Exponentiation

Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n, and pronounced as "b raised to the power of n".[1] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:[1]

The exponent is usually shown as a superscript to the right of the base. In that case, bn is called "b raised to the nth power", "b raised to the power of n", "the nth power of b", "b to the nth power",[2] or most briefly as "b to the nth".

One has b1 = b, and, for any positive integers m and n, one has bnbm = bn+m. To extend this property to non-positive integer exponents, b0 is defined to be 1, and bn (with n a positive integer and b not zero) is defined as 1/bn. In particular, b−1 is equal to 1/b, the reciprocal of b.

The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.

Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.

The term power (Latin: potentia, potestas, dignitas) is a mistranslation[3][4] of the ancient Greek δύναμις (dúnamis, here: "amplification"[3]) used by the Greek mathematician Euclid for the square of a line,[5] following Hippocrates of Chios.[6] In The Sand Reckoner, Archimedes discovered and proved the law of exponents, 10a ⋅ 10b = 10a+b, necessary to manipulate powers of 10.[citation needed] In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms مَال (māl, "possessions", "property") for a square—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"[7]—and كَعْبَة (kaʿbah, "cube") for a cube, which later Islamic mathematicians represented in mathematical notation as the letters mīm (m) and kāf (k), respectively, by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī.[8]

In the late 16th century, Jost Bürgi used Roman numerals for exponents.[9]

Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel in the 16th century. The word exponent was coined in 1544 by Michael Stifel.[10][11] Samuel Jeake introduced the term indices in 1696.[5] In the 16th century, Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth).[7] Biquadrate has been used to refer to the fourth power as well.

Early in the 17th century, the first form of our modern exponential notation was introduced by René Descartes in his text titled La Géométrie; there, the notation is introduced in Book I.[12]

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

Another historical synonym,[clarification needed] involution, is now rare[13] and should not be confused with its more common meaning.

In 1748, Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing:

"consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant."[14]

The expression b2 = bb is called "the square of b" or "b squared", because the area of a square with side-length b is b2.

Similarly, the expression b3 = bbb is called "the cube of b" or "b cubed", because the volume of a cube with side-length b is b3.

When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, 35 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power.

The word "raised" is usually omitted, and sometimes "power" as well, so 35 can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiation bn can be expressed as "b to the power of n", "b to the nth power", "b to the nth", or most briefly as "b to the n".

A formula with nested exponentiation, such as 357 (which means 3(57) and not (35)7), is called a tower of powers, or simply a tower.

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

The definition of the exponentiation as an iterated multiplication can be formalized by using induction,[15] and this definition can be used as soon one has an associative multiplication:

The associativity of multiplication implies that for any positive integers m and n,

This definition is the only possible that allows extending the formula

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

Exponentiation with negative exponents is defined by the following identity, which holds for any integer n and nonzero b:

The following identities, often called exponent rules, hold for all integer exponents, provided that the base is non-zero:[1]

Unlike addition and multiplication, exponentiation is not commutative. For example, 23 = 8 ≠ 32 = 9. Also unlike addition and multiplication, exponentiation is not associative. For example, (23)2 = 82 = 64, whereas 2(32) = 29 = 512. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up[17][18][19][20] (or left-associative). That is,

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

However, this formula is true only if the summands commute (i.e. that ab = ba), which is implied if they belong to a structure that is commutative. Otherwise, if a and b are, say, square matrices of the same size, this formula cannot be used. It follows that in computer algebra, many algorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose computer algebra systems use a different notation (sometimes ^^ instead of ^) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

For nonnegative integers n and m, the value of nm is the number of functions from a set of m elements to a set of n elements (see cardinal exponentiation). Such functions can be represented as m-tuples from an n-element set (or as m-letter words from an n-letter alphabet). Some examples for particular values of m and n are given in the following table:

In the base ten (decimal) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, 103 = 1000 and 10−4 = 0.0001.

Exponentiation with base 10 is used in scientific notation to denote large or small numbers. For instance, 299792458 m/s (the speed of light in vacuum, in metres per second) can be written as 2.99792458×108 m/s and then approximated as 2.998×108 m/s.

SI prefixes based on powers of 10 are also used to describe small or large quantities. For example, the prefix kilo means 103 = 1000, so a kilometre is 1000 m.

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

Powers of 2 appear in set theory, since a set with n members has a power set, the set of all of its subsets, which has 2n members.

Integer powers of 2 are important in computer science. The positive integer powers 2n give the number of possible values for an n-bit integer binary number; for example, a byte may take 28 = 256 different values. The binary number system expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a binary point, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and the negative exponents are determined by the rank on the right of the point.

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

Because of this, powers of −1 are useful for expressing alternating sequences. For a similar discussion of powers of the complex number i, see § Powers of complex numbers.

The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound:

This can be read as "b to the power of n tends to +∞ as n tends to infinity when b is greater than one".

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

If b < –1, bn, alternates between larger and larger positive and negative numbers as n alternates between even and odd, and thus does not tend to any limit as n grows.

If the exponentiated number varies while tending to 1 as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is

Other limits, in particular those of expressions that take on an indeterminate form, are described in § Limits of powers below.

See § Real exponents and § Non-integer powers of complex numbers for details on the way these problems may be handled.

For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (§ Limits of rational exponents, below), or in terms of the logarithm of the base and the exponential function (§ Powers via logarithms, below). The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to complex exponents.

On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values (see § Real exponents with negative bases). One may choose one of these values, called the principal value, but there is no choice of the principal value for which the identity

is true; see § Failure of power and logarithm identities. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.

Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by continuity with the rule[22]

where the limit is taken over rational values of r only. This limit exists for every positive b and every real x.

The definition of ex as the exponential function allows defining bx for every positive real numbers b, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function ex means that one has

If b is a positive real number, exponentiation with base b and complex exponent z is defined by means of the exponential function with complex argument (see the end of § The exponential function, above) as

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

The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an nth root of a complex number can be obtained by taking the nth root of the absolute value and dividing its argument by n:

The nth roots of unity are the n complex numbers such that wn = 1, where n is a positive integer. They arise in various areas of mathematics, such as in discrete Fourier transform or algebraic solutions of algebraic equations (Lagrange resolvent).

The nth roots of unity allow expressing all nth roots of a complex number z as the n products of a given nth roots of z with a 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 all cases, the complex logarithm is used to define complex exponentiation as

Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions. For example:

If b is a positive real algebraic number, and x is a rational number, then bx is an algebraic number. This results from the theory of algebraic extensions. This remains true if b is any algebraic number, in which case, all values of bx (as a multivalued function) are algebraic. If x is irrational (that is, not rational), and both b and x are algebraic, Gelfond–Schneider theorem asserts that all values of bx are transcendental (that is, not algebraic), except if b equals 0 or 1.

An algebraic structure consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a monoid. In such a monoid, exponentiation of an element x is defined inductively by

Exponentiation with integer exponents obeys the following laws, for x and y in the algebraic structure, and m and n integers:

These definitions are widely used in many areas of mathematics, notably for groups, rings, fields, square matrices (which form a ring). They apply also to functions from a set to itself, which form a monoid under function composition. This includes, as specific instances, geometric transformations, and endomorphisms of any mathematical structure.

A multiplicative group is a set with as associative operation denoted as multiplication, that has an identity element, and such that every element has an inverse.

Order of elements play a fundamental role in group theory. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the order of the group). The possible orders of group elements are important in the study of the structure of a group (see Sylow theorems), and in the classification of finite simple groups.

These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups.[31] Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the heat equation, Schrödinger equation, wave equation, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.

A field is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is associative and every nonzero element has a multiplicative inverse. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of 0. Common examples are the complex numbers and their subfields, the rational numbers and the real numbers, which have been considered earlier in this article, and are all infinite.

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

This means the functor "exponentiation to the power T" is a right adjoint to the functor "direct product with T".

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

Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 00. The limits in these examples exist, but have different values, showing that the two-variable function xy has no limit at the point (0, 0). One may consider at what points this function does have a limit.

In fact, f has a limit at all accumulation points of D, except for (0, 0), (+∞, 0), (1, +∞) and (1, −∞).[32] Accordingly, this allows one to define the powers xy by continuity whenever 0 ≤ x ≤ +∞, −∞ ≤ y ≤ +∞, except for 00, (+∞)0, 1+∞ and 1−∞, which remain indeterminate forms.

These powers are obtained by taking limits of xy for positive values of x. This method does not permit a definition of xy when x < 0, since pairs (x, y) with x < 0 are not accumulation points of D.

On the other hand, when n is an integer, the power xn is already meaningful for all values of x, including negative ones. This may make the definition 0n = +∞ obtained above for negative n problematic when n is odd, since in this case xn → +∞ as x tends to 0 through positive values, but not negative ones.

Computing bn using iterated multiplication requires n − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2100, apply Horner's rule to the exponent 100 written in binary:

Then compute the following terms in order, reading Horner's rule from right to left.

Programming languages generally express exponentiation either as an infix operator or as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is the caret (^). The original version of ASCII included an uparrow symbol (), intended for exponentiation, but this was replaced by the caret in 1967, so the caret became usual in programming languages.[38] The notations include:

In most programming languages with an infix exponentiation operator, it is right-associative, that is, a^b^c is interpreted as a^(b^c).[42] This is because (a^b)^c is equal to a^(b*c) and thus not useful. In some languages, it is left-associative, notably in Algol, Matlab and the Microsoft Excel formula language.

Still others only provide exponentiation as part of standard libraries: