) z = x ( y z ) (associative) x − n = ( x − 1 ) n x m − n = x m x − n displaystyle begin aligned xx^ -1 &=x^ -1 x=1quad text (two-sided inverse) \(xy)z&=x(yz)quad text (associative) \x^ -n &=left(x^ -1 right)^ n \x^ m-n &=x^ m x^ -n end aligned If the multiplication operation is commutative (as for instance in abelian groups), then the following holds: ( x y ) n = x n y n displaystyle (xy)^ n =x^ n y^ n If the binary operation is written additively, as it often is for
abelian groups, then "exponentiation is repeated multiplication" can
be reinterpreted as "multiplication is repeated addition". Thus, each
of the laws of exponentiation above has an analogue among laws of
multiplication.
When there are several power-associative binary operations defined on
a set, any of which might be iterated, it is common to indicate which
operation is being repeated by placing its symbol in the superscript.
Thus, x∗n is x ∗ ... ∗ x, while x#n is x # ... # x, whatever the
operations ∗ and # might be.
⨁ i ∈ N V i displaystyle bigoplus _ iin mathbb N V_ i where each Vi is a vector space.
Then if Vi = V for each i, the resulting direct sum can be written in
exponential notation as V⊕N, or simply VN with the understanding
that the direct sum is the default. We can again replace the set N
with a cardinal number n to get Vn, although without choosing a
specific standard set with cardinality n, this is defined only up to
isomorphism. Taking V to be the field R of real numbers (thought of as
a vector space over itself) and n to be some natural number, we get
the vector space that is most commonly studied in linear algebra, the
real vector space Rn.
If the base of the exponentiation operation is a set, the
exponentiation operation is the
S N ≡ f : N → S displaystyle S^ N equiv fcolon Nto S This fits in with the exponentiation of cardinal numbers, in the sense
that SN = SN, where X is the cardinality of X. When "2" is
defined as 0, 1 , we have 2X = 2X, where 2X, usually denoted by
P(X), is the power set of X; each subset Y of X corresponds uniquely
to a function on X taking the value 1 for x ∈ Y and 0 for x ∉ Y.
In category theory[edit]
Main article: Cartesian closed category
In a Cartesian closed category, the exponential operation can be used
to raise an arbitrary object to the power of another object. This
generalizes the
x+∞ = +∞ and x−∞ = 0, when 1 < x ≤ +∞. x+∞ = 0 and x−∞ = +∞, when 0 ≤ x < 1. 0y = 0 and (+∞)y = +∞, when 0 < y ≤ +∞. 0y = +∞ and (+∞)y = 0, when −∞ ≤ y < 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 < 0, since pairs (x, y) with x < 0 are not accumulation points of D. On the other hand, when n is an integer, the power xn is already meaningful for all values of x, including negative ones. This may make the definition 0n = +∞ obtained above for negative n problematic when n is odd, since in this case xn → +∞ as x tends to 0 through positive values, but not negative ones. Efficient computation with integer exponents[edit] Computing bn using iterated multiplication requires n − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2100, note that 100 = 64 + 32 + 4. Compute the following in order: 22 = 4 (22)2 = 24 = 16 (24)2 = 28 = 256 (28)2 = 216 = 65,536 (216)2 = 232 = 4,294,967,296 (232)2 = 264 = 18,446,744,073,709,551,616 264 232 24 = 2100 = 1,267,650,600,228,229,401,496,703,205,376 This series of steps only requires 8 multiplication operations instead
of 99 (since the last product above takes 2 multiplications).
In general, the number of multiplication operations required to
compute bn can be reduced to Θ(log n) by using exponentiation by
squaring or (more generally) addition-chain exponentiation. Finding
the minimal sequence of multiplications (the minimal-length addition
chain for the exponent) for bn is a difficult problem for which no
efficient algorithms are currently known (see
x ↑ y: Algol, Commodore BASIC
x ^ y: AWK, BASIC, J, MATLAB,
Many other programming languages lack syntactic support for exponentiation, but provide library functions: pow(x, y): C, C++ Math.Pow(x, y): C# math:pow(X, Y): Erlang For certain exponents there are special ways to compute xy much faster than through generic exponentiation. These cases include small positive and negative integers (prefer x*x over x2; prefer 1/x over x−1) and roots (prefer sqrt(x) over x0.5, prefer cbrt(x) over x1/3). See also[edit]
Double exponential function Exponential decay Exponential growth List of exponential topics Modular exponentiation Scientific notation Unicode subscripts and superscripts xy=yx Zero to the power of zero References[edit] ^ a b O'Connor, John J.; Robertson, Edmund F., "Etymology of some
common mathematical terms", MacTutor History of
Earliest Known Uses of Some of the Words of Mathematics Michael Stifel, Arithmetica integra (Nuremberg ("Norimberga"), (Germany): Johannes Petreius, 1544), Liber III (Book 3), Caput III (Chapter 3): De Algorithmo numerorum Cossicorum. (On algorithms of algebra.), page 236. Stifel was trying to conveniently represent the terms of geometric progressions. He devised a cumbersome notation for doing that. On page 236, he presented the notation for the first eight terms of a geometric progression (using 1 as a base) and then he wrote: "Quemadmodum autem hic vides, quemlibet terminum progressionis cossicæ, suum habere exponentem in suo ordine (ut 1ze habet 1. 1ʓ habet 2 &c.) sic quilibet numerus cossicus, servat exponentem suæ denominationis implicite, qui ei serviat & utilis sit, potissimus in multiplicatione & divisione, ut paulo inferius dicam." (However, you see how each term of the progression has its exponent in its order (as 1ze has a 1, 1ʓ has a 2, etc.), so each number is implicitly subject to the exponent of its denomination, which [in turn] is subject to it and is useful mainly in multiplication and division, as I will mention just below.) [Note: Most of Stifel's cumbersome symbols were taken from Christoff Rudolff, who in turn took them from Leonardo Fibonacci's Liber Abaci (1202), where they served as shorthand symbols for the Latin words res/radix (x), census/zensus (x2), and cubus (x3).] ^ Quinion, Michael. "
Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest; Clifford
Stein (2001). Introduction to Algorithms (second ed.). MIT Press.
ISBN 0-262-03293-7. Online resource Archived 2007-09-30 at
the Wayback Machine.
Paul Cull; Mary Flahive; Robby Robson (2005). Difference Equations:
From Rabbits to Chaos (Undergraduate Texts in
^
External links[edit] "Introducing 0th power". PlanetMath. Laws of Exponents with deriv |