Exponentiation is a mathematical operation, written aⁿ, involving two numbers, the base a and the exponent n. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values radix|basis (topologyIn Arithmetic, the base refers to the number b in an expression of the form b n. When n is a positive integer, exponentiation corresponds to repeated multiplication:

just as multiplication by a whole number corresponds to repeated addition:

The exponent is usually shown as a superscript to the right of the base. A negative number is a Number that is less than zero, such as −2 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Addition is the mathematical process of putting things together This article is about the terms 'subscript' and 'superscript' as used in typography The exponentiation aⁿ can be read as: a raised to the n-th power or a raised to the power [of] n, or more briefly: a to the n-th power or a to the power [of] n, or even more briefly: a to the n. Some exponents can be read in a certain way; for example a² is usually read as a squared and a³ as a cubed.

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 e^x is fundamental for this definition. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) It enables the functions of trigonometry to be expressed by exponentiation. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. 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. 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 In Mathematics, a linear differential equation is a Differential equation of the form Ly = f \ where the Differential

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 is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on Population growth is the change in Population over time and can be quantified as the change in the number of individuals in a population using "per unit time" for Chemical kinetics, also known as reaction kinetics is the study of rates of chemical processes A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. Public-key cryptography, also known as asymmetric cryptography, is a form of Cryptography in which the key used to encrypt a message differs from the key

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

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 &mdash the result of multiplying it by itself three times

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. "Difference equation" redirects here It should not be confused with a Differential equation.

Exponents one and zero

Notice that 3¹ is the 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

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

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. In Mathematics, an empty product, or nullary product, is the result of multiplying no numbers The case of 0⁰ is discussed below.

Combinatorial interpretation

For non-negative integers n and m, the power n^m 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. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple

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. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple

See also exponentiation over sets.

Negative integer exponents

Raising a nonzero number to the −1 power produces its reciprocal. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which

Thus:

Raising 0 to a negative power would imply division by 0, and so is undefined. In

A negative integer exponent can also be seen as repeated division by the base. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. Thus .

Identities and properties

The most important identity satisfied by integer exponentiation is:

This identity has the consequence:

for a ≠ 0, and

. In Mathematics, the term identity has several different important meanings An identity is an equality that remains true regardless of the values of

Another basic identity is

.

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, commutativity is the ability to change the order of something without changing the end result

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. In Mathematics, associativity is a property that a Binary operation can have Without parentheses to modify the order of calculation, the order is usually understood to be from right to left:

Powers of ten

See Scientific notation

Powers of 10 are easily computed in the base ten (decimal) number system. Scientific notation, also sometimes known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be The decimal ( base ten or occasionally denary) Numeral system has ten as its base. For example, 10⁸ = 100000000.

Exponentiation with base 10 is used in scientific notation to describe large or small numbers. Scientific notation, also sometimes known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be 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. An approximation (represented by the symbol ≈ is an inexact representation of something that is still close enough to be useful 998·10⁸, (or sometimes as 299. 8·10⁶, or 299. 8E+6, especially in computer software).

SI prefixes based on powers of 10 are also used to describe small or large quantities. 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 For example, the prefix kilo means 10³ = 1000, so a kilometre is 1000 metres. Kilo- (symbol k is a prefix in the SI and other systems of units denoting 103 or 1000 The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International

Powers of two

The positive powers of 2 are important in computer science because there are 2ⁿ possible values for an n-bit variable. In Mathematics, a power of two is any of the Integer powers of the number two; in other words two multiplied by itself a certain Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. See Binary numeral system. 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. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S)

The negative powers of 2 are commonly used, and the first two have special names: half, and quarter.

Powers of one

The integer powers of one are one: 1ⁿ = 1.

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 (0⁻ⁿ, 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. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation See for example Root of unity#Periodicity. In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power

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. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational 718. It can be approximated by large positive or negative powers of numbers close to one, such as

or

and defined as the limit

Any nonzero integer power of e can be computed like this:

The exponential function, defined by

has applications in many areas of mathematics and science. The limit of a sequence is one of the oldest concepts in Mathematical analysis. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) This definition of e^x matches the definition of e^k when x is an integer, but it also applies for fractional, real, or complex values of x, and even when x is a square matrix, which is used in ordinary differential equations. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally 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

Another popular formula is the power series

. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 +

Powers of real numbers

Exponentiation with various bases; from top to bottom, base 10 (green), base e (red), base 2 (blue), base ½ (cyan). The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line 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: x^1/8, x^1/4, x^1/2, x¹, x², x⁴, x⁸.

Raising a positive real number to a power that is not an integer can be accomplished in two ways.

• Rational number exponents can be defined in terms of n^th roots, and arbitrary nonzero exponents can then be defined by continuity. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, an n th root of a Number a is a number b such that bn = a.
• The natural logarithm can be used to define real exponents using the exponential function. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational

The identities and properties shown above are true for non-integer exponents as well.

Principal n-th root

Main article: n-th root

An nth root of a number a is a number b such that bⁿ = a. In Mathematics, an n th root of a Number a is a number b such that bn = a. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring.

When referring to the n-th root of a real number a it is assumed that what is desired is the principal n-th root of the number. In Mathematics, the real numbers may be described informally in several different ways 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

is called the principal n-th root of a, and is denoted ⁿ√a using the radical symbol √ . In Mathematics, an n th root of a Number a is a number b such that bn = a. The musical instrument is spelled Cymbal. A symbol is something --- such as an object, Picture, written word a sound a piece

.

For example: 4^1/2 = 2, 8^1/3 = 2, (−8)^1/3 = −2.

Note that if n is even, negative numbers do not have a principal n-th root. In Mathematics, the parity of an object states whether it is even or odd A negative number is a Number that is less than zero, such as −2

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 Non-integer rational numbers (commonly called fractions

For example, 8^2/3 = 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

where the limit is taken only over rational values of r. In Mathematics, the real numbers may be described informally in several different ways 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, if

then

Real powers of positive real numbers

The natural logarithm ln(x) is the inverse of the exponential function e^x. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B It is defined for every positive real number b and satisfies the equation

Assuming b^x is already defined, logarithm and exponent rules give the equality

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

This definition of the real number power b^x 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 a^k 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 a^m/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 a^m/n when n is even and m is odd must use the imaginary unit i, as described more fully in the next section. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

The logarithm method cannot be used to define a^k as a real number when a < 0 because e^x 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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output The function f(r) = a^r 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.

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 −1^q = 1 is dense in the rational numbers, as is the set of q for which −1^q = −1. In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if This means that the function (−1)^q is not continuous at any rational number q where it is defined.

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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Consider the right triangle (0, 1, 1+i·x/n). Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised For big values of n the triangle is almost a circular sector with a small central angle equal to x/n radian. A circular sector or circle sector, is the portion of a Circle enclosed by two radii and an arc. The triangles (0, (1+i·x/n)^k, (1+i·x/n)^k+1) are mutually similar for all values of k. Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other 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. In Mathematics, a unit circle is The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 The polar coordinates of this point are (r,θ) = (1,x), and the cartesian coordinates are (cos(x), sin(x)). In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane So e ^i·x = cos(x) + i·sin(x), and this is Euler's formula, connecting algebra to trigonometry by means of complex numbers. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

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

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

.

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

More simply: e^iπ=−1; e^x+iy = e^x(cos y+i sin y) .

Trigonometric functions

Main article: Euler's formula

It follows from Euler's formula that the trigonometric functions cosine and sine are

Historically, cosine and sine were defined geometrically before the invention of complex numbers. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic The above formula reduces the complicated formulas for trigonometric functions of a sum into the simple exponentiation formula

Using exponentiation with complex exponents one need not study trigonometry. In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables

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) of a Real number is its numerical value without regard to its sign. Direction is the information contained in the relative position of one point with respect to another point without the Distance information

Complex powers of positive real numbers

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

2 ⁱ = e ^i·ln(2) = cos(ln(2))+i·sin(ln(2)) = 0. 7692+i·0. 63896

e ⁱ = 0. 54030+i·0. 84147

10 ⁱ = −0. 66820+i·0. 74398

(e ^2·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 e^x 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. In Mathematics, a multivalued function (shortly multifunction, other names set-valued function, set-valued map, multi-valued map None of these options are entirely satisfactory.

The rational power of a complex number must be the solution to an algebraic equation. For example, w = z^1/2 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 z^1/2. Choosing one of the two solutions as the principal value of z^1/2 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 e^z = a, is called the principal value of the complex logarithm. In considering complex Multiple-valued functions in Complex analysis, the principal values of a function are the values along one chosen branch of that It is the unique solution whose imaginary part lies in the interval (−π, π]. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set 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. In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued 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.

Complex power of a complex number

The general complex power a^b of a nonzero complex number a is defined as

Using the identity from Euler's formula,

When the exponent is a rational number the power z = a^n/m is a solution to the equation z^m = aⁿ . This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions

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. In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power 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 zⁿ = 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. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

The number e^{2πi (1/n)} is the primitive nth root of unity with the smallest positive complex argument. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted (It is sometimes called the principal nth root of unity, although this terminology is not universal and should not be confused with the principal value of ⁿ√1, which is 1. In considering complex Multiple-valued functions in Complex analysis, the principal values of a function are the values along one chosen branch of that ^[1])

The other nth roots of unity are given by

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 a^z 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 a^1/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 a^1/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 a^1/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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Every complex number z can be written in the polar form

where r is a non-negative real number and θ is the (real) argument of z. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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. In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued The polar form has a simple geometric interpretation: if a complex number u + iv is thought of as representing a point (u, v) in the complex plane using Cartesian coordinates, then (r, θ) is the same point in polar coordinates. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by That is, r is the "radius" r² = u² + v² and θ is the "angle" θ = atan2(v, u). In Computing, atan2 is a two-argument function that makes it easy to find the angle round the origin of a point 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].

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

.

Then

and thus

If b is decomposed as c + di, then the formula for a^b can be written more explicitly as

This final formula allows complex powers to be computed easily from decompositions of the base into polar form and the exponent into Cartesian form. It is shown here both in polar form and in Cartesian form (via Euler's identity).

The following examples use the principal value, the branch cut which causes θ to be in the interval (−π, π]. To compute iⁱ, write i in polar and Cartesian forms:

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

Similarly, to find (−2)^{3 + 4i}, compute the polar form of −2,

and use the formula above to compute

The value of a complex power depends on the branch used. For example, if the polar form i = 1e^i(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(a^b) = 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:

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

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

and

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(x^y) 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.

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. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In many other settings, 0⁰ is left undefined. In calculus, 0⁰ is an indeterminate form, which must be analyzed rather than evaluated. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Calculus and other branches of Mathematical analysis, an indeterminate form is an Algebraic expression obtained in the context of Limits In general, mathematical analysis treats 0⁰ as undefined^[3] in order that the exponential function be continuous. 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:

• When 0⁰ is regarded as an empty product of zeros, its value is 1. In Mathematics, an empty product, or nullary product, is the result of multiplying no numbers
• The combinatorial interpretation of 0⁰ is the number of empty tuples of elements from the empty set. There is exactly one empty tuple. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the 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. In Mathematics, an empty function is a function whose domain is the Empty set.
• It greatly simplifies the theory of polynomials and power series that a constant term can be written ax⁰ for an arbitrary x. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + For example:
  • The formula for the coefficients in a product of polynomials would lose much of its simplicity if constant terms had to be treated specially.
  • A power series such as is not valid for x = 0 unless 0⁰, which appears in the numerator of the first term of the series, is 1. Otherwise one would need to use the longer identity .
  • The binomial theorem is not valid for x = 0, unless 0⁰ = 1. In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says ^[4] By defining 0⁰ to be 1, a special case of the theorem can be eliminated.
• In differential calculus, the power rule is not valid for n=1 at x=0 unless 0⁰ = 1. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change This article concerns power rules for computing the Derivative in Calculus In Mathematics, the power rule is a method for differentiating Defining it this way eliminates the need for a special case for the power rule.

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:

• 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, and . A more abstract way of saying this is the following: The real function x^y 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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output ^[5] That is, the function x^y has no continuous extension from the open first quadrant to include the point (0,0). ^[6] The rule in calculus, that whenever both sides of the equation are defined, would fail if 0⁰ were defined.
• The function z^z, viewed as a function of a complex number variable z and defined as e^z log z is undefined at z = 0 because log z is undefined at z = 0. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Moreover, because z^z has a logarithmic branch point at z = 0, it is not common to extend the domain of z^z to the origin in this context. In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued ^[7]

Treatment in programming languages and calculators

The computer programming languages that evaluate 0⁰ to be 1^[8] include bc, Haskell, J, Java, MATLAB, ML, Perl, Python, R, Ruby, Scheme, and SQL. Haskell is a standardized Purely functional Programming language with non-strict semantics, named after the Logician Haskell Curry Not to be confused with the J++ or J# programming languages The J programming language, developed in the early 1990s by MATLAB is a numerical computing environment and Programming language. 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 NOTES FOR EDITORS "Perl" is not an acronym (read the "Name" section below Python is a general-purpose High-level programming language. Its design philosophy emphasizes programmer productivity and code readability The R programming language, sometimes described as GNU S, is a programming language and software environment for statistical computing and Ruby is a dynamic, reflective, general purpose Object-oriented programming language that combines syntax inspired by Perl with Smalltalk Scheme is a Multi-paradigm programming language. It is one of the two main dialects of Lisp and supports a number of programming paradigms but is In the .NET Framework, the method System. In Object-oriented programming, the term method refers to a Subroutine that is exclusively associated either with a class (called class methods Math. Pow treats 0⁰ to be 1. Microsoft Excel issues an error when it evaluates 0⁰. In Computing, Microsoft Excel (full name Microsoft Office Excel) consists of a proprietary Spreadsheet -application written and distributed

Microsoft Windows' Calculator and Google search when used for its calculator function^[9] evaluate 0⁰ to 1. Microsoft Windows is a series of Software Operating systems and Graphical user interfaces produced by Microsoft. Google search is a Web search engine owned by Google Inc, and it is the most used search engine on the Web.

Maple simplifies a⁰ to 1 and 0^a to 0, even if no constraints are placed on a, and evaluates 0⁰ to 1. Maple is a general-purpose commercial Computer algebra system.

Mathematica simplifies a⁰ to 1, even if no constraints are placed on a. Mathematica is a computer program used widely in scientific engineering and mathematical fields It does not simplify 0^a, and it takes 0⁰ to be an indeterminate form.

Powers with infinity

Exponential expressions involving infinity may be thought of as generalizations of more familiar kinds of exponentiation, but there are at least two sharply distinct kinds of generalization to the infinite case. On the one hand, there is the combinatorial or set theoretic interpretation; see exponentiation of cardinal numbers.

On the other hand, one can find expressions such as ∞⁰ and 1^∞ arising 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 ∞; then, the limit of the power can be anything, not predictable from the limits of f and g.

However, if you have a number x^∞, where |x|<1, then x^∞ converges to zero.

It does make sense to say that ∞^∞ = ∞ if this is simply interpreted as an abbreviation for the theorem that if f and g both approach infinity as x approaches a, then lim f(x)^g(x) is also infinite. (Likewise, 7^∞ = ∞, (1. 3)^∞ = ∞, etc. )

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. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments Exponentiating by squaring is an Algorithm used for the fast computation of large Integer powers of a Number. In Mathematics and Computer science, optimal addition-chain exponentiation is a method of Exponentiation by positive Integer powers that requires 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.

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. In Mathematics, a composite function represents the application of one function to the results of another Thus f³(x) may mean f(f(f(x))); in particular, f ^-1(x) usually denotes the inverse function of f. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B

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)).

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 subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules

Let X be a set with a power-associative binary operation, which we will write multiplicatively. In Abstract algebra, power associativity is a weak form of Associativity. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two 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 natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Abstract algebra, power associativity is a weak form of Associativity.

Now additionally suppose that the operation has an identity element 1. 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 Then we can define x⁰ to be equal to 1 for any x. Now xⁿ is defined for any natural number n, including 0.

Finally, suppose that the operation has inverses, and that the multiplication is associative (so that the magma is a group). In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Then we can define x⁻ⁿ 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.

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

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. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. 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.

If in addition the multiplication operation is commutative (so that the set X is an abelian group), then we have some additional laws:

• (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". In Mathematics, commutativity is the ability to change the order of something without changing the end result An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Addition is the mathematical process of putting things together Thus, each of the laws of exponentiation above has an analogue among laws of multiplication. Analogy is both the cognitive process of transferring Information from a particular subject (the analogue or source to another particular subject (the target and

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. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class That is, g^h = h⁻¹gh, where g and h are elements of some group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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. In Mathematics, racks and quandles are sets with a Binary operation satisfying axioms analogous to the Reidemeister moves of knot diagram In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations,

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 (a₀, a₁, a₂, . . . , a_n−1) represents the function that sends i to a_i.

For an infinite cardinal number κ and a set A, the notation A^κ is also used to denote the set of all functions from a set of size κ to A. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. This is sometimes written ^κA 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. In Mathematics, a structure on a set, or more generally a type, consists of additional Mathematical objects that in some manner attach to the For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. Linear algebra is the branch of Mathematics concerned with The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added That is, we can speak of

where each V_i is a vector space. Then if V_i = V for each i, the resulting direct sum can be written in exponential notation as V^(+)N, or simply V^N 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. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose 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 Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

If the base of the exponentiation operation is a set, the exponentiation operation is the Cartesian product unless otherwise stated. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. Since multiple Cartesian products produce an n-tuple, which can be represented by a function on a set of appropriate cardinality, S^N becomes simply the set of all functions from N to S in this case:

This fits in with the exponentiation of cardinal numbers, in the sense that |S^N| = |S|^|N|, where |X| is the cardinality of X. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function When N=2={0,1}, we have |2^X| = 2^|X|, where 2^X, 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 \mathcal{P}(S P ( S)

Exponentiation in category theory

Main article: Cartesian closed category

In a Cartesian closed category, the exponential operation can be used to raise an arbitrary object to the power of another object. 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 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 This generalizes the Cartesian product in the category of sets. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

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. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In the mathematical field of Set theory, ordinal arithmetic describes the three usual operations on Ordinal numbers addition multiplication and exponentiation This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set.

If κ and λ are cardinal numbers, the expression κ^λ 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 Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In the ordinal numbers, exponentiation is defined by transfinite induction. Transfinite induction is an extension of Mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. For ordinals α and β, the exponential α^β is the supremum of the ordinal product α^γα over all γ < β.

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. In Mathematics, tetration (also known as hyper -4 Iterating tetration leads to another operation, and so on. This sequence of operations is captured by the Ackermann function and Knuth's up-arrow notation. In Recursion theory, the Ackermann function or Ackermann-Péter function is a simple example of a general recursive function that is not primitive In Mathematics, Knuth's up-arrow notation is a method of notation of very large Integers introduced by Donald Knuth in 1976

Exponentiation in programming languages

The superscript notation x^y is convenient in handwriting but inconvenient for typewriters and computer terminals that align the baselines of all characters on each line. A typewriter is a mechanical or Electromechanical device with a set of "keys" that when pressed cause characters to be printed on a medium 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 Many programming languages have alternate ways of expressing exponentiation that do not use superscripts:

• x ↑ y: Algol, Commodore BASIC
• x ^ y: BASIC, J, Matlab, R, Microsoft Excel, TeX (and its derivatives), Haskell (for integer exponents), and most computer algebra systems
• x ** y: Ada, Bash, Fortran, FoxPro, Perl, Python, Ruby, SAS, ABAP, Haskell (for floating-point exponents), Turing
• x * y: APL
• Power(x, y): Microsoft Excel, Delphi/Pascal (declared in "Math"-unit)
• pow(x, y): C, C++, PHP
• Math. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. Algol (β Per / Beta Persei known colloquially as the Demon Star, is a bright Star in the Constellation Perseus. Commodore BASIC, also known as PET BASIC, is the dialect of the BASIC programming language used in Commodore International 's 8-bit Home In Computer programming, BASIC (an Acronym for Beginner's All-purpose Symbolic Instruction Code) is a family of High-level programming languages Not to be confused with the J++ or J# programming languages The J programming language, developed in the early 1990s by MATLAB is a numerical computing environment and Programming language. The R programming language, sometimes described as GNU S, is a programming language and software environment for statistical computing and In Computing, Microsoft Excel (full name Microsoft Office Excel) consists of a proprietary Spreadsheet -application written and distributed TeX (ˈtɛx as in Greek, often /ˈtɛk/ in English; written with a lowercase 'e' in imitation of the logo is a Typesetting system designed and mostly Haskell is a standardized Purely functional Programming language with non-strict semantics, named after the Logician Haskell Curry 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 Bash is a Free software Unix shell written for the GNU Project. Fortran (previously FORTRAN) is a general-purpose, procedural, imperative Programming language that is especially suited to NOTES FOR EDITORS "Perl" is not an acronym (read the "Name" section below Python is a general-purpose High-level programming language. Its design philosophy emphasizes programmer productivity and code readability Ruby is a dynamic, reflective, general purpose Object-oriented programming language that combines syntax inspired by Perl with Smalltalk SAS (pronounced "sass" originally Statistical Analysis System) is an integrated system of software products provided by SAS Institute that enables the programmer ABAP ( A dvanced B usiness A pplication P rogramming originally A llgemeiner B erichts- A ufbereitungs- P rozessor Turing is a Pascal -like Programming language developed in 1982 by Ric Holt and James Cordy, then of University of Toronto, Canada tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured C++ (" C Plus Plus " ˌsiːˌplʌsˈplʌs is a general-purpose Programming language. PHP is a computer Scripting language. Originally designed for producing Dynamic web pages it has evolved to include a Command line interface capability pow(x, y): Java, JavaScript, Modula-3
• Math. JavaScript is a Scripting language most often used for Client-side web development Modula-3 is a Programming language conceived as a successor to an upgraded version of Modula-2. Pow(x, y): C#
• (expt x y): Common Lisp, Scheme

In Bash, C, C++, C#, Java, JavaScript, PHP, Python and Ruby, the symbol ^ represents bitwise XOR. C# (pronounced C Sharp is a Multi-paradigm Common Lisp, commonly abbreviated CL, is a dialect of the Lisp Programming language, published in ANSI standard document Information Scheme is a Multi-paradigm programming language. It is one of the two main dialects of Lisp and supports a number of programming paradigms but is In Pascal, it represents indirection. 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. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel. Nicolas Chuquet (1445 but some sources say c 1455 &ndash 1488 some sources say c Henricus Grammateus (also known as Henricus Scriptor, Heinrich Schreyber or Heinrich Schreiber) (1495 - 1525 or 1526 was a German Mathematician Michael Stifel or Styfel ( Esslingen 1486 or 1487 – April 19, 1567, Jena) was an Augustinian monk who became an early 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). Robert Recorde (c 1510 &ndash 1558 was a Welsh Physician and Mathematician. The zenzizenzizenzic of a number is its eighth power. This term was suggested by Robert Recorde, a 16th century Welsh writer of popular Mathematics Biquadrate has been used to refer to the fourth power as well.

Another historical synonym, involution,^[11] is now rare and should not be confused with its more common meaning.

References

See also

• Exponential growth and Exponential decay
• List of exponential topics
• Logarithm
• Tetration
• Modular exponentiation
• Nth root
• Unicode subscripts and superscripts

External links

• sci.math FAQ: What is 0⁰?
• Introducing 0th power on PlanetMath
• Laws of Exponents with derivation and examples
• 1058 Powers of Two