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{{short description|Mathematical operation}}{{Redirect|Exponent}}Image:Expo02.svg|thumb|315px|Graphs of {{math|1=y = bx}} for various bases b:{{legend-line|inline=yes|green solid 2px|base 10}},{{legend-line|inline=yes|red solid 2px|base e}},{{legend-line|inline=yes|blue solid 2px|base 2base 2{{Calculation results}}Exponentiation is a mathematical operation, written as {{math|b'n}}, involving two numbers, the base {{mvar|b}} and the exponent or power {{mvar|n}}. When {{mvar|n}} is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, {{math|b'n}} is the product of multiplying {{mvar|n}} bases:
b^n = underbrace{b times dots times b}_{n, textrm{times}}.
The exponent is usually shown as a superscript to the right of the base. In that case, {{math|bn}} is called "b raised to the n-th power", "b raised to the power of n", "the n-th power of b", "b to the nth", or most briefly as "b to the n".For any positive integers {{mvar|m}} and {{mvar|n}}, one has {{math|1=b'n â‹… b'm = b'n+m}}. To extend this property to non-positive integer exponents, {{math|b0}} is defined to be 1, and {{math|bâˆ’n}} with {{mvar|n}} a positive integer and {{mvar|b}} not zero is defined as {{math|{{sfrac|1|b'n}}}}. In particular, {{math|bâˆ’1}} is equal to {{math|{{sfrac|1|b}}}}, the reciprocal of {{mvar|b}}.The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.

## History of the notation

The term power was used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios.W. W. Rouse Ball, A Short Account of the History of Mathematics (1888) p. 36. Archimedes discovered and proved the law of exponents, {{math|1=10a â‹… 10b = 10a+b}}, necessary to manipulate powers of {{math|10}}.For further analysis see The Sand Reckoner.{{better source|date=July 2018}} In the 9th century, the Persian mathematician Muhammad ibn MÅ«sÄ al-KhwÄrizmÄ« used the terms mal for a square and kahb for a cube, which later Islamic mathematicians represented in mathematical notation as m and k, respectively, by the 15th century, as seen in the work of AbÅ« al-Hasan ibn AlÄ« al-QalasÄdÄ«.{{MacTutor|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}In the late 16th century, Jost BÃ¼rgi used Roman numerals for exponents.Cajori, Florian (2007). A History of Mathematical Notations; Vol I. Cosimo Classics. p. 344 {{ISBN|1-60206-684-1}}Early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La GÃ©omÃ©trie; there, the notation is introduced in Book I.RenÃ© Descartes, Discourse de la MÃ©thode ... (Leiden, (Netherlands): Jan Maire, 1637), appended book: La GÃ©omÃ©trie, book one, page 299. From page 299: " ... Et aa, ou a2, pour multiplier a par soy mesme; Et a3, pour le multiplier encore une fois par a, & ainsi a l'infini ; ... " ( ... and aa, or a2, in order to multiply a by itself; and a3, in order to multiply it once more by a, and thus to infinity ; ... )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.See:
• 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 cossicÃ¦, suum habere exponentem in suo ordine (ut 1ze habet 1. 1Ê“ habet 2 &c.) sic quilibet numerus cossicus, servat exponentem suÃ¦ 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, 1Ê“ 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).] Samuel Jeake introduced the term indices in 1696.{{MacTutor|class=Miscellaneous|id=Mathematical_notation|title=Etymology of some common mathematical terms}} In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth).WEB,weblink Zenzizenzizenzic â€“ the eighth power of a number, World Wide Words, Michael, Quinion, 2010-03-19, Biquadrate has been used to refer to the fourth power as well.
Some mathematicians (e.g., 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 {{math|ax + bxx + cx3 + d}}.Another historical synonym, involution,This definition of "involution" appears in the OED second edition, 1989, and Merriam-Webster online dictionary weblink. The most recent usage in this sense cited by the OED is from 1806. is now rare and should not be confused with its more common meaning.In 1748 Leonhard Euler wrote "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."Leonhard Euler (1748) Introduction to the Analysis of the Infinite, English version, page 75 With this introduction of transcendental functions, Euler laid the foundation for the modern introduction of natural logarithm as the inverse function for the natural exponential function, {{math|1=f(x) = ex}}.

## Terminology

The expression {{math|1=b2 = b â‹… b}} is called "the square of b" or "b squared" because the area of a square with side-length {{math|b}} is {{math|b2}}.The expression {{math|1=b3 = b â‹… b â‹… b}} is called "the cube of b" or "b cubed" because the volume of a cube with side-length {{math|b}} is {{math|b3}}.When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, {{math|1=35 = 3 â‹… 3 â‹… 3 â‹… 3 â‹… 3 = 243}}. The base {{math|3}} appears {{math|5}} times in the repeated multiplication, because the exponent is {{math|5}}. Here, {{math|3}} is the base, {{math|5}} is the exponent, and {{math|243}} is the power or, more specifically, 3 raised to the 5th power.The word "raised" is usually omitted, and sometimes "power" as well, so {{math|35}} can also be read "3 to the 5th" or "3 to the 5". Therefore, the exponentiation {{math|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".

## Integer exponents

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

### Positive exponents

Formally, powers with positive integer exponents may be defined by the initial conditionBOOK,weblink Abstract Algebra: an inquiry based approach, Jonathan K., Hodge, Steven, Schlicker, Ted, Sundstorm, 94, 2014, CRC Press, 978-1-4665-6706-1,
b^1 = b
and the recurrence relation
b^{n+1} = b^n cdot b.
From the associativity of multiplication, it follows that for any positive integers {{mvar|m}} and {{mvar|n}},
b^{m+n} = b^m cdot b^n.

### Zero exponent

Any nonzero number raised to the {{math|0}} power is {{math|1}}:BOOK,weblink Technical Shop Mathematics, Thomas, Achatz, 101, 2005, 3rd, Industrial Press, 978-0-8311-3086-2,
b^0=1.
One interpretation of such a power is as an empty product.The case of {{math|00}} is more complicated, and the choice of whether to assign it a value and what value to assign may depend on context. {{Crossref|For more details, see Zero to the power of zero.}}

### Negative exponents

The following identity holds for an arbitrary integer {{mvar|n}} and nonzero {{mvar|b}}:
b^{-n} = frac{1}{b^n}.
Raising 0 to a negative exponent is undefined, but in some circumstances, it may be interpreted as infinity ({{math|âˆž}}).The identity above may be derived through a definition aimed at extending the range of exponents to negative integers.For non-zero {{mvar|b}} and positive {{mvar|n}}, the recurrence relation above can be rewritten as
b^n = frac{b^{n+1}}{b}, quad n ge 1 .
By defining this relation as valid for all integer {{mvar|n}} and nonzero {{mvar|b}}, it follows that
begin{align}
b^0 &= frac{b^1}{b} = 1, [3pt]
b^{-1} &= frac{b^0}{b} = frac{1}{b},
end{align}and more generally for any nonzero {{mvar|b}} and any nonnegative integer {{mvar|n}},
b^{-n} = frac{1}{b^n}.
This is then readily shown to be true for every integer {{mvar|n}}.

### Identities and properties

The following identities hold for all integer exponents, provided that the base is non-zero:
begin{align}
b^{m + n} &= b^m cdot b^n
left(b^mright)^n &= b^{m cdot n}
(b cdot c)^n &= b^n cdot c^n
end{align}Unlike addition and multiplication:{{bulleted list|0^5 = 0|none|1^4 = 1|(1,1,1,1)|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)|3^2 = 9|(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)|4^1 = 4|(1),(2),(3),(4)|5^0 = 1|()
commutative. For example, {{math>1=23 = 8 â‰  32 = 9}}.associative. For example, {{math>1=(23)4 = 84 {{=}} 4096}}, whereas {{math2417851639229258349412352}}}}. Without parentheses, the conventional order of operations in superscript notation is top-down (or right-associative), not bottom-upJOURNAL
, A report on primes of the form k Â· 2n + 1 and on factors of Fermat numbers
, Raphael M. Robinson
, Proc. Amer. Math. Soc.
, 9
, 5
, 677
, 1958
,weblink
, 10.1090/s0002-9939-1958-0096614-7
, (or left-associative). That is,
b^{p^q} = b^{left(p^qright)},
which, in general, is different from
left(b^pright)^q = b^{p q} .}}

### Combinatorial interpretation

{{see also|#Exponentiation over sets|l1=Exponentiation over sets}}For nonnegative integers {{mvar|n}} and {{mvar|m}}, the value of {{math|nm}} is the number of functions from a set of {{mvar|m}} elements to a set of {{mvar|n}} elements (see cardinal exponentiation). Such functions can be represented as {{mvar|m}}-tuples from an {{mvar|n}}-element set (or as {{mvar|m}}-letter words from an {{mvar|n}}-letter alphabet). Some examples for particular values of {{mvar|m}} and {{mvar|n}} are given in the following table:
{| class="wikitable"
!{{math|nm}}!The {{math|nm}} possible {{mvar|m}}-tuples of elements from the set {{math|{{mset|1, ..., n}}}}

### Particular bases

#### {{anchor|Base 10}}Powers of ten

{{see also|Scientific notation}}In the base ten (decimal) number system, integer powers of {{math|10}} are written as the digit {{math|1}} followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, {{math|1={{val|e=3}} = {{val|1000}}}} and {{math|1={{val|e=-4}} = {{val|0.0001}}}}.Exponentiation with base {{math|10}} is used in scientific notation to denote large or small numbers. For instance, {{val|299792458|u=m/s}} (the speed of light in vacuum, in metres per second) can be written as {{val|2.99792458|e=8|u=m/s}} and then approximated as {{val|2.998|e=8|u=m/s}}.SI prefixes based on powers of {{math|10}} are also used to describe small or large quantities. For example, the prefix kilo means {{math|1={{val|e=3}} = {{val|1000}}}}, so a kilometre is {{val|1000|u=metres}}.

#### {{anchor|Base 2}}Powers of two

The first negative powers of {{math|2}} are commonly used, and have special names, e.g.: half and quarter.Powers of {{math|2}} appear in set theory, since a set with {{math|n}} members has a power set, the set of all of its subsets, which has {{math|2n}} members.Integer powers of {{math|2}} are important in computer science. The positive integer powers {{math|2n}} give the number of possible values for an {{math|n}}-bit integer binary number; for example, a byte may take {{math|1=28 = 256}} different values. The binary number system expresses any number as a sum of powers of {{math|2}}, and denotes it as a sequence of {{math|0}} and {{math|1}}, separated by a binary point, where {{math|1}} indicates a power of {{math|2}} that appears in the sum; the exponent is determined by the place of this {{math|1}}: the nonnegative exponents are the rank of the {{math|1}} on the left of the point (starting from {{math|0}}), and the negative exponents are determined by the rank on the right of the point.

#### Powers of one

The powers of one are all one: {{math|1=1n = 1}}.

#### Powers of zero

If the exponent {{mvar|n}} is positive ({{math|n > 0}}), the {{mvar|n}}th power of zero is zero: {{math|1=0n = 0}}.If the exponent {{mvar|n}} is negative ({{math|n < 0}}), the {{mvar|n}}th power of zero {{math|0n}} is undefined, because it must equal 1/0^{-n} with {{math|-n > 0}}, and this would be 1/0 according to above.The expression {{math|00}} is either defined as 1, or it is left undefined (see Zero to the power of zero).

#### Powers of negative one

If {{math|n}} is an even integer, then {{math|1=(âˆ’1)n = 1}}.If {{math|n}} is an odd integer, then {{math|1=(âˆ’1)n = âˆ’1}}.Because of this, powers of {{math|âˆ’1}} are useful for expressing alternating sequences. For a similar discussion of powers of the complex number {{math|i}}, see {{section link||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:
{{math|bn â†’ âˆž}} as {{math|n â†’ âˆž}} when {{math|b > 1}}
This can be read as "b to the power of n tends to +âˆž as n tends to infinity when b is greater than one".Powers of a number with absolute value less than one tend to zero:
{{math|bn â†’ 0}} as {{math|n â†’ âˆž}} when {{math|{{abs|b}} < 1}}
Any power of one is always one:
{{math|1=bn = 1}} for all {{math|n}} if {{math|1=b = 1}}
Powers of {{math|â€“1}} alternate between {{math|1}} and {{math|â€“1}} as {{math|n}} alternates between even and odd, and thus do not tend to any limit as {{math|n}} grows.If {{math|b < â€“1}}, {{math|1=bn}}, alternates between larger and larger positive and negative numbers as {{math|n}} alternates between even and odd, and thus does not tend to any limit as {{math|n}} grows.If the exponentiated number varies while tending to {{math|1}} as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is
{{math|(1 + 1/n)n â†’ e}} as {{math|n â†’ âˆž}}
See {{section link||The exponential function}} below.Other limits, in particular those of expressions that take on an indeterminate form, are described in {{section link||Limits of powers}} below.

### Power functions

(File:Potenssi 1 3 5.png|thumb|left|Power functions for n=1,3,5 )(File:Potenssi 2 4 6.png|thumb|Power functions for n=2,4,6)Real functions of the form f(x) = cx^n, where c ne 0, are sometimes called power functions.{{cn|date=November 2017}} When n is an integer and n ge 1, two primary families exist: for n even, and for n odd. In general for c > 0, when n is even f(x) = cx^n will tend towards positive infinity with increasing x, and also towards positive infinity with decreasing x. All graphs from the family of even power functions have the general shape of y=cx^2, flattening more in the middle as n increases.BOOK, Anton, Howard, Bivens, Irl, Davis, Stephen, Calculus: Early Transcendentals, John Wiley & Sons, 28, 9th, Functions with this kind of symmetry {{nobr|(f(-x)= f(x))}} are called even functions.When n is odd, f(x)'s asymptotic behavior reverses from positive x to negative x. For c > 0, f(x) = cx^n will also tend towards positive infinity with increasing x, but towards negative infinity with decreasing x. All graphs from the family of odd power functions have the general shape of y=cx^3, flattening more in the middle as n increases and losing all flatness there in the straight line for n=1. Functions with this kind of symmetry {{nobr|(f(-x)= -f(x))}} are called odd functions.For c < 0, the opposite asymptotic behavior is true in each case.BOOK, Anton, Howard, Bivens, Irl, Davis, Stephen, Calculus: Early Transcendentals, John Wiley & Sons, 28, 9th,

### List of whole-number powers{|class"wikitable" style"text-align:right"

!n !!n2 !!n3 !!n4 !!n5 !!n6 !!n7 !!n8 !!n9 !!n102 width="8%"8 width="8%"32 width="10%"128 width="12%"512 width="14%"|1,0243 >|59,0494 >|1,048,5765 >|9,765,6256 >|60,466,1767 >|282,475,2498 >|1,073,741,8249 >|3,486,784,40110>|10,000,000,000

## Rational exponents

right|thumb|300px|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 {{math|1=xn = b}}.If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to {{math|1=xn = b}}. This solution is called the principal nth root of b. It is denoted {{radic|b|n}}, where {{radic|  }} is the radical symbol; alternatively, the principal root may be written b1/n. For example: {{math|1=91/2 = {{radic|9}} = 3}} and {{math|1=81/3 = {{radic|8|3}} = 2}}.The fact that x = b^frac{1}{n} solves x^n = b follows from noting that
begin{align}
x^n &= left(b^frac{1}{n}right)^n
= underbrace{b^frac{1}{n} times b^frac{1}{n} times cdots times b^frac{1}{n}}_{n , textrm{times}}
&= b^{underbrace{left(frac{1}{n} + frac{1}{n} + cdots + frac{1}{n}right)}_{n , textrm{times}}}
= b^frac{n}{n} = b^1 = b.
end{align}If n is even and b is positive, then {{math|1=xn = b}} has two real solutions, which are the positive and negative nth roots of b, that is, {{math|b1/n > 0}} and {{math|âˆ’(b1/n) < 0.}} If n is even and b is negative, the equation has no solution in real numbers.If n is odd, then {{math|1=xn = b}} has exactly one real solution, which is positive if b is positive ({{math|b1/n > 0}}) and negative if b is negative ({{math|b1/n < 0}}).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^frac{u}{v} = left(b^uright)^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 b'u is positive; and yields a negative real result, if u and v are both odd, because then b'u is negative. The case of even v (and, hence, odd u) cannot be treated this way within the reals, since there is no real number x such that {{math|1=x2k = âˆ’1}}, the value of bu/v in this case must use the imaginary unit i, as described more fully in the section Â§ Powers of complex numbers.Thus we have {{math|1=(âˆ’27)1/3 = âˆ’3}} and {{math|1=(âˆ’27)2/3 = 9}}. The number 4 has two 3/2 powers, namely 8 and âˆ’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 v/u-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)^{left(left(frac{2}{3}right)left(frac{3}{2}right)right)} = left((-27)^frac{2}{3}right)^frac{3}{2} = 9^frac{3}{2} = 27
is clearly wrong. The problem starts already in the first equality by introducing a standard notation for an inherently ambiguous situation â€“asking for an even rootâ€“ 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:
left((-27)^frac{2}{3}right)^frac{3}{2} = sqrt{left(sqrt{(-27)^2}right)^3} = sqrt{(-27)^{2}} ne -27
instead of
left((-27)^frac{2}{3}right)^frac{3}{2} = -sqrt{left(sqrt{(-27)^2}right)^3} = -sqrt{(-27)^2} = -27.
In general the same sort of problems occur for complex numbers as described in the section {{section link||Failure of power and logarithm identities}}.

## Real exponents

Exponentiation to real powers of positive real numbers can be defined either by extending the rational powers to reals by continuity, or more usually as given in {{section link||Powers via logarithms}} below. The result is always a positive real number, and the identities and properties shown above for integer exponents are true for positive real bases with non-integer exponents as well. 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 {{section link||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 an identity such as
left(b^rright)^s = b^{rcdot s}
is true; see {{section link||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

(File:Continuity of the Exponential at 0.svg|thumb|Because the exponential function is continuous we find lim_{ntoinfty} e^{x_n} = e^{lim_{ntoinfty} x_n} for convergent sequences (xn). This is shown here for xn = {{sfrac|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 ruleBOOK, Elements of Real Analysis, Denlinger, Charles G., Jones and Bartlett, 2011, 278â€“283, 978-0-7637-7947-4,
b^x = lim_{r (inmathbb{Q})to x} b^rquad(b inmathbb{R}^+,,xinmathbb{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 (ÎµÎ´)-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 {{mvar|x}} so all the rational powers in the interval are within the desired accuracy.For example, if {{math|1=x = Ï€}}, the nonterminating decimal representation {{math|1=Ï€ = 3.14159â€¦}} can be used (based on strict monotonicity of the rational power) to obtain the intervals bounded by rational powers
left[b^3, b^4right], 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], â€¦
The bounded intervals converge to a unique real number, denoted by b^pi. This technique can be used to obtain the power of a positive real number b for any irrational exponent. The function {{math|1=f'b(x) = b'x}} is thus defined for any real number x.

### The exponential function

The important mathematical constant {{mvar|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 e'x 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 rightarrow infty} left(1+frac x n right)^n
Among other properties, exp satisfies the exponential identity
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 e'x for integer x, and it also follows that rational powers denote (positive) roots as usual, so exp(x) coincides with the e'x definitions in the previous section for all real x by continuity.

### Powers via logarithms

When {{math|e'x}} is defined as the exponential function, {{math|b'x}} can be defined, for other positive real numbers {{math|b}}, in terms of {{math|e'x}}. Specifically, the natural logarithm {{math|ln(x)}} is the inverse of the exponential function {{math|e'x}}. It is defined for {{math|b > 0}}, and satisfies
b = e^{ln b}
If bx is to preserve the logarithm and exponent rules, then one must have
b^x = left(e^{ln b}right)^x = e^{x cdotln b}
for each real number {{math|x}}.This can be used as an alternative definition of the real number power {{math|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

Powers of a positive real number are always positive real numbers. The solution of x2 = 4, however, can be either 2 or âˆ’2. The principal value of 41/2 is 2, but âˆ’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 b'r as a real number for a negative real number b and an arbitrary real number r. Indeed, e'r is positive for every real number r, so ln(b) is not defined as a real number for {{math|b â‰¤ 0}}.The rational exponent method cannot be used for negative values of b because it relies on continuity. The function {{math|1=f(r) = br}} has a unique continuous extension from the rational numbers to the real numbers for each {{math|b > 0}}. But when {{math|b < 0}}, the function f is not even continuous on the set of rational numbers r for which it is defined.For example, consider {{math|1=b = âˆ’1}}. The nth root of âˆ’1 is âˆ’1 for every odd natural number n. So if n is an odd positive integer, {{math|1=(âˆ’1)(m/n) = âˆ’1}} if m is odd, and {{math|1=(âˆ’1)(m/n) = 1}} if m is even. Thus the set of rational numbers q for which {{math|1=(âˆ’1)q = 1}} is dense in the rational numbers, as is the set of q for which {{math|1=(âˆ’1)q = âˆ’1}}. This means that the function (âˆ’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

If b is a positive real algebraic number, and x is a rational number, it has been shown above that b'x is an algebraic number. This remains true even if one accepts any algebraic number for b, with the only difference that b'x may take several values (a finite number, see below), which are all algebraic. The Gelfondâ€“Schneider theorem provides some information on the nature of bx when x is irrational (that is, not rational). It states:

## Complex exponents with a positive real base

If {{mvar|b}} is a positive real number, and {{mvar|z}} is any complex number, the power {{math|bz}} is defined as
b^z = left(e^{ln b}right)^z = e^{(zln b)},
where {{math|x {{=}} ln(b)}} is the unique real solution to the equation {{math|ex {{=}} b}}, and the complex power of {{math|e}} is defined by the exponential function, which is the unique function of a complex variable that is equal to its derivative and takes the value 1 for {{math|1=x = 0}}.As, in general, {{math|b'z}} is not a real number, an expression such as {{math|(b'z)w}} is not defined by the previous definition. It must be interpreted via the rules for powers of complex numbers, and, unless {{mvar|z}} is real or {{mvar|w}} is integer, does not generally equal {{math|bzw}}, as one might expect.There are various definitions of the exponential function but they extend compatibly to complex numbers and satisfy the exponential property. For any complex numbers z and w, the exponential function satisfies e^{z+w} = e^z e^w. In particular, for any complex number z = x+iy
e^z = e^{x+iy} = e^x cdot e^{iy},
The second term e^{iy} has a value given by Euler's formula
e^{iy} = cos y + i sin y.
This formula links problems in trigonometry and algebra.Therefore, for any complex number z = x+iy,
e^z = e^{x+iy} = e^x cdot e^{iy} = e^x (cos y + i sin y).
Because of the Pythagorean trigonometric identity, the absolute value of cos y + i sin y is {{math|1}}. Therefore the real factor e^x is the absolute value of e^z and the imaginary part y of the exponent identifies the argument (angle) of the complex number e^z.

### Series definition

The exponential function being equal to its derivative and satisfying e^0=1, its Taylor series must be
e^z = sum_{n=0}^infty {z^n over n!} = 1 + z + frac{z^2}{2!} + frac{z^3}{3!} + frac{z^4}{4!} + cdots.
This infinite series, which is often taken as the definition of the exponential function {{math|ez}} for arbitrary complex exponents, is absolutely convergent for all complex numbers {{mvar|z.}}When z is purely imaginary, that is, z = iy for a real number y, the series above becomes
e^{iy} = 1 + iy + frac{(iy)^2}{2!} + frac{(iy)^3}{3!} + frac{(iy)^4}{4!} + cdots,
which (because it converges absolutely) may be reordered to
e^{iy} = left(1 - frac{y^2}{2!} + frac{y^4}{4!} - frac{y^6}{6!} + cdotsright) + i left(y - frac{y^3}{3!} + frac{y^5}{5!} - cdotsright).
The real and the imaginary parts of this expression are Taylor expansions of cosine and sine, respectively, centered at zero, implying Euler's formula:
e^{iy} = cos y + i sin y.

### Limit definition

missing image!
- ExpIPi.gif -
This animation shows by repeated multiplications in the complex plane for values of {{mvar|n}} (denoted as {{math|N}} in the picture), increasing from {{math|1}} to {{math|100,}} how (1 + ipi/n)^n approaches {{math|-1.
The values of (1 + ipi/n)^k, for k = 0 ... n, are the vertices of a polygonal path whose leftmost endpoint is (1 + ipi/k)^k for the actual {{mvar|k.}} It can be seen that as {{mvar|k}} gets larger (1 + ipi/k)^k approaches the limit {{math|âˆ’1,}} illustrating (Euler's identity]]: e^{ipi} = -1.)Another characterization of the exponential function e^z is as the limit of (1 + z/n)^n, as n approaches infinity. By thinking of the nth power in this definition as repeated multiplication in polar form, it can be used to visually illustrate Euler's formula. Any complex number can be represented in polar form as (r,theta), where r is the absolute value and Î¸ is its argument. The product of two complex numbers left(r_1, theta_1right) and left(r_2, theta_2right) is left(r_1 r_2, theta_1 + theta_2right).Consider the right triangle in the complex plane which has 0, 1, and 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 form (1, x/n). So, in the limit as n approaches infinity, (1 + ix/n)^n approaches (1, x/n)^n = left(1^n, nx/nright) = (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 + i sin x; this is âˆ’againâˆ’ Euler's formula, allowing for the same connections to the trigonometric functions as elaborated with the series definition.

### Periodicity

The solutions to the equation e^z = 1 are the integer multiples of 2 pi i:
left{ z : e^z = 1 right} = { 2kpi i : k in mathbb{Z} }
Thus, if v is a complex number such that e^v = w, then every z that also satisfies e^z = w can be obtained from e^z = e^vcdot 1 = e^{v+i2kpi}, i.e., by adding an arbitrary integer multiple of 2 pi i to v:
left{ z : e^z = w right} = { v + 2kpi i : k in mathbb{Z} }
That is, the complex exponential function e^z = exp(z) = exp(z + 2kpi i) for any integer {{mvar|k}} is a periodic function with period 2 pi i.

### Examples

begin{align}
2^i &= e^{i ln(2)} = cos(ln(2)) + i sin(ln(2)) approx 0.76924 + 0.63896 i
e^i &= cos(1) + isin(1) approx 0.54030 + 0.84147 i
left(e^{2pi}right)^i &= {(535.49165dots)}^i = e^{ilnleft(e^{2pi}right)} = e^{i(2pi)} = cos(2pi) + isin(2pi)= 1.
end{align}

## Powers of complex numbers

Integer powers of nonzero complex numbers are defined by repeated multiplication or division as above. If i is the imaginary unit and n is an integer, then in equals 1, i, âˆ’1, or âˆ’i, according to whether the integer n is congruent to 0, 1, 2, or 3 modulo 4. Because of this, the powers of i are useful for expressing sequences of period 4.Complex powers of positive reals are defined via ex as in section Complex exponents with positive real bases above. These are continuous functions.Trying to extend these functions to the general case of noninteger powers of complex numbers that are not positive reals leads to difficulties. Either we define discontinuous functions or multivalued functions. Neither of these options is entirely satisfactory.The rational power of a complex number must be the solution to an algebraic equation. Therefore, it always has a finite number of possible values. For example, {{math|1=w = z1/2}} must be a solution to the equation {{math|1=w2 = z}}. But if w is a solution, then so is âˆ’w, because {{math|1=(âˆ’1)2 = 1}}. A unique but somewhat arbitrary solution called the principal value can be chosen using a general rule which also applies for nonrational powers.Complex powers and logarithms are more naturally handled as single valued functions on a Riemann surface. Single valued versions are defined by choosing a sheet. The value has a discontinuity along a branch cut. Choosing one out of many solutions as the principal value leaves us with functions that are not continuous, and the usual rules for manipulating powers can lead us astray.Any nonrational power of a complex number has an infinite number of possible values because of the multi-valued nature of the complex logarithm. The principal value is a single value chosen from these by a rule which, amongst its other properties, ensures powers of complex numbers with a positive real part and zero imaginary part give the same value as does the rule defined above for the corresponding real base.Exponentiating a real number to a complex power is formally a different operation from that for the corresponding complex number. However, in the common case of a positive real number the principal value is the same.The powers of negative real numbers are not always defined and are discontinuous even where defined. In fact, they are only defined when the exponent is a rational number with the denominator being an odd integer. When dealing with complex numbers the complex number operation is normally used instead.

### Complex exponents with complex bases

For complex numbers w and z with {{math|w â‰  0}}, the notation wz is ambiguous in the same sense that log w is.To obtain a value of wz, first choose a logarithm of w; call it {{math|log w}}. Such a choice may be the principal value {{math|Log w}} (the default, if no other specification is given), or perhaps a value given by some other branch of log w fixed in advance. Then, using the complex exponential function one defines
w^z = e^{z log w}
because this agrees with the earlier definition in the case where w is a positive real number and the (real) principal value of {{math|log w}} is used.If z is an integer, then the value of wz is independent of the choice of {{math|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 {{math|z > 0}}, then the countably infinitely many choices of {{math|log w}} yield only n different values for w'z; these values are the n complex solutions s to the equation {{math|1=s'n = wm}}.If z is an irrational number, then the countably infinitely many choices of {{math|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

(File:One3Root.svg|thumb|right|The three 3rd roots of 1)A complex number w such that {{math|1=w'n = 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 {{math|1=w'n = 1}} but {{math|w'k â‰  1}} for all natural numbers k such that {{math|0 < k < n}}, then w is called a primitive nth root of unity. The negative unit âˆ’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 âˆ’i.The number e{{sfrac|2Ï€i|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 {{radic|1|n}}, which is 1.This definition of a principal root of unity can be found in:
• BOOK, Introduction to Algorithms, second, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein, MIT Press, 2001, 978-0-262-03293-3, Online resource {{webarchive|url=https://web.archive.org/web/20070930201902weblink |date=2007-09-30 }}
• BOOK, Difference Equations: From Rabbits to Chaos, Undergraduate Texts in Mathematics, Paul Cull, Mary Flahive, Mary Flahive, Robby Robson, 2005, Springer, 978-0-387-23234-8, Defined on p. 351
• "Principal root of unity", MathWorld.)
The other nth roots of unity are given by
left( e^{ frac{2}{n} pi i } right) ^k = e^{ frac{2}{n} pi i k }
for {{math|2 â‰¤ k â‰¤ n}}.

### Roots of arbitrary complex numbers

Although there are infinitely many possible values for a general complex logarithm, there are only a finite number of values for the power {{math|wq}} in the important special case where {{math|1=q = 1/n}} and {{math|n}} is a positive integer. These are the {{math|n}}th roots of {{math|w}}; they are solutions of the equation {{math|1=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 {{math|w1/n}} as the principal value of the root, which is, conventionally, the {{math|n}}th root whose argument has the smallest absolute value. When {{math|w}} is a positive real number, this is coherent with the usual convention of defining {{math|w1/n}} as the unique positive real {{math|n}}th root. On the other hand, when {{math|w}} is a negative real number, and {{mvar|n}} is an odd integer, the unique real {{math|n}}th root is not one of the two {{math|n}}th roots whose argument has the smallest absolute value. In this case, the meaning of {{math|w1/n}} may depend on the context, and some care may be needed for avoiding errors.The set of {{math|n}}th roots of a complex number {{math|w}} is obtained by multiplying the principal value {{math|w1/n}} by each of the {{math|n}}th roots of unity. For example, the fourth roots of 16 are 2, âˆ’2, 2{{math|i}}, and âˆ’2{{math|i}}, because the principal value of the fourth root of 16 is 2 and the fourth roots of unity are 1, âˆ’1, {{math|i}}, and âˆ’{{math|i}}.

### Computing complex powers

It is often easier to compute complex powers by writing the number to be exponentiated in polar form. Every complex number z can be written in the polar form
z = re^{itheta} = e^{log(r) + itheta}
where r is a nonnegative real number and Î¸ is the (real) argument of z. The polar form has a simple geometric interpretation: if a complex number {{math|u + iv}} is thought of as representing a point {{math|(u, v)}} in the complex plane using Cartesian coordinates, then {{math|(r, Î¸)}} is the same point in polar coordinates. That is, r is the "radius" {{math|1=r2 = u2 + v2}} and Î¸ is the "angle" {{math|1=Î¸ = atan2(v, u)}}. The polar angle Î¸ is ambiguous since any integer multiple of 2Ï€ could be added to Î¸ without changing the location of the point. Each choice of Î¸ 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 Î¸ chosen in the interval {{open-closed|âˆ’Ï€, Ï€}}. 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^{itheta}
Then
log(w) = log(r) + i theta
and thus
w^z = e^{z log(w)} = e^{z(log(r) + itheta)}
If z is decomposed as {{math|c + di}}, then the formula for wz can be written more explicitly as
left( r^c e^{-dtheta} right) e^{i (d log(r) + ctheta)} = left( r^c e^{-dtheta} right) left[ cos(d log(r) + ctheta) + i sin(d log(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 Î¸ to be in the interval {{math|(âˆ’Ï€, Ï€]}}. To compute ii, write i in polar and Cartesian forms:
begin{align}
i &= 1 cdot e^{frac{1}{2} i pi}
i &= 0 + 1i
end{align}Then the formula above, with {{math|1=r = 1}}, {{math|1=Î¸ = {{sfrac|Ï€|2}}}}, {{math|1=c = 0}}, and {{math|1=d = 1}}, yields:
i^i = left( 1^0 e^{-frac{1}{2}pi} right) e^{i left[1 cdot log(1) + 0 cdot frac{1}{2}pi right]} = e^{-frac{1}{2}pi} approx 0.2079
Similarly, to find {{math|(âˆ’2)3 + 4i}}, compute the polar form of âˆ’2,
-2 = 2e^{i pi}
and use the formula above to compute
(-2)^{3 + 4i} = left( 2^3 e^{-4pi} right) e^{i[4log(2) + 3pi]} approx (2.602 - 1.006 i) cdot 10^{-5}
The value of a complex power depends on the branch used. For example, if the polar form {{math|1=i = 1e5Ï€i/2}} is used to compute i'i, the power is found to be eâˆ’5Ï€/2; the principal value of i'i, computed above, is eâˆ’Ï€/2. The set of all possible values for i'i is given by:Complex number to a complex power may be real at Cut The Knot gives some references to i'i
begin{align}
i &= 1 cdot e^{frac{1}{2} ipi + i 2 pi k} mid k isin mathbb{Z}
i^i &= e^{i left(frac{1}{2} ipi + i 2 pi kright)}
&= e^{-left(frac{1}{2} pi + 2 pi kright)}
end{align}So there is an infinity of values which are possible candidates for the value of i'i, one for each integer k. All of them have a zero imaginary part so one can say i'i has an infinity of valid real values.

### 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:{{bulleted list1=log(bx) = x â‹… logâ€‰b}} holds whenever {{mvarx}} is a real number. But for the principal branch of the complex logarithm one has
ipi = log(-1) = logleft[(-i)^2right] 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) equiv z cdot log(w) pmod{2 pi i}
This identity does not hold even when considering log as a multivalued function. The possible values of {{math|log(wz)}} contain those of {{math|z â‹… logâ€‰w}} as a subset. Using {{math|Log(w)}} for the principal value of {{math|log(w)}} and {{mvar|m}}, {{mvar|n}} as any integers the possible values of both sides are:
begin{align}
left{log(w^z)right} &= left{ z cdot operatorname{Log}(w) + z cdot 2 pi i n + 2 pi i m right}
left{z cdot log(w)right} &= left{ z cdot operatorname{Log}(w) + z cdot 2 pi i n right}
end{align}
1=(bc)x = b'x'cx}} and {{mathb/c)x = b'x/c'x}} are valid when {{mvar>b}} and {{mvarx}} is a real number. But a calculation using principal branches shows that
1 = (-1 cdot -1)^frac{1}{2} not = (-1)^frac{1}{2}(-1)^frac{1}{2} = -1
and
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 {{mvar|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 {{math|(âˆ’1 â‹… âˆ’1)1/2}} are {{math|{1, âˆ’1}}}. The identity holds, but saying {{math|1={1} = {(âˆ’1 â‹… âˆ’1)1/2}}} is wrong.1=(e'x)y = e'xy}} holds for real numbers {{mvary}}, but assuming its truth for complex numbers leads to the following Mathematical fallacy, discovered in 1827 by Thomas Clausen (mathematician)>Clausen:STEINER J, CLAUSEN T, ABEL NH >TITLE=AUFGABEN UND LEHRSÃ¤TZE, ERSTERE AUFZULÃ¶SEN, LETZTERE ZU BEWEISEN URL=HTTP://GDZ.SUB.UNI-GOETTINGEN.DE/NO_CACHE/DMS/LOAD/IMG/?IDDOC=270662 CRELLE'S JOURNAL>JOURNAL FÃ¼R DIE REINE UND ANGEWANDTE MATHEMATIK YEAR=1827 date=August 2019 fix-attempted=yes }}For any integer {{mvar|n}}, we have:
1. e^{1 + 2 pi i n} = e^1 e^{2 pi i n} = e cdot 1 = e
2. left(e^{1 + 2pi i n}right)^{1 + 2 pi i n} = eqquad (taking the (1 + 2 pi i n)-th power of both sides)
3. e^{1 + 4 pi i n - 4 pi^2 n^2} = eqquad (using left(e^xright)^y = e^{xy} and expanding the exponent)
4. e^1 e^{4 pi i n} e^{-4 pi^2 n^2} = eqquad (using e^{x+y} = e^x e^y)
5. e^{-4 pi^2 n^2} = 1qquad (dividing by {{mvar|e}})
but this is false when the integer {{mvar|n}} is nonzero.The error is the following: by definition, e^y is a notation for exp(y), a true function, and x^y is a notation for exp(ylog x), which is a multi-valued function. Thus the notation is ambiguous when {{math|1=x = e}}. Here, before expanding the exponent, the second line should be
expleft((1 + 2pi i n)log exp(1 + 2pi i n)right) = exp(1 + 2pi i n).
Therefore, when expanding the exponent, one has implicitly supposed that log exp z =z for complex values of {{mvar|z}}, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity {{math|1=(e'x)y = e'xy}} must be replaced by the identity
left(e^xright)^y = e^{ylog e^x},
which is a true identity between multivalued functions.}}

## Generalizations

### Monoids

Exponentiation with integer exponents can be defined in any multiplicative monoid.BOOK, Nicolas Bourbaki, AlgÃ¨bre, 1970, Springer, , I.2 A monoid is an algebraic structure consisting of a set X together with a rule for composition ("multiplication") satisfying an associative law and a multiplicative identity, denoted by 1. Exponentiation is defined inductively by:
• x^0=1 for all xin X
• x^{n+1}=x^nx for all xin X and non-negative integers n
• If n is a negative integer then x^n is only definedBOOK, David M. Bloom, Linear Algebra and Geometry, 1979, 978-0-521-29324-2, 45, if 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

If A is a square matrix, then the product of A with itself n times is called the matrix power. Also A^0 is defined to be the identity matrix,Chapter 1, Elementary Linear Algebra, 8E, Howard Anton and if A is invertible, then 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.{{citation|first=Gilbert|last=Strang|title=Linear algebra and its applications|publisher=Brooks-Cole|year=1988|edition=3rd}}, Chapter 5. This is the standard interpretation of a Markov chain, for example. Then A^2x is the state of the system after two time steps, and so forth: A^nx is the state of the system after n time steps. The matrix power 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/dx, which is a linear operator acting on functions f(x) to give a new function (d/dx)f(x) = f'(x). The n-th power of the differentiation operator is the n-th derivative:
left(frac{d}{dx}right)^nf(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.E Hille, R S Phillips: Functional Analysis and Semi-Groups. American Mathematical Society, 1975. 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, SchrÃ¶dinger equation, wave equation, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.

### Finite fields

A field is an algebraic structure in which multiplication, addition, subtraction, and division are 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} with addition defined by 0+1=1+0=1 and 0+0=1+1=0, and multiplication 0cdot 0=1cdot 0 = 0cdot 1=0 and 1cdot 1=1.Exponentiation in finite fields has applications in public key cryptography. For example, the Diffieâ€“Hellman key exchange uses the fact that exponentiation is computationally inexpensive in finite fields, whereas the discrete logarithm (the inverse of exponentiation) is computationally expensive.Any finite field F has the property that there is a unique prime number p such that px=0 for all x in F; that is, x added to itself p times is zero. For example, in F_2, the prime number {{math|1=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 that raises each element of F to the power p. This is called the Frobenius automorphism of F. It is an automorphism of the field because of the Freshman's dream identity (x+y)^p = x^p+y^p. The Frobenius automorphism is important in number theory because it generates the Galois group of F over its prime subfield.

### In abstract algebra

Exponentiation for integer exponents can be defined for quite general structures in abstract algebra.Let X be a set with a power-associative binary operation which is written multiplicatively. Then xn is defined for any element x of X and any nonzero natural number n as the product of n copies of x, which is recursively defined by
begin{align}
x^1 &= x
x^n &= x^{n-1}x quadhbox{for }n>1
end{align}One has the following properties
begin{align}
left(x^i x^jright) x^k &= x^i left(x^j x^kright) & &text{(power-associative property)}
x^{m+n} &= x^m x^n
left(x^mright)^n &= x^{mn}
end{align}If the operation has a two-sided identity element 1, then x0 is defined to be equal to 1 for any x.
begin{align}
x1 &= 1x = x & &text{(two-sided identity)}
x^0 &= 1
end{align}{{citation needed|date=April 2014}}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 xâˆ’1 and follows all the usual rules for exponents.
begin{align}
x x^{-1} &= x^{-1} x = 1 & &text{(two-sided inverse)}
(x y) z &= x (y z) & &text{(associative)}
x^{-n} &= left(x^{-1}right)^n
x^{m-n} &= x^m x^{-n}
end{align}If the multiplication operation is commutative (as for instance in abelian groups), then the following holds:
(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, xâˆ—n is {{math|x âˆ— ... âˆ— x}}, while x#n is {{math|x # ... # x}}, whatever the operations âˆ— and # might be.Superscript notation is also used, especially in group theory, to indicate conjugation. That is, {{math|1=gh = hâˆ’1gh}}, where g and h are elements of some group. Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role.

### {{Anchor|Exponentiation over sets}}Over sets

If n is a natural number and A is an arbitrary set, the expression A'n is often used to denote the set of ordered n-tuples of elements of A. This is equivalent to letting A'n denote the set of functions from the set {{math|{0, 1, 2, ..., nâˆ’1}{{null}}}} to the set A; the n-tuple {{math|(a0, a1, a2, ..., anâˆ’1)}} represents the function that sends i to ai.For an infinite cardinal number Îº and a set A, the notation AÎº is also used to denote the set of all functions from a set of size Îº to A. This is sometimes written ÎºA to distinguish it from cardinal exponentiation, defined below.This generalized exponential can also be defined for operations on sets or for sets with extra structure. For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of
bigoplus_{i in mathbb{N}} V_{i}
where each Vi is a vector space.Then if V'i = V for each i, the resulting direct sum can be written in exponential notation as VâŠ•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 V'n, 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 Cartesian product unless otherwise stated. Since multiple Cartesian products produce an n-tuple, which can be represented by a function on a set of appropriate cardinality, SN becomes simply the set of all functions from N to S in this case:
S^N equiv { fcolon N to S }
This fits in with the exponentiation of cardinal numbers, in the sense that {{math|1={{abs|S'N}} = {{abs|S}}{{abs|N}}}}, where {{abs|X}} is the cardinality of X. When "2" is defined as {{math|{0, 1},}} we have {{math|1={{abs|2X}} = 2{{abs|X}}}}, 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 {{math|x âˆˆ Y}} and 0 for {{math|x âˆ‰ Y''}}.

### In category theory

In a Cartesian closed category, the exponential operation can be used to raise an arbitrary object to the power of another object. This generalizes the Cartesian product in the category of sets. If 0 is an initial object in a Cartesian closed category, then the exponential object 00 is isomorphic to any terminal object 1.

### Of cardinal and ordinal numbers

In set theory, there are exponential operations for cardinal and ordinal numbers.If Îº and Î» are cardinal numbers, the expression ÎºÎ» represents the cardinality of the set of functions from any set of cardinality Î» to any set of cardinality Îº.N. Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.Â§3.5. If Îº and Î» are finite, then this agrees with the ordinary arithmetic exponential operation. For example, the set of 3-tuples of elements from a 2-element set has cardinality {{math|1=8 = 23}}. In cardinal arithmetic, Îº0 is always 1 (even if Îº is an infinite cardinal or zero).Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.

## Repeated exponentiation

Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation. This sequence of operations is expressed by the Ackermann function and Knuth's up-arrow notation. Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at {{math|(3, 3)}}, the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and {{val|7625597484987}} ({{math|1== 327 = 333 = 33}}) respectively.

## Limits of powers

Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 00. The limits in these examples exist, but have different values, showing that the two-variable function {{math|xy}} has no limit at the point {{math|(0, 0)}}. One may consider at what points this function does have a limit.More precisely, consider the function {{math|1=f(x, y) = x'y}} defined on {{math|1=D = {(x, y) âˆˆ R2 : x > 0}.}} Then {{math|D}} can be viewed as a subset of {{math|{{overline|R}}2}} (that is, the set of all pairs {{math|(x, y)}} with {{math|x}}, {{math|y}} belonging to the extended real number line {{math|1={{overline|R'}} = [âˆ’âˆž, +âˆž]}}, endowed with the product topology), which will contain the points at which the function {{math|f''}} has a limit.In fact, {{math|f}} has a limit at all accumulation points of {{math|D}}, except for {{math|(0, 0)}}, {{math|(+âˆž, 0)}}, {{math|(1, +âˆž)}} and {{math|(1, âˆ’âˆž)}}.N. Bourbaki, Topologie gÃ©nÃ©rale, V.4.2. Accordingly, this allows one to define the powers {{math|xy}} by continuity whenever {{math|0 â‰¤ x â‰¤ +âˆž}}, {{math|âˆ’âˆž â‰¤ y â‰¤ +âˆž}}, except for 00, (+âˆž)0, 1+âˆž and 1âˆ’âˆž, which remain indeterminate forms.Under this definition by continuity, we obtain:
• {{math|1=x+âˆž = +âˆž}} and {{math|1=xâˆ’âˆž = 0}}, when {hide}math|1
< x â‰¤ +âˆž{edih}.
• {{math|1=x+âˆž = 0}} and {{math|1=xâˆ’âˆž = +âˆž}}, when {{math|0 â‰¤ x < 1}}.
• {{math|1=0y = 0}} and {{math|1=(+âˆž)y = +âˆž}}, when {{math|0 < y â‰¤ +âˆž}}.
• {{math|1=0y = +âˆž}} and {{math|1=(+âˆž)y = 0}}, when {{math|âˆ’âˆž â‰¤ y < 0}}.
These powers are obtained by taking limits of {{math|x'y}} for positive values of {{math|x}}. This method does not permit a definition of {{math|x'y}} when {{math|x < 0}}, since pairs {{math|(x, y)}} with {{math|x < 0}} are not accumulation points of {{math|D}}.On the other hand, when {{math|n}} is an integer, the power {{math|x'n}} is already meaningful for all values of {{math|x}}, including negative ones. This may make the definition {{math|1=0n = +âˆž}} obtained above for negative {{math|n}} problematic when {{math|n}} is odd, since in this case {{math|x'n â†’ +âˆž}} as {{math|x}} tends to {{math|0}} through positive values, but not negative ones.

## Efficient computation with integer exponents

Computing bn using iterated multiplication requires {{math|n âˆ’ 1}} multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2100, note that {{math|1=100 = 64 + 32 + 4}}. Compute the following in order:
1. 22 = 4
2. (22)2 = 24 = 16
3. (24)2 = 28 = 256
4. (28)2 = 216 = 65,536
5. (216)2 = 232 = 4,294,967,296
6. (232)2 = 264 = 18,446,744,073,709,551,616
7. 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 b'n can be reduced to Î˜(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 b'n is a difficult problem for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available.JOURNAL, Gordon, D. M., 10.1006/jagm.1997.0913, A Survey of Fast Exponentiation Methods, Journal of Algorithms, 27, 129â€“146, 1998,weblink 10.1.1.17.7076,

## Exponential notation for function names

Placing an integer superscript after the name or symbol of a function, as if the function were being raised to a power, commonly refers to repeated function composition rather than repeated multiplication. Thus, {{math|f{{i sup|3}}(x)}} may mean {{math|f(f(f(x)))}}; in particular, {{math|f{{i sup|âˆ’1}}(x)}} usually denotes the inverse function of {{math|f}}. Iterated functions are of interest in the study of fractals and dynamical systems. Babbage was the first to study the problem of finding a functional square root {{math|f{{i sup|1/2}}(x)}}.For historical reasons, this notation applied to the trigonometric and hyperbolic functions has a specific and diverse interpretation: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of {{math|âˆ’1}} denotes the inverse function. That is, {{math|sin2 x}} is just a shorthand way to write {{math|(sin x)2}} without using parentheses, whereas {{math|sinâˆ’1 x}} refers to the inverse function of the sine, also called {{math|arcsin x}}. Each trigonometric and hyperbolic has its own name and abbreviation both for the reciprocal; for example, {{math|1=1/(sin x) = (sin x)âˆ’1 = csc x}}, as well as for its inverse, for example {{math|1=coshâˆ’1 x = arcosh x}}. A similar convention applies to logarithms, where {{math|log2 x}} usually means {{math|(log x)2}}, not {{math|log log x}}.

## In programming languages

Programming languages generally express exponentiation either as an infix operator or as a (prefix) function, as they are linear notations which do not support superscripts:
Many other programming languages lack syntactic support for exponentiation, but provide library functions:
• 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 xâˆ’1) and roots (prefer sqrt(x) over x0.5, prefer cbrt(x) over x1/3).Not all programming languages adhere to the same association convention for exponentiation: while the Wolfram language, Google Search and others use right-association (i.e. a^b^c is evaluated as a^(b^c)), many computer programs such as Microsoft Office Excel and Matlab associate to the left (i.e. a^b^c is evaluated as (a^b)^c).

## See also

{{div col|colwidth=20em}} {{div col end}}

{{Reflist}}

## External links

• {{planetmath reference|id=3948|title=Introducing 0th power}}
• Laws of Exponents with derivation and examples
{{Good article}}{{Hyperoperations}}

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