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Encyclopedia > Exponentiation

Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication: Look up exponent in Wiktionary, the free dictionary. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ... In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... The integers are commonly denoted by the above symbol. ... In mathematics, multiplication is an elementary arithmetic operation. ...

a^n = underbrace{a times cdots times a}_n,

just as multiplication by a whole number corresponds to repeated addition: 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ...

a times n = underbrace{a + cdots + a}_n.

The exponent is usually shown as a superscript to the right of the base. This notation is invented by René Descartes. The exponentiation an can be read as: a raised to the n-th power or a raised to the power n, or more briefly: a to the n-th power or a to the power n. Some exponents can be read in a certain way; for example a2 is usually read as a squared and a3 as a cubed. This article is about the term superscript as used in typography. ... “Descartes” redirects here. ...


The power an can also be defined when the exponent n is a negative integer. When the base a is a positive real number, exponentiation is defined for real and even complex exponents n. The special exponential function ex is fundamental for this definition. It enables the functions of trigonometry to be expressed by exponentiation. However, when the base a is not a positive real number and the exponent n is not an integer, then an cannot be defined as a unique continuous function of a. The exponential function is one of the most important functions in mathematics. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...


Exponentiation where the exponent is a matrix is used for solving systems of linear differential equations. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ...


Exponentiation is used pervasively in many other fields as well, 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. Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ... Theoretical Human population increase from 10,000 BC – 2000 AD. Population growth is the change in population over time, and can be quantified as the change in the number of individuals in a population per unit time. ... Chemical substances in a system may increase or decrease in concentration with time due to chemical reactions. ... A wave is a disturbance that propagates through space or spacetime, transferring energy and momentum and sometimes angular momentum. ... Public key cryptography is a form of cryptography which generally allows users to communicate securely without having prior access to a shared secret key, by using a pair of cryptographic keys, designated as public key and private key, which are related mathematically. ...

Contents

Exponentiation with integer exponents

The exponentiation operation with integer exponents only requires elementary algebra. Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. ...


Positive integer exponents

a2 = a·a is called the square of a because the area of a square with side-length a is a2. In algebra, the square of a number is that number multiplied by itself. ...


a3 = a·a·a is called the cube, because the volume of a cube with side-length a is a3. In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. ...


So 32 is pronounced "three squared",and 23 is "two cubed".


The exponent says how many copies of the base are multiplied together. For example, 35 = 3·3·3·3·3 = 243. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the base, 5 is the exponent, and 243 is the power or, more specifically, the fifth power of 3 or 3 raised to the fifth power.


The word "raised" is usually omitted, and most often "power" as well, so 35 is typically pronounced "three to the fifth" or "three to the five".


Formally, powers with positive integer exponents may be defined by the initial condition a1 = a and the recurrence relation an+1 = a·an. In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...


Exponents one and zero

Notice that 31 is product of only one 3, which is evidently 3.


Also note that 35 = 3·34. Also 34 = 3·33. Continuing this trend, we should have

31 = 3·30

Another way of saying this is that when n, m, and n - m are positive (and if x is not equal to zero), one can see by counting the number of occurrences of x that

 frac{x^n}{x^m} = x^{n - m}.

Extended to the case that n and m are equal, the equation would read

 1 = frac{x^n}{x^n} = x^{n - n} = x^0

since both the numerator and the denominator are equal. Therefore we take this as the definition of x0.


Therefore we define 30 = 1 so that the above equality holds. This leads to the following rule:

  • Any number to the power 1 is itself.
  • Any nonzero number to the power 0 is 1; one interpretation of these powers is as empty products. The case of 00 is discussed below.

In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ...

Combinatorial interpretation

For non-negative integers n and m, the power nm equals the cardinality of the set of m-tuples from an n-element set, or the number of m-letter words from an n-letter alphabet. In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

05 = | {} | = 0. There is no 5-tuple from the empty set.
14 = | { (1,1,1,1) } | = 1. There is one 4-tuple from a one-element set.
23 = | { (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2) } | = 8. There are 8 3-tuples from a two-element set.
32 = | { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) } | = 9. There are 9 2-tuples from a three-element set.
41 = | { (1), (2), (3), (4) } | = 4. There are 4 1-tuples from a four-element set.
50 = | { () } | = 1. There is exactly one empty tuple.

See also exponentiation over sets. In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ... “Exponent” redirects here. ...


Negative integer exponents

Raising a nonzero number to the −1 power produces its reciprocal. The reciprocal function: y = 1/x. ...

a^{-1} = frac{1}{a}

Thus:

a^{-n} = (a^n)^{-1} = frac{1}{a^n}

Raising 0 to a negative power would imply division by 0, and so is undefined. In mathematics, a division is called a division by zero if the divisor is zero. ...


A negative integer exponent can also be seen as repeated division by the base. Thus 3^{-4} = (((1/3)/3)/3)/3 = frac{1}{81} = frac{1}{3^{4}}. In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...


Identities and properties

The most important identity satisfied by integer exponentiation is: In mathematics, the term identity has several important uses: An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ...

 a^{m + n} = a^m cdot a^n

This identity has the consequence:

a^{m - n} =frac{a^m}{a^n}

for a ≠ 0, and

(a^m)^n = a^{m cdot n}.

While addition and multiplication are commutative (for example, 2+3 = 5 = 3+2 and 2·3 = 6 = 3·2), exponentiation is not commutative: 23 = 8, but 32 = 9. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...


Similarly, while addition and multiplication are associative (for example, (2+3)+4 = 9 = 2+(3+4) and (2·3)·4 = 24 = 2·(3·4), exponentiation is not associative either: 23 to the 4th power is 84 or 4096, but 2 to the 34 power is 281 or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, the order is usually understood to be from right to left: In mathematics, associativity is a property that a binary operation can have. ...

a^{b^c}=a^{(b^c)}ne (a^b)^c=a^{(bcdot c)}=a^{bcdot c}

Powers of ten

See Scientific notation

Powers of 10 are easily computed in the base ten (decimal) number system. For example, 108 = 100000000. Scientific notation, also known as standard form, is a notation for writing numbers that is often used by scientists and mathematicians to make it easier to write large and small numbers. ... For other uses, see Decimal (disambiguation). ...


Exponentiation with base 10 is used in scientific notation to describe large or small numbers. For instance, 299,792,458 (the speed of light in a vacuum, in meters per second) can be written as 2.99792458·108 and then approximated as 2.998·108, (or sometimes as 299.8·106, or 299.8E+6, especially in computer software). This article is about the number 10. ... Scientific notation, also known as standard form, is a notation for writing numbers that is often used by scientists and mathematicians to make it easier to write large and small numbers. ... “Lightspeed” redirects here. ... It has been suggested that this article or section be merged with estimation. ...


SI prefixes based on powers of 10 are also used to describe small or large quantities. For example, the prefix kilo means 103 = 1000, so a kilometre is 1000 metres. An SI prefix (also known as a metric prefix) is a name or associated symbol that precedes a unit of measure (or its symbol) to form a decimal multiple or submultiple. ... Kilo (symbol: k) is a prefix in the SI system denoting 103 or 1000. ... This article is about the unit of length. ...


Powers of two

The positive powers of 2 are important in computer science because there are 2n possible values for an n-bit variable. See Binary numeral system. In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... This article is about the unit of information. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ...


Powers of 2 are important in set theory since a set with n members has a power set, or set of all subsets of the original set, with 2n members. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... “Superset” redirects here. ...


The negative powers of 2 are commonly used, and the first two have special names: half, and quarter. One half is the fraction resulting from dividing one by two (½), or any number by its double; multiplication by one half is equivalent to division by two. ... Look up quarter in Wiktionary, the free dictionary. ...


Powers of zero

If the exponent is positive, the power of zero is zero: 0n = 0, where n > 0.


If the exponent is negative, the power of zero (0n, where n > 0) remains undefined, because division by zero is implied.


If the exponent is zero, some authors define 00=1, whereas others leave it undefined, as discussed below.


Powers of minus one

The powers of minus one are useful for expressing alternating sequences.


If the exponent is even, the power of minus one is one: (−1)2n = 1.


If the exponent is odd, the power of minus one is minus one: (−1)2n+1 = −1.


Powers of the imaginary unit

The powers of the imaginary unit i are useful for expressing sequences of period 4. See for example Root of unity#Periodicity. In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In mathematics, the nth roots of unity, or de Moivre numbers, are all the complex numbers which yield 1 when raised to a given power n. ...

i^{4n+1}=i !
i^{4n+2}=-1 !
i^{4n+3}=-i !
i^{4n}=1 !

Powers of e

Main article: Exponential function

The number e, the base of the natural logarithm, is a well studied constant approximately equal to 2.718. The function ex, known as the exponential function, has applications in many areas of mathematics and science. The exponential function is one of the most important functions in mathematics. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...


The function ex can be computed in many ways, including the limit The limit of a sequence is one of the oldest concepts in mathematical analysis. ...

 e^x = lim_{n rightarrow infty} left(1+frac{x}{n} right) ^n,

and the power series In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

e^x = 1 + x+ frac{x^2}2+ frac{x^3}6+cdots+frac{x^n}{n!}+cdots ,.

These formulas for ex only require that x can be raised to positive integer powers, and thus the formulas can be used not only for integer values of x, but for fractional, real, or complex values. For this reason, the exponential function can be used to define ab for general real or complex values of a and b, as discussed below. A further generalization is the computation of eA where A is a square matrix, which is used to solve ordinary differential equations. For the square matrix section, see square matrix. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...


Powers of real numbers

Exponentiation with various bases; from top to bottom, base 10 (green), base e (red), base 2 (blue), base ½ (cyan). Note how all of the curves pass through the point (0, 1). This is because, in accordance with the properties of exponentiation, any non-zero number raised to the power 0 is 1. Also note that at x=1, the y value equals the base. This is because any number raised to the power 1 is that same number.
Exponentiation with various bases; from top to bottom, base 10 (green), base e (red), base 2 (blue), base ½ (cyan). Note how all of the curves pass through the point (0, 1). This is because, in accordance with the properties of exponentiation, any non-zero number raised to the power 0 is 1. Also note that at x=1, the y value equals the base. This is because any number raised to the power 1 is that same number.
From top to bottom: x1/8, x1/4, x1/2, x1, x2, x4, x8.
From top to bottom: x1/8, x1/4, x1/2, x1, x2, x4, x8.

Raising a positive real number to a power that is not an integer can be accomplished in two ways. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... “Exponent” redirects here. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

  • Rational number exponents can be defined in terms of nth roots, and arbitrary nonzero exponents can then be defined by continuity.
  • The natural logarithm can be used to define real exponents using the exponential function.

The identities and properties shown above are true for non-integer exponents as well. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an nth root of a number a is a number b such that bn=a. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...


Principal n-th root

Main article: n-th root

An nth root of a number a is a number b such that bn = a. ... For other uses, see Number (disambiguation). ...


When referring to the nth root of a real number a it is assumed that what is desired is the principal nth root of the number. If a is a real number, and n is a positive integer, then the unique real solution with the same sign as a to the equation In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

 x^n = a

is called the principal nth root of a, and is denoted sqrt[n]{a} using the radical symbol (sqrt{,,}).. In mathematics, an nth root of a number a is a number b such that bn=a. ...

 x=a^{frac{1}{n}} = sqrt[n]{a}.

For example: 41/2 = 2, 81/3 = 2, (-8)1/3 = -2, .


Note that if n is even, negative numbers will not have a principal nth root. Look up een, even in Wiktionary, the free dictionary. ... A negative number is a number that is less than zero, such as −3. ...


Rational powers of positive real numbers

Exponentiation with a rational exponent m/n can be defined as In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

a^{frac{m}{n}} = left(a^mright)^{frac{1}{n}} = (sqrt[n]{a^m}).

For example, 82/3 = 4.


Since any real number can be approximated by rational numbers, exponentiation to an arbitrary real exponent k can be defined by continuity with the rule In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

 a^k = lim_{r to k} a^r,

where the limit is taken only over rational values of r.


For example, if

k approx 1.732

then

5^k approx 5^{1.732}.

Real powers of positive real numbers

The natural logarithm ln(x) is the inverse of the exponential function ex. It is defined for every positive real number b and satisfies the equation The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

b = e^{ln b}.,

Assuming bx is already defined, logarithm and exponent rules give the equality

b^x = (e^{ln b})^x = e^{x cdotln b}.,

This equality can be used to define exponentiation with any positive real base b as

b^x = e^{xcdotln b}.,

This definition of the real number power bx 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.


Some rational powers of negative real numbers

Neither the logarithm method nor the fractional exponent method can be used to define ak as a real number for a negative real number a and an arbitrary real number k. In some special cases, a definition is possible: integral powers of negative real numbers are real numbers, and rational powers of the form am/n where n is odd can be computed using roots. But since there is no real number x such that x2 = −1, the definition of am/n when n is even and m is odd must use the imaginary unit i, as described more fully in the next section. In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...


The logarithm method cannot be used to define ak as a real number when a < 0 because ex is nonnegative for every real number x, so log(a) cannot be a real number.


The rational exponent method cannot be used for negative values of a because it relies on continuity. The function f(r) = ar has a unique continuous extension from the rational numbers to the real numbers for each a > 0. But when a < 0, the function f is not even continuous on the set of rational numbers r for which it is defined. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...


For example, take a = −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 −1q = 1 is dense in the rational numbers, as is the set of q for which −1q = −1. This means that the function (−1)q is not continuous at any rational number q where it is defined. In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of...


Imaginary powers of e

The geometric interpretation of the operations on complex numbers and the definition of powers of e is the clue to understanding e i·x for real x. Consider the right triangle (0, 1, 1+i·x/n). For big values of n the triangle is almost a circular sector with a small central angle equal to x/n radian. The triangles (0, (1+i·x/n)k, (1+i·x/n)k+1) are mutually similar for all values of k. So for big values of n the limiting point of (1+ix/n)n is the point on the unit circle whose angle from the positive real axis is x radians. The polar coordinates of this point are (r,θ) = (1,x), and the cartesian coordinates are (cos(x), sin(x)). So e i·x = cos(x) + i·sin(x), and this is Euler's formula, connecting algebra to trigonometry by means of complex numbers. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... “Exponent” redirects here. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A circular sector or circle sector also known as a pie piece is the portion of a circle enclosed by two radii and an arc. ... // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ... Illustration of a unit circle. ... Some common angles, measured in radians. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... This article is about the branch of mathematics. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...


The solutions to the equation ez = 1 are the integer multiples of 2·π·i :

{ z : e^z=1 } = { kcdot 2cdot picdot i : k in mathbb{Z} }.

More generally, if eb = a, then every solution to ez = a can be obtained by adding an integer multiple of 2·π·i to b:

{ z : e^z=a } = { b+kcdot 2cdotpicdot i : k in mathbb{Z} } .

Thus the complex exponential function is a periodic function with period 2·π·i. In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ...


Trigonometric functions

Main article: Euler's formula

It follows from Euler's formula that the trigonometric functions cosine and sine are Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

cos(z) = frac{e^{icdot z} + e^{-icdot z}}{2} qquad sin(z) = frac{e^{icdot z} - e^{-icdot z}}{2cdot i}.,

Historically, cosine and sine were defined geometrically before the invention of complex numbers. The above formula reduces the complicated formulas for trigonometric functions of a sum into the simple exponentiation formula In mathematics, trigonometric identities are equalities that involve trigonometric functions that are true for all values of the occurring variables. ...

e^{icdot (x+y)}=e^{icdot x}cdot e^{icdot y}.,

Using exponentiation with complex exponents one need not study trigonometry.


Complex powers of e

The power e x+i·y is computed e x · e i·y. The real factor e x is the absolute value of e x+i·y and the complex factor e i·y identifies the direction of e x+i·y. In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... The shape of each panel of this road sign, and the broken lines at the ends, represents an arrow; a space-consuming central bar of the arrow sign is dispensed with. ...


Complex powers of positive real numbers

If a is a positive real number, and z is any complex number, the power az is defined as ez·ln(a), where x = ln(a) is the unique real solution to the equation ex = a. So the same method working for real exponents also work for complex exponents. For example:

2 i = e i·log(2) = cos(ln(2))+i·sin(ln(2)) = 0.7692+i·0.63896
e i = 0.54030+i·0.84147
10 i = −0.66820+i·0.74398
(e π) i = 535.49 i = 1

Powers of complex numbers

Integer powers of complex numbers are defined by repeated multiplication or division as above. Complex powers of positive reals are defined via ex as above. These are continuous functions. Trying to extend these functions to the general case of non-integer powers of complex numbers that are not positive reals leads to difficulties. Either we define discontinuous functions or multivalued functions. None of these options are entirely satisfactory. This diagram does not represent a true function, because the element 3 in X is associated with two elements, b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...


The rational power of a complex number must be the solution to an algebraic equation. For example, w = z1/2 must be a solution to the equation w2 = z. But if w is a solution, then so is −w, because (−1)2 = 1 . So the algebraic equation w2 = z is not sufficient for defining z1/2 . Choosing one of the two solutions as the principal value of z1/2 leaves us with a function that is not continuous, and the usual rules for manipulating powers lead us astray.


The logarithm of a complex numbers

One solution, z=log a, to the equation ez = a, is called the principal value of the complex logarithm. It is the unique solution whose imaginary part lies in the interval (-pi, pi],. For example, log(1) = 0, log(−1) = πi, log(i) = πi/2, and log(−i) = −πi/2. The principal value of the logarithm is known as a branch of the logarithm; other branches can be specified by choosing a different range for the imaginary part of the logarithm. The boundary between branches is known as a branch cut. The principal value has a branch cut extending from the origin along the negative real axis, and is discontinuous at each point of the branch cut. See also Cauchy principal value for its use in describing improper integrals In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ...


Complex power of a complex number

The general complex power ab of a nonzero complex number a is defined as

a^b = e^{log(a^b)} = e^{bcdot log a}.,

When the exponent is a rational number the power z=an/m is a solution to the equation zm = an . In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...


The computation of complex powers is facilitated by converting the base a to polar form, as described in detail below.


Complex roots of unity

Main article: Root of unity

A complex number a such that an = 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. In mathematics, the nth roots of unity, or de Moivre numbers, are all the complex numbers which yield 1 when raised to a given power n. ...


If z n = 1 but z k ≠ 1 for all natural numbers k such that 0 < k < n, then z 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 4'th roots of unity; the other one is −i. In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...


The number e2πi (1/n) is the primitive nth root of unity with the smallest positive complex 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 sqrt[n]{1}, which is 1.[1]) Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... See also Cauchy principal value for its use in describing improper integrals In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued. ...


The other nth roots of unity are given by

big ( e^{2pi i (1/n)} big ) ^k = e^{2pi i k/n}

for 2 ≤ k ≤ 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 az in the important special case where z = 1/n and n is a positive integer. These are the nth roots of a; they are solutions of the equation xn = a. 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 conventional in mathematics to define a1/n as the principal value of the root. If a is a positive real number, it is also conventional to select a positive real number as the principal value of the root a1/n. For general complex numbers, the nth root with the smallest argument is often selected as the principal value of the nth root operation, as with principal values of roots of unity.


The set of nth roots of a complex number a is obtained by multiplying the principal value a1/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

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 In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

z = re^{itheta} = e^{log(r) + itheta} ,,

where r is a non-negative real number and θ is the (real) argument of z. The argument, like the complex logarithm, has many possible values for each z and so a branch cut is used to choose a specific value. The polar form has a simple geometric interpretation: if a complex number u+i v 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 branch cut corresponds to the notion that a polar angle θ is ambiguous, since any multiple of 2π could be added to θ without changing the location of the point. The principal value (the most common branch cut), as mentioned above, corresponds to θ chosen in the interval (−π, π]. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Fig. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... Atan2 is a two-parameter function for computing the arctangent in the C programming language. ...


In order to compute the complex power ab, write a in polar form:

a = r e^{itheta} ,.

Then

log a = log r + i theta ,,

and thus

a^b = e^{b log a} = e^{b(log r + itheta)}. ,

If b is decomposed as c + di, then the formula for ab 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 θ to be in the interval (−π, π]). To compute i i, write i in polar and Cartesian forms:

i = 1 cdot e^{i pi/2},,
i = 0 + 1i.,

Then the formula above, with r = 1, θ = π/2, c = 0, and d = 1, yields:

 i^i = left( 1^0 e^{-pi/2} right) e^{i(1cdot log 1 + 0 cdot pi/2)} = e^{-pi/2} approx 0.2079.

Similarly, to find (−2)3 + 4i, compute the polar form of −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 i = 1ei(5π/2) is used to compute i i, the power is found to be e−5π/2; the principal value of i i, computed above, is e−π/2.


Failure of power and logarithm identities

Identities for powers and logarithms that hold for positive real numbers may fail when the positive real numbers are replaced by arbitrary complex numbers. There is no method to define complex powers or the complex logarithm as complex-valued functions while preserving the identities these operations possess in the positive real numbers.


An example involving logarithms concerns the rule log(ab) = b·log(a), which holds whenever a is a positive real number and b is a real number. The following calculation shows that this identity does not hold in general for the principal value of the complex logarithm when a is not a positive real number:

 ipi = log(-1) = log((-i)^2) not = 2log(-i) = 2(-ipi/2) = -ipi.

Regardless of which branch of the logarithm is used, a similar failure of the identity will always exist.


An example involving power rules concerns the identities

(ab)^c = a^cb^c, qquad left ( frac{a}{b}right)^c = frac{a^c}{b^c}.

These identities are valid when a and b are positive real numbers and c is a real number. But a calculation using principal values shows that

1 = (-1cdot -1)^{1/2} not = (-1)^{1/2}(-1)^{1/2} = -1,

and

i = (-1)^{1/2} = left (frac{1}{-1}right )^{1/2} not = frac{1^{1/2}}{(-1)^{1/2}} = frac{1}{i} = -i.

These examples illustrate that complex powers and logarithms do not behave the same way as their real counterparts, and so caution is required when working with the complex versions of these operations.


Zero to the zero power

Plot of z = abs(xy) with different curves (red) showing how 00 can evaluate to different values. The green curves all have a limit of 1.

The evaluation of 00 presents a problem, because different mathematical reasoning leads to different results. The best choice for its value depends on the context. According to Benson (1999), "The choice whether to define 00 is based on convenience, not on correctness."[2] There are two principal treatments in practice, one from discrete mathematics and the other from analysis. Image File history File links Size of this preview: 620 × 600 pixel Image in higher resolution (953 × 922 pixel, file size: 179 KB, MIME type: image/png) 3D plot of z = abs(x^y) showing how 0^0 can evaluate to many different values. ... Image File history File links Size of this preview: 620 × 600 pixel Image in higher resolution (953 × 922 pixel, file size: 179 KB, MIME type: image/png) 3D plot of z = abs(x^y) showing how 0^0 can evaluate to many different values. ...


In many settings, especially in foundations and combinatorics, 00 is defined to be 1. This definition arises in foundational treatments of the natural numbers as finite cardinals, and is useful for shortening combinatorial identities and removing special cases from theorems, as illustrated below. In many other settings, 00 is left undefined. In calculus, 00 is an indeterminate form, which must be analyzed rather than evaluated. In general, mathematical analysis treats 00 as undefined[3] in order that the exponential function be continuous. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... For other uses, see Calculus (disambiguation). ... In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression whose limit cannot be evaluated by substituting the limits of the subexpressions. ... Analysis has its beginnings in the rigorous formulation of calculus. ... In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...


Justifications for defining 00 = 1 include:

  • x^x to 1 , as x to 0,
  • When 00 is regarded as an empty product of zeros, its value is 1.
  • The combinatorial interpretation of 00 is the number of empty tuples of elements from the empty set. There is exactly one empty tuple.
  • Equivalently, the set-theoretic interpretation of 00 is the number of functions from the empty set to the empty set. There is exactly one such function, the empty function.
  • A power series identity with nonzero constant term, such as textstyle e^{x} = sum_{n=0}^{infty} frac{x^n}{n!} , is not valid for x = 0 unless 00, which appears in the numerator of the first term of the series, is 1. Thus defining 00 to be 1 allows this identity to be used instead of the longer identity textstyle e^{x} = 1 + sum_{n=1}^{infty} frac{x^n}{n!} .
  • The binomial theorem textstyle(1+x)^n = sum_{k = 0}^n binom{n}{k} x^k is not valid for x = 0, unless 00 = 1.[4] By defining 00 to be 1, a special case of the theorem can be eliminated.

In contexts where the exponent may vary continuously, it is generally best to treat 00 as an ill-defined quantity. Justifications for treating it as undefined include: In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ... “Exponent” redirects here. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ... In mathematics, an empty or nullary function, is a function whose domain is the empty set. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

  • The value 00 often arises as the formal limit of exponentiated functions, f(x)g(x), when f(x) and g(x) approach 0 as x approaches a (a constant or infinity). There, 00 suggests [lim f(x)]lim g(x), which is a well defined quantity and is the correct value of lim f(x)g(x) when f and g approach nonzero constants, but is not well defined when f and g approach 0. The same reasoning applies to certain powers involving infinity, infty^0 and 1^infty.
A more abstract way of saying this is the following: The real function xy of the two nonnegative real variables x and y is not continuous at the point (x, y) = (0, 0), and so 00 is not determined by continuity.[5] That is, the function xy has no continuous extension from the open first quadrant to include the point (0,0).[6] A discontinuous extension would cause the function to lose a number of desirable properties. For example, it is ordinarily taken as a rule in calculus that lim_{x to a} f(x)^{g(x)} = (lim_{x to a} f(x))^{lim_{x to a} g(x)} whenever both sides of the equation are defined; this rule would necessarily fail if 00 were defined.
  • The function zz, viewed as a function of a complex number variable z and defined as ez log z is undefined at z = 0 because log z is undefined at z = 0. Moreover, because zz has a logarithmic branch point at z = 0, it is not common to extend the domain of zz to the origin in this context.[7]

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ...

Treatment in programming languages

The evaluation of 00 is possible in several computer programming languages. Many languages, including Java, Python, Ruby, Haskell, ML, Scheme, MATLAB, Microsoft Windows' Calculator, and others (especially when using IEEE floating-point arithmetic, but also for integer arithmetic), evaluate 00 to be 1.[8] Mathematica simplifies a0 to 1, even if no constraints are placed on a, but does not simplify 0a, and takes 00 to be an indeterminate form. Maple simplifies a0 to 1 and 0a to 0, even if no constraints are placed on a, and evaluates 00 to 1. Google search when used for its calculator function evaluates 00 to 1 [9] “Java language” redirects here. ... Python is a high-level programming language first released by Guido van Rossum in 1991. ... Ruby is a reflective, dynamic, object-oriented programming language. ... Haskell is a standardized purely functional programming language with non-strict semantics, named after the logician Haskell Curry. ... ML is a general-purpose functional programming language developed by Robin Milner and others in the late 1970s at the University of Edinburgh, whose syntax is inspired by ISWIM. Historically, ML stands for metalanguage as it was conceived to develop proof tactics in the LCF theorem prover (the language of... Scheme is a multi-paradigm programming language. ... Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ... The IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754) is the most widely-used standard for floating-point computation, and is followed by many CPU and FPU implementations. ... For other uses, see Mathematica (disambiguation). ... Maple is a general-purpose commercial mathematics software package. ...


Powers with infinity

The expressions infty^0 and 1^infty arise in analysis for the same reason as 00, and they are undefined for the same reason. That is, it is true that lim f(x)]lim g(x) = lim f(x)g(x) when f and g approach nonzero finite constants, but not when they approach 0 or infinity; then, the limit of the power can be anything, not predictable from the limits of f and g.


There is one exception. If f and g both approach infinity as x approaches a, then lim f(x)g(x) does equal infinity. Thus, it makes sense to say the expression infty^infty is well defined but, by Cantor's theorem, does not equal the same infty; instead we have infty < 2^infty and hence infty < infty^infty. Thus exponentiation defines an infinite hierarchy of infinities, usually expressed as Beth numbers. In Zermelo-Fränkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ... In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ...


Efficiently computing a power

The simplest method of computing an requires n−1 multiplication operations, but it can be computed more efficiently as illustrated by the following example. To compute 2100, note that 100 = 96 + 4 and 96 = 3*32. 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
232 232 23224 = 2100

This series of steps only requires 8 multiplication operations instead of 99.


In general, the number of multiplication operations required to compute an 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 an is a difficult problem for which no efficient algorithms are currently known, but many reasonably efficient heuristic algorithms are available. In mathematical analysis, and in particular in the analysis of algorithms, to classify the growth of functions one has recourse to asymptotic notations. ... Exponentiating by squaring is an algorithm used for the fast computation of large integer powers of a number. ... In mathematics, addition-chain exponentiation is a method of exponentiation by positive integer powers that requires a minimal number of multiplications. ...


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 f3(x) may mean f(f(f(x))); in particular, f -1(x) usually denotes the inverse function of f. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...


However, for historical reasons, a special syntax applies to the trigonometric functions: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of −1 denotes the inverse function. That is, sin2x is just a shorthand way to write (sin x)2 without using parentheses, whereas sin−1x refers to the inverse function of the sine, also called arcsin x. There is no need for a shorthand for the reciprocals of trigonometric functions since each has its own name and abbreviation, for example 1 / sin(x) = (sin x)−1 is csc x. A similar convention applies to logarithms, where log2(x) = (log (x))2 and there is no common abbreviation for log(log(x)). [citation needed] In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...


Generalizations of exponentiation

Exponentiation in abstract algebra

Exponentiation for integer exponents can be defined for quite general structures in abstract algebra. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...


Let X be a set with a power-associative binary operation, which we will write multiplicatively. In this very general situation, we can define xn for any element x of X and any nonzero natural number n, by simply multiplying x by itself n times; by definition, power associativity means that it doesn't matter in which order we perform the multiplications. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In abstract algebra, power associativity is a weak form of associativity. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In abstract algebra, power associativity is a weak form of associativity. ...


Now additionally suppose that the operation has an identity element 1. Then we can define x0 to be equal to 1 for any x. Now xn is defined for any natural number n, including 0. In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...


Finally, suppose that the operation has inverses, and that the multiplication is associative (so that the magma is a group). Then we can define xn to be the inverse of xn when n is a natural number. Now xn is defined for any integer n and any x in the group. In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ... This picture illustrates how the hours on a clock form a group under modular addition. ...


Exponentiation in this purely algebraic sense satisfies the following laws (whenever both sides are defined):

  •  x^{m+n}=x^mx^n
  •  x^{m-n}=x^m/x^n
  •  x^{-n}=1/x^n
  •  x^0=1
  •  x^1=x
  •  x^{-1}=1/x
  •  (x^m)^n=x^{mn}

Here, we use a division slash ("/") to indicate multiplying by an inverse, in order to reserve the symbol x−1 for raising x to the power −1, rather than the inverse of x. However, as one of the laws above states, x−1 is always equal to the inverse of x, so the notation doesn't matter in the end. In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...


If in addition the multiplication operation is commutative (so that the set X is an abelian group), then we have some additional laws: In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesnt matter. ...

  • (xy)n = xnyn
  • (x/y)n = xn/yn

If we take this whole theory of exponentiation in an algebraic context but write the binary operation additively, 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. In mathematics, multiplication is an elementary arithmetic operation. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... Analogy is both the cognitive process of transferring information from a particular subject (the analogue or source) to another particular subject (the target), and a linguistic expression corresponding to such a process. ...


When one has several operations around, any of which might be repeated using exponentiation, 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.


Superscript notation is also used, especially in group theory, to indicate conjugation. That is, gh = h−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. Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a rack and a quandle in knot theory are sets with a binary operation mimicking the three Reidemeister moves of diagram manipulation. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ...


Exponentiation over sets

If n is a natural number and A is an arbitrary set, the expression An is often used to denote the set of ordered n-tuples of elements of A. This is equivalent to letting An denote the set of functions from the set {0, 1, 2, ..., n−1} to the set A; the n-tuple (a0a1a2, ..., an−1) represents the function that sends i to ai.


For an infinite cardinal number κ and a set A, the notation Aκ is also used to denote the set of all functions from a set of size κ to A. This is sometimes written κA to distinguish it from cardinal exponentiation, defined below. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ...


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 In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

bigoplus_{i in 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 Euclidean space Rn. In mathematics, the term up to xxxx is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...


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: In mathematics, the Cartesian product is a direct product of sets. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

S^N equiv { f: N to S },

This fits in with the exponentiation of cardinal numbers, in the sense that |SN| = |S||N|, where |X| is the cardinality of X. When N=2={0,1}, we have |2X| = 2|X|, where 2X, usually denoted by PX, 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 mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... “Superset” redirects here. ...


Exponentiation 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. In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. ... In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. ... In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. ... In mathematics, the Cartesian product is a direct product of sets. ...


Exponentiation of cardinal and ordinal numbers

Main articles: cardinal arithmetic and ordinal arithmetic

In set theory, there are exponential operations for cardinal and ordinal numbers. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In the mathematical field of set theory, there are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...


If κ and λ are cardinal numbers, the expression κλ represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ. If κ and λ are finite then this agrees with the ordinary exponential operation. For example, the set of 3-tuples of elements from a 2-element set has cardinality 8.


Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process. In the ordinal numbers, exponentiation is defined by transfinite induction. For ordinals α and β, the exponential αβ is the supremum of the ordinal product αγα over all γ < β. In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as... Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. ...


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 tetration. Iterating tetration leads to another operation, and so on. This sequence of operations is captured by the Ackermann function. Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ... In recursion theory, the Ackermann function or Ackermann-Péter function is a simple example of a computable function that is not primitive recursive. ...


Exponentiation in programming languages

The superscript notation xy is convenient in handwriting but inconvenient for typewriters and computer terminals that align the baselines of all characters on each line. Many programming languages have alternate ways of expressing exponentiation that do not use superscripts: Mechanical desktop typewriters, such as this Underwood Five, were long time standards of government agencies, newsrooms, and sales offices. ... A computer terminal is an electronic or electromechanical hardware device that is used for entering data into, and displaying data from, a computer or a computing system. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ...

In Bash, C, C++, C#, Java, JavaScript and Python, the symbol ^ represents bitwise XOR. In Pascal, it represents indirection. ALGOL (short for ALGOrithmic Language) is a programming language originally developed in the mid 1950s which became the de facto standard way to report algorithms in print for almost the next 30 years. ... Commodore BASIC is the dialect of BASIC used in Commodore Internationals 8-bit home computer line, stretching from the PET of 1977 to the C128 of 1985. ... This article is about the programming language. ... The J programming language, developed in the early 1990s by Ken Iverson and Roger Hui, is a synthesis of APL (also by Iverson) and the FP and FL functional programming languages created by John Backus (of FORTRAN, ALGOL, and BNF fame). ... Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ... Microsoft Excel (full name Microsoft Office Excel) is a spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS. It features calculation and graphing tools which, along with aggressive marketing, have made Excel one of the most popular microcomputer applications to date. ... TeX (IPA: as in Greek, often in English; written with a lowercase e in imitation of the logo) is a typesetting system created by Donald Knuth. ... // Haskell may refer ro: Haskell (programming language), a standardized pure functional programming language with non-strict semantics Haskell Indian Nations University, a four year degree granting university in Lawrence, Kansas which offers free tuition to members of registered Native American tribes in the United States Haskell Free Library and Opera... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... Ada is a structured, statically typed, imperative, and object-oriented high-level computer programming language. ... -1... Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ... Visual FoxPro is a data-centric object-oriented and procedural programming language by Microsoft. ... Wikibooks has a book on the topic of Perl Programming Perl is a dynamic programming language created by Larry Wall and first released in 1987. ... Python is a high-level programming language first released by Guido van Rossum in 1991. ... Ruby is a reflective, dynamic, object-oriented programming language. ... Overview The SAS System is an integrated system of software products (provided by the SAS Institute) that enables the programmer to perform: data entry, retrieval, and management report writing and graphics statistical and mathematical analysis business planning, forecasting, and decision support operations research and project management quality improvement applications development. ... ABAP (Advanced Business Application Programming) is a high level programming language created by the German software company SAP. It is currently positioned, alongside the more recently introduced Java, as the language for programming SAPs Web Application Server, part of its NetWeaver platform for building business applications. ... // Haskell may refer ro: Haskell (programming language), a standardized pure functional programming language with non-strict semantics Haskell Indian Nations University, a four year degree granting university in Lawrence, Kansas which offers free tuition to members of registered Native American tribes in the United States Haskell Free Library and Opera... APL (for A Programming Language) is an array programming language based on a notation invented in 1957 by Kenneth E. Iverson while at Harvard University. ... C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ... C++ (pronounced see plus plus, IPA: ) is a general-purpose programming language with high-level and low-level capabilities. ... For other uses, see PHP (disambiguation). ... “Java language” redirects here. ... It has been suggested that Client-side JavaScript be merged into this article or section. ... This article or section does not adequately cite its references or sources. ... The title given to this article is incorrect due to technical limitations. ... Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard X3. ... Scheme is a multi-paradigm programming language. ... Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ... In computer programming, indirection is the ability to reference something using a name, reference, or container instead of the value itself. ...


History of the notation

The term power was used by Euclid for the square of a line. Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel. Samuel Jeake introduced the term indices in 1696.[10] In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), surfolide (fifth), zenzicube (sixth), second surfolide (seventh) and Zenzizenzizenzic (eighth). For other uses, see Euclid (disambiguation). ... Nicolas Chuquet (born 1445 (some sources say c. ... Henricus Grammateus (also known as Henricus Scriptor, Heinrich Schreyber or Heinrich Schreiber) (1495 - 1525 or 1526[1]) was a German mathematician. ... Michael Stifel (1487 - 1567) was a German mathematician. ... The year 1696 had the earliest equinoxes and solstices for 400 years in the Gregorian calendar, because this year is a leap year and the Gregorian calendar would have behaved like the Julian calendar since March 1500 had it have been in use that long. ... Robert Recorde (c. ... Zenzizenzizenzic is the eighth power or exponent of a number. ...


Another historical synonym, involution,[11] is now rare and should not be confused with its more common meaning. In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...


References

  1. ^ This definition of a principal root of unity can be found in:
    • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2001). Introduction to Algorithms, second edition, MIT Press. ISBN 0262032937.  Online resource
    • Paul Cull, Mary Flahive, and Robby Robson (2005). Difference Equations: From Rabbits to Chaos, Undergraduate Texts in Mathematics, Springer. ISBN 0387232346.  Defined on page 351, available on Google books.
    • "Principal root of unity", MathWorld.
  2. ^ Benson, Donald C. The Moment of Proof : Mathematical Epiphanies. New York Oxford University Press (UK), 1999. ISBN 9780195117219
  3. ^ Examples of this include Edwards and Penny (1994). Calculus, 4th ed,, Prentice-Hall, p. 466, and Keedy, Bittinger, and Smith (1982). Algebra Two. Addison-Wesley, p. 32.
  4. ^ "Some textbooks leave the quantity 00 undefined, because the functions x0 and 0x have different limiting values when x decreases to 0. But this is a mistake. We must define x0 = 1, for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = −y. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0x is quite unimportant".Ronald Graham, Donald Knuth, and Oren Patashnik (1989-01-05). "Binomial coefficients", Concrete Mathematics, 1st edition, Addison Wesley Longman Publishing Co, 162. ISBN 0-201-14236-8. 
  5. ^ L. J. Paige (March 1954). "A note on indeterminate forms". American Mathematical Monthly 61 (3): 189–190. 
  6. ^ Along the x-axis the limit is 1, along the y-axis the limit is 0, and any intermediate limit a can be obtained using the curve y = log(a)/log(x). However, if y is an analytic function of x, or if there exists a positive constant, a, such that y < ax, then the limit is 1.
  7. ^ "... Let's start at x=0. Here xx is undefined." Mark D. Meyerson "The Xx Spindle." Mathematics Magazine, v. 69 n. 3, Jun 1996, pp. 198-206.
  8. ^ For example, see John Benito (April 2003). "Rationale for International Standard — Programming Languages — C". Revision 5.10.
  9. ^ http://www.google.co.uk/search?q=0%5E0
  10. ^ O'Connor, John J; Edmund F. Robertson "Etymology of some common mathematical terms". MacTutor History of Mathematics archive.  
  11. ^ This definition of "involution" appears in the OED second edition, 1989, and Merriam-Webster online dictionary [1]. The most recent usage in this sense cited by the OED is from 1806.

Ronald L. Graham (born October 31, 1935) is a mathematician credited by the American Mathematical Society with being one of the principle architects of the rapid development worldwide of discrete mathematics in recent years[1]. He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. ... Donald Ervin Knuth ( or Ka-NOOTH[1], Chinese: [2]) (b. ... Oren Patashnik (born 1954) is a computer scientist. ... Year 1989 (MCMLXXXIX) was a common year starting on Sunday (link displays 1989 Gregorian calendar). ... is the 5th day of the year in the Gregorian calendar. ... Concrete Mathematics by Ronald L. Graham, Donald E. Knuth and Oren Patashnik is a textbook that provides its readers with mathematical background that can be especially useful in computer science. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ...

See also

Wikimedia Commons has media related to:

Image File history File links Commons-logo. ... In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... This is a list of exponential topics, by Wikipedia page. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... Modular exponentiation is a type of exponentiation performed over a modulus. ... In mathematics, an nth root of a number a is a number b such that bn=a. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

External links


  Results from FactBites:
 
Exponentiation - definition of Exponentiation in Encyclopedia (1776 words)
In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition.
Exponentiation can also be understood purely in terms of abstract algebra, if we limit the exponents to integers.
Exponential notation is also used, especially in group theory, to indicate conjugation.
Exponentiation - Wikipedia, the free encyclopedia (2356 words)
In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition.
Exponentiation with various bases: red is to base e, green is to base 10, and purple is to base 1.7.
Exponentiation is a basic mathematical operation that is used pervasively in other fields as well, including physics, chemistry, biology, computer science and economics, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.
  More results at FactBites »

 
 

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