Exponentiation

{{#invoke:Hatnote|hatnote}}Template:Main other

B=1/2}} (cyan). Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself. Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent (or power) n. When n is a natural number (i.e., a positive integer), exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases: ${\displaystyle b^{n}=\underbrace {b\times \cdots \times b} _{n}}$ The exponent is usually shown as a superscript to the right of the base. Some common exponents have their own names: the exponent 2 (or 2nd power) is called the square of b (b2) or b squared; the exponent 3 (or 3rd power) is called the cube of b (b3) or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b. When n is a negative integer and b is not zero, bn is naturally defined as 1/bn, preserving the property bn × bm = bn + m. Exponentiation for integer exponents can 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. Background and terminology The expression b2 = b·b is called the square of b because the area of a square with side-length b is b2. It is pronounced "b squared". The expression b3 = b·b·b is called the cube of b because the volume of a cube with side-length b is b3. It is pronounced "b 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, 3 raised to the fifth power, or 3 to the power of 5. The word "raised" is usually omitted, and very often "power" as well, so 35 is typically pronounced "three to the fifth" or "three to the five". The exponentiation bn can be read as b raised to the n-th power, or b raised to the power of n, or b raised by the exponent of n, or most briefly as b to the n. Exponentiation may be generalized from integer exponents to more general types of numbers. The word "exponent" was coined in 1544 by Michael Stifel.[1] The modern notation for exponentiation was introduced by René Descartes in his Géométrie of 1637.[2][3] Integer exponents The exponentiation operation with integer exponents requires only elementary algebra. Positive integer exponents Formally, powers with positive integer exponents may be defined by the initial condition[4] ${\displaystyle b^{1}=b}$ and the recurrence relation ${\displaystyle b^{n+1}=b^{n}\cdot b}$ From the associativity of multiplication, it follows that for any positive integers m and n, ${\displaystyle b^{m+n}=b^{m}\cdot b^{n}}$ Zero exponent Any nonzero number raised by the exponent 0 is 1;[5] one interpretation of such a power is as an empty product. The case of 00 is discussed below. Negative exponents The following identity holds for an arbitrary integer n and nonzero b: ${\displaystyle b^{-n}=1/b^{n}}$ Raising 0 by a negative exponent is left undefined. The identity above may be derived through a definition aimed at extending the range of exponents to negative integers. For non-zero b and positive n, the recurrence relation from the previous subsection can be rewritten as ${\displaystyle b^{n}={b^{n+1}}/{b},\quad n\geq 1.}$ By defining this relation as valid for all integer n and nonzero b, it follows that {\displaystyle {\begin{aligned}b^{0}&={b^{1}}/{b}=1\\b^{-1}&={b^{0}}/{b}={1}/{b}\end{aligned}}} and more generally for any nonzero b and any nonnegative integer n, ${\displaystyle b^{-n}={1}/{b^{n}}.}$ This is then readily shown to be true for every integer n. Combinatorial interpretation For nonnegative 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.  05 = │ { } │ = 0 There is no 5-tuple from the empty set. 15 = │ { (1,1,1,1,1) } │ = 1 There is one 5-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 eight 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 nine 2-tuples from a three-element set. 41 = │ { (1), (2), (3), (4) } │ = 4 There are four 1-tuples from a four-element set. 50 = │ { () } │ = 1 There is exactly one 0-tuple. {{#invoke:see also|seealso}} Identities and properties The following identities hold for all integer exponents, provided that the base is non-zero: {\displaystyle {\begin{aligned}b^{m+n}&=b^{m}\cdot b^{n}\\(b^{m})^{n}&=b^{m\cdot n}\\(b\cdot c)^{n}&=b^{n}\cdot c^{n}\end{aligned}}} Exponentiation is not commutative. This contrasts with addition and multiplication, which are. For example, 2 + 3 = 3 + 2 = 5 and 2 · 3 = 3 · 2 = 6, but 23 = 8, whereas 32 = 9. Exponentiation is not associative either. Addition and multiplication are. For example, (2 + 3) + 4 = 2 + (3 + 4) = 9 and (2 · 3) · 4 = 2 · (3 · 4) = 24, but 23 to the 4 is 84 or 4,096, whereas 2 to the 34 is 281 or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up: ${\displaystyle b^{p^{q}}=b^{(p^{q})}\neq (b^{p})^{q}=b^{(p\cdot q)}=b^{p\cdot q}.}$ Particular bases Powers of ten {{#invoke:see also|seealso}} 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, Template:Val = 1,000 and Template:Val = 0.0001. Exponentiation with base 10 is used in scientific notation to denote large or small numbers. For instance, 299,792,458 m/s (the speed of light in vacuum, in metre per second) can be written as Template:Val and then approximated as Template:Val. SI prefixes based on powers of 10 are also used to describe small or large quantities. For example, the prefix kilo means Template:Val = 1,000, so a kilometre is 1,000 metres. Powers of two The positive powers of 2 are important in computer science because there are 2n possible values for an n-bit binary register. 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. The negative powers of 2 are commonly used, and the first two have special names: half, and quarter. In the base 2 (binary) number system, integer powers of 2 are written as 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, two to the power of three is written as 1000 in binary. Powers of one The integer powers of one are all 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 (0n, where n < 0) is 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 If n is an even integer, then (−1)n = 1. If n is an odd integer, then (−1)n = −1. Because of this, powers of −1 are useful for expressing alternating sequences. For a similar discussion of powers of the complex number i, see the section on Powers of complex numbers. Large exponents The limit of a sequence of powers of a number greater than one diverges, in other words they grow without bound: bn → ∞ as n → ∞ when b > 1 This can be read as "b to the power of n tends to +∞ as n tends to infinity when b is greater than one". Powers of a number with absolute value less than one tend to zero: bn → 0 as n → ∞ when |b| < 1 Any power of one is always itself: bn = 1 for all n if b = 1 If the number b varies tending to 1 as the exponent tends to infinity then the limit is not necessarily one of those above. A particularly important case is (1 + 1/n)ne as n → ∞ See the section below, The exponential function. Other limits, in particular of those tending to indeterminate forms, are described in limits of powers below. Rational exponents {{#invoke:main|main}} From top to bottom: x1/8, x1/4, x1/2, x1, x2, x4, x8. An n-th root of a number b is a number x such that xn = b. If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to xn = b. This solution is called the principal n-th root of b. It is denoted nb, where √ is the radical symbol; alternatively, it may be written b1/n. For example: 41/2 = 2, 81/3 = 2. This follows from noting that ${\displaystyle x^{n}=\underbrace {b^{\frac {1}{n}}\times b^{\frac {1}{n}}\times \cdots \times b^{\frac {1}{n}}} _{n}=b^{\left({\frac {1}{n}}+{\frac {1}{n}}+\cdots +{\frac {1}{n}}\right)}=b^{\frac {n}{n}}=b^{1}=b}$ If n is even, then xn = b has two real solutions if b is positive, which are the positive and negative nth roots. The equation has no solution in real numbers if b is negative. If n is odd, then xn = b has one real solution. The solution is positive if b is positive and negative if b is negative. Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for negative b if m and n are odd, and can be either sign if b is positive and n is even. (−27)1/3 = −3, (−27)2/3 = 9, and 43/2 has two roots 8 and −8, however by convention 43/2 denotes the principal root which is 8. Since there is no real number x such that x2 = −1, the definition of bm/n when b is negative and n is even must use the imaginary unit i, as described more fully in the section Powers of complex numbers. A power of a positive real number b with a rational exponent m/n in lowest terms satisfies ${\displaystyle b^{\frac {m}{n}}=\left(b^{m}\right)^{\frac {1}{n}}={\sqrt[{n}]{b^{m}}}}$ where m is an integer and n is a positive integer. Care needs to be taken when applying the power law identities with negative nth roots. For instance, −27 = (−27)((2/3)⋅(3/2)) = ((−27)2/3)3/2 = 93/2 = 27 is clearly wrong. The problem here occurs in taking the positive square root rather than the negative one at the last step, but in general the same sorts of problems occur as described for complex numbers in the section Failure of power and logarithm identities. Real exponents The identities and properties shown above for integer exponents are true for positive real numbers with non-integer exponents as well. However the identity ${\displaystyle (b^{r})^{s}=b^{r\cdot s}}$ cannot be extended consistently to cases where b is a negative real number (see Real exponents with negative bases). The failure of this identity is the basis for the problems with complex number powers detailed under failure of power and logarithm identities. The extension of exponentiation to real powers of positive real numbers can be done either by extending the rational powers to reals by continuity, or more usually as given in the section Powers via logarithms below. Limits of rational exponents 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[6] ${\displaystyle b^{x}=\lim _{r(\in \mathbb {Q} )\to x}b^{r}\quad (b\in \mathbb {R} ^{+},\,x\in \mathbb {R} )}$ where the limit as r gets close to x is taken only over rational values of r. This limit only exists for positive b. The (ε, δ)-definition of limit is used, this involves showing that for any desired accuracy of the result ${\displaystyle \scriptstyle b^{x}}$ one can choose a sufficiently small interval around Template:Mvar so all the rational powers in the interval are within the desired accuracy. For example, if ${\displaystyle \scriptstyle x\;=\;\pi }$, the nonterminating decimal representation ${\displaystyle \scriptstyle \pi \;=\;3.14159\ldots }$ can be used (based on strict monotonicity of the rational power) to obtain the intervals bounded by rational powers ${\displaystyle [b^{3},b^{4}]}$, ${\displaystyle [b^{3.1},b^{3.2}]}$, ${\displaystyle [b^{3.14},b^{3.15}]}$, ${\displaystyle [b^{3.141},b^{3.142}]}$, ${\displaystyle [b^{3.1415},b^{3.1416}]}$, ${\displaystyle [b^{3.14159},b^{3.14160}]}$, … The bounded intervals converge to a unique real number, denoted by ${\displaystyle \scriptstyle b^{\pi }}$. This technique can be used to obtain any irrational power of Template:Mvar. The function ${\displaystyle \scriptstyle f(x)\;=\;b^{x}}$ is thus defined for any real number Template:Mvar. The exponential function {{#invoke:main|main}} The important mathematical constant [[E (mathematical constant)|Template:Mvar]], sometimes called Euler's number, is approximately equal to 2.718 and is the base of the natural logarithm. Although exponentiation of e could, in principle, be treated the same as exponentiation of any other real number, such exponentials turn out to have particularly elegant and useful properties. Among other things, these properties allow exponentials of e to be generalized in a natural way to other types of exponents, such as complex numbers or even matrices, while coinciding with the familiar meaning of exponentiation with rational exponents. As a consequence, the notation ex usually denotes a generalized exponentiation definition called the exponential function, exp(x), which can be defined in many equivalent ways, for example by: ${\displaystyle \exp(x)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}}$ Among other properties, exp satisfies the exponential identity: ${\displaystyle \exp(x+y)=\exp(x)\cdot \exp(y)}$ The exponential function is defined for all integer, fractional, real, and complex values of Template:Mvar. In fact, the matrix exponential is well-defined for square matrices (in which case the exponential identity only holds when Template:Mvar and Template:Mvar commute), and is useful for solving systems of linear differential equations. Since exp(1) is equal to Template:Mvar and exp(x) satisfies the exponential identity, it immediately follows that exp(x) coincides with the repeated-multiplication definition of ex for integer x, and it also follows that rational powers denote (positive) roots as usual, so exp(x) coincides with the ex definitions in the previous section for all real x by continuity. Powers via logarithms The natural logarithm ln(x) is the inverse of the exponential function ex. It is defined for b > 0, and satisfies ${\displaystyle b=e^{\ln b}}$ If bx is to preserve the logarithm and exponent rules, then one must have ${\displaystyle b^{x}=(e^{\ln b})^{x}=e^{x\cdot \ln b}}$ for each real number x. This can be used as an alternative definition of the real number power bx and agrees with the definition given above using rational exponents and continuity. The definition of exponentiation using logarithms is more common in the context of complex numbers, as discussed below. Real exponents with negative bases Powers of a positive real number are always positive real numbers. The solution of x2 = 4, however, can be either 2 or −2. The principal value of 41/2 is 2, but −2 is also a valid square root. If the definition of exponentiation of real numbers is extended to allow negative results then the result is no longer well behaved. Neither the logarithm method nor the rational exponent method can be used to define br as a real number for a negative real number b and an arbitrary real number r. Indeed, er is positive for every real number r, so ln(b) is not defined as a real number for b ≤ 0. The rational exponent method cannot be used for negative values of b because it relies on continuity. The function f(r) = br has a unique continuous extension[6] from the rational numbers to the real numbers for each b > 0. But when b < 0, the function f is not even continuous on the set of rational numbers r for which it is defined. For example, consider b = −1. The nth root of −1 is −1 for every odd natural number n. So if n is an odd positive integer, (−1)(m/n) = −1 if m is odd, and (−1)(m/n) = 1 if m is even. Thus the set of rational numbers q for which (−1)q = 1 is dense in the rational numbers, as is the set of q for which (−1)q = −1. This means that the function (−1)q is not continuous at any rational number q where it is defined. On the other hand, arbitrary complex powers of negative numbers b can be defined by choosing a complex logarithm of b. Complex exponents with positive real bases Imaginary exponents with base e {{#invoke:main|main}} {{ safesubst:#invoke:Unsubst||N=Technical |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} The exponential function ez can be defined as the limit of (1 + z/N)N, as N approaches infinity, and thus e is the limit of (1 + /N)N. In this animation N takes values increasing from 1 to 100. The computation of (1 + /N)N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + /N)N. It can be seen that as N gets larger (1 + /N)N approaches a limit of −1. Therefore, e = −1, which is known as Euler's identity. The geometric interpretation of the operations on complex numbers and the definition of the exponential function is the clue to understanding eix for real x. In particular, for two complex numbers z1, z2 with polar coordinates (r1, θ1), (r2, θ2), their product z1z2 is equal to (r1r2, θ1 + θ2). Consider the right triangle in the complex plane which has 0, 1, 1 + ix/n as vertices. For large values of n, the triangle is almost a circular sector with a radius of 1 and a small central angle equal to x/n radians. 1 + ix/n may then be approximated by the number with polar coordinates (1, x/n). So, in the limit as n approaches infinity, (1 + ix/n)n approaches (1, x/n)n = (1n, nx/n) = (1, x), the point on the unit circle whose angle from the positive real axis is x radians. The cartesian coordinates of this point are (cos x, sin x). So e ix = cos x + isin x; this is Euler's formula, connecting algebra to trigonometry by means of complex numbers. The solutions to the equation ez = 1 are the integer multiples of 2πi: ${\displaystyle \{z:e^{z}=1\}=\{2k\pi i:k\in {\mathbb {Z} }\}}$ More generally, if ev = w, then every solution to ez = w can be obtained by adding an integer multiple of 2πi to v: ${\displaystyle \{z:e^{z}=w\}=\{v+2k\pi i:k\in {\mathbb {Z} }\}}$ Thus the complex exponential function is a periodic function with period 2πi. More simply: e = −1; ex + iy = ex(cos y + i sin y). Trigonometric functions {{#invoke:main|main}} It follows from Euler's formula stated above that the trigonometric functions cosine and sine are ${\displaystyle \cos(z)={\frac {e^{iz}+e^{-iz}}{2}};\qquad \sin(z)={\frac {e^{iz}-e^{-iz}}{2i}}}$ 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 ${\displaystyle e^{i(x+y)}=e^{ix}\cdot e^{iy}}$ Using exponentiation with complex exponents may reduce problems in trigonometry to algebra. Complex exponents with base e The power z = ex + iy can be computed as ex · eiy. The real factor ex is the absolute value of z and the complex factor eiy identifies the direction of z. Complex exponents with positive real bases If b is a positive real number, and z is any complex number, the power bz is defined as ez·ln(b), where x = ln(b) is the unique real solution to the equation ex = b. So the same method working for real exponents also works for complex exponents. For example: 2i = e i·ln(2) = cos(ln(2)) + i·sin(ln(2)) ≈ 0.76924 + 0.63896i ei ≈ 0.54030 + 0.84147i 10i ≈ −0.66820 + 0.74398i (e)i ≈ 535.49i ≈ 1 The identity ${\displaystyle (b^{z})^{u}=b^{zu}}$ is not generally valid for complex powers. A simple counterexample is given by: ${\displaystyle (e^{2\pi i})^{i}=1^{i}=1\neq e^{-2\pi }=e^{2\pi i\cdot i}}$ The identity is, however, valid when ${\displaystyle z}$ is a real number, and also when ${\displaystyle u}$ is an integer. Powers of complex numbers Integer powers of nonzero complex numbers are defined by repeated multiplication or division as above. If i is the imaginary unit and n is an integer, then in equals 1, i, −1, or −i, according to whether the integer n is congruent to 0, 1, 2, or 3 modulo 4. Because of this, the powers of i are useful for expressing sequences of period 4. Complex powers of positive reals are defined via ex as in section Complex powers of positive real numbers above. These are continuous functions. Trying to extend these functions to the general case of noninteger powers of complex numbers that are not positive reals leads to difficulties. Either we define discontinuous functions or multivalued functions. Neither of these options is entirely satisfactory. The rational power of a complex number must be the solution to an algebraic equation. Therefore it always has a finite number of possible values. For example, 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. A unique but somewhat arbitrary solution called the principal value can be chosen using a general rule which also applies for nonrational powers. Complex powers and logarithms are more naturally handled as single valued functions on a Riemann surface. Single valued versions are defined by choosing a sheet. The value has a discontinuity along a branch cut. Choosing one out of many solutions as the principal value leaves us with functions that are not continuous, and the usual rules for manipulating powers can lead us astray. Any nonrational power of a complex number has an infinite number of possible values because of the multi-valued nature of the complex logarithm. The principal value is a single value chosen from these by a rule which, amongst its other properties, ensures powers of complex numbers with a positive real part and zero imaginary part give the same value as for the corresponding real numbers. Exponentiating a real number to a complex power is formally a different operation from that for the corresponding complex number. However in the common case of a positive real number the principal value is the same. The powers of negative real numbers are not always defined and are discontinuous even where defined. In fact, they are only defined when the exponent is a rational number with the denominator being an odd integer. When dealing with complex numbers the complex number operation is normally used instead. Complex exponents with complex bases For complex numbers w and z with w ≠ 0, the notation wz is ambiguous in the same sense that log w is. To obtain a value of wz, first choose a logarithm of w; call it log w. Such a choice may be the principal value Log w (the default, if no other specification is given), or perhaps a value given by some other branch of log w fixed in advance. Then, using the complex exponential function one defines ${\displaystyle w^{z}=e^{z\log w}}$ because this agrees with the earlier definition in the case where w is a positive real number and the (real) principal value of log w is used. If z is an integer, then the value of wz is independent of the choice of log w, and it agrees with the earlier definition of exponentation with an integer exponent. If z is a rational number m/n in lowest terms with z > 0, then the infinitely many choices of log w yield only n different values for wz; these values are the n complex solutions s to the equation sn = wm. If z is an irrational number, then the infinitely many choices of log w lead to infinitely many distinct values for wz. The computation of complex powers is facilitated by converting the base w to polar form, as described in detail below. A similar construction is employed in quaternions. Complex roots of unity {{#invoke:main|main}} The three 3rd roots of 1 A complex number w such that wn = 1 for a positive integer n is an nth root of unity. Geometrically, the nth roots of unity lie on the unit circle of the complex plane at the vertices of a regular n-gon with one vertex on the real number 1. If wn = 1 but wk ≠ 1 for all natural numbers k such that 0 < k < n, then w is called a primitive nth root of unity. The negative unit −1 is the only primitive square root of unity. The imaginary unit i is one of the two primitive 4-th roots of unity; the other one is −i. The number e{{ safesubst:#invoke:Unsubst||$B=2πi/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 n1, which is 1.[7])

The other nth roots of unity are given by

${\displaystyle \left(e^{{\frac {2}{n}}\pi i}\right)^{k}=e^{{\frac {2}{n}}\pi ik}}$

for 2 ≤ kn.

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 wq in the important special case where q = 1/n and n is a positive integer. These are the nth roots of w; they are solutions of the equation zn = w. As with real roots, a second root is also called a square root and a third root is also called a cube root.

It is conventional in mathematics to define w1/n as the principal value of the root. If w is a positive real number, it is also conventional to select a positive real number as the principal value of the root w1/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 w is obtained by multiplying the principal value w1/n by each of the nth roots of unity. For example, the fourth roots of 16 are 2, −2, 2i, and −2i, because the principal value of the fourth root of 16 is 2 and the fourth roots of unity are 1, −1, i, and −i.

Computing complex powers

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

${\displaystyle z=re^{i\theta }=e^{\ln(r)+i\theta }}$

where r is a nonnegative real number and θ is the (real) argument of z. The polar form has a simple geometric interpretation: if a complex number u + iv is thought of as representing a point (u, v) in the complex plane using Cartesian coordinates, then (r, θ) is the same point in polar coordinates. That is, r is the "radius" r2 = u2 + v2 and θ is the "angle" θ = atan2(v, u). The polar angle θ is ambiguous since any integer multiple of 2π could be added to θ without changing the location of the point. Each choice of θ gives in general a different possible value of the power. A branch cut can be used to choose a specific value. The principal value (the most common branch cut), corresponds to θ chosen in the interval (−π, π]. For complex numbers with a positive real part and zero imaginary part using the principal value gives the same result as using the corresponding real number.

In order to compute the complex power wz, write w in polar form:

${\displaystyle w=re^{i\theta }}$

Then

${\displaystyle \log w=\log r+i\theta }$

and thus

${\displaystyle w^{z}=e^{z\log w}=e^{z(\log r+i\theta )}}$

If z is decomposed as c + di, then the formula for wz can be written more explicitly as

${\displaystyle \left(r^{c}e^{-d\theta }\right)e^{i(d\log r+c\theta )}=\left(r^{c}e^{-d\theta }\right)\left[\cos(d\log r+c\theta )+i\sin(d\log r+c\theta )\right]}$

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

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

{\displaystyle {\begin{aligned}i&=1\cdot e^{{\frac {1}{2}}i\pi }\\i&=0+1i\end{aligned}}}

Then the formula above, with r = 1, θ = {{ safesubst:#invoke:Unsubst||$B=π/2}}, c = 0, and d = 1, yields: ${\displaystyle i^{i}=\left(1^{0}e^{-{\frac {1}{2}}\pi }\right)e^{i\left[1\cdot \log 1+0\cdot {\frac {1}{2}}\pi \right]}=e^{-{\frac {1}{2}}\pi }\approx 0.2079}$ Similarly, to find (−2)3 + 4i, compute the polar form of −2, ${\displaystyle -2=2e^{i\pi }}$ and use the formula above to compute ${\displaystyle (-2)^{3+4i}=\left(2^{3}e^{-4\pi }\right)e^{i[4\log(2)+3\pi ]}\approx (2.602-1.006i)\cdot 10^{-5}}$ The value of a complex power depends on the branch used. For example, if the polar form i = 1e{{ safesubst:#invoke:Unsubst||$B=5πi/2}} is used to compute i i, the power is found to be e−{{ safesubst:#invoke:Unsubst||$B=5π/2}}; the principal value of i i, computed above, is e−{{ safesubst:#invoke:Unsubst||$B=π/2}}. The set of all possible values for i i is given by:[8]

{\displaystyle {\begin{aligned}i&=1\cdot e^{{\frac {1}{2}}i\pi +i2\pi k}{\big |}k\in {\mathbb {Z} }\\i^{i}&=e^{i\left({\frac {1}{2}}i\pi +i2\pi k\right)}\\&=e^{-\left({\frac {1}{2}}\pi +2\pi k\right)}\end{aligned}}}

So there is an infinity of values which are possible candidates for the value of ii, one for each integer k. All of them have a zero imaginary part so one can say ii has an infinity of valid real values.

Failure of power and logarithm identities

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

• The identity (ex)y = exy holds for real numbers x and y, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:[9]
For any integer n, we have:
but this is false when the integer n is nonzero.
There are a number of problems in the reasoning:
The major error is that changing the order of exponentiation in going from line two to three changes what the principal value chosen will be.
From the multi-valued point of view, the first error occurs even sooner. Implicit in the first line is that e is a real number, whereas the result of e1+2πin is a complex number better represented as e+0i. Substituting the complex number for the real on the second line makes the power have multiple possible values. Changing the order of exponentiation from lines two to three also affects how many possible values the result can have. ${\displaystyle \scriptstyle (e^{z})^{w}\;\neq \;e^{zw}}$, but rather ${\displaystyle \scriptstyle (e^{z})^{w}\;=\;e^{(z\,+\,2\pi in)w}}$ multivalued over integers n.

Zero to the power of zero

Discrete exponents

There are many widely used formulas having terms involving natural-number exponents that require 00 to be evaluated to 1. For example:

However, not all sources define 00 to be 1, particularly in the context of continuously varying exponents.

Continuous exponents

Plot of z = xy. The red curves (with z constant) yield different limits as (x,y) approaches (0,0). The green curves (of finite constant slope, y = ax) all yield a limit of 1.

When the form 00 arises as a limit of ${\displaystyle \scriptstyle f(x)^{g(x)}}$, it must be handled as an indeterminate form.

• Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form.[12] In fact, when f(t) and g(t) are real-valued functions both approaching 0 (as t approaches a real number or ±∞), with f(t) > 0, the function f(t)g(t) need not approach 1; depending on f and g, the limit of f(t)g(t) can be any nonnegative real number or +∞, or it can be undefined. For example, the functions below are of the form f(t)g(t) with f(t),g(t) → 0 as t → 0+, but the limits are different:
${\displaystyle \lim _{t\to 0^{+}}{t}^{t}=1,\quad \lim _{t\to 0^{+}}\left(e^{-{\frac {1}{t^{2}}}}\right)^{t}=0,\quad \lim _{t\to 0^{+}}\left(e^{-{\frac {1}{t^{2}}}}\right)^{-t}=+\infty ,\quad \lim _{t\to 0^{+}}\left(e^{-{\frac {1}{t}}}\right)^{at}=e^{-a}}$.
So 00 is an indeterminate form. This behavior shows that the two-variable function xy, though continuous on the set {(x,y): x > 0}, cannot be extended to a continuous function on any set containing (0,0), no matter how 00 is defined.[13] However, under certain conditions, such as when f and g are both analytic functions and f is positive on the open interval (0,b) for some positive b, the limit approaching from the right is always 1.[14][15][16]
• In the complex domain, the function zw is defined for nonzero z by choosing a branch of log z and setting zw := ew log z, but there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.[17][18][19]

History of differing points of view

The debate over the definition of 00 has been going on at least since the early 19th century. At that time, most mathematicians agreed that 00 = 1, until in 1821 Cauchy[20] listed 00 along with expressions like {{ safesubst:#invoke:Unsubst||$B=0/0}} in a table of indeterminate forms. In the 1830s Libri[21][22] published an unconvincing argument for 00 = 1, and Möbius[23] sided with him, erroneously claiming that ${\displaystyle \scriptstyle \lim _{t\to 0^{+}}f(t)^{g(t)}\;=\;1}$ whenever ${\displaystyle \scriptstyle \lim _{t\to 0^{+}}f(t)\;=\;\lim _{t\to 0^{+}}g(t)\;=\;0}$. A commentator who signed his name simply as "S" provided the counterexample of ${\displaystyle \scriptstyle (e^{-1/t})^{t}}$, and this quieted the debate for some time. More historical details can be found in Knuth (1992).[24] More recent authors interpret the situation above in different ways: Treatment on computers IEEE floating point standard The IEEE 754-2008 floating point standard is used in the design of most floating point libraries. It recommends a number of different functions for computing a power:[27] • pow treats 00 as 1. This is the oldest defined version. If the power is an exact integer the result is the same as for pown, otherwise the result is as for powr (except for some exceptional cases). • pown treats 00 as 1. The power must be an exact integer. The value is defined for negative bases; e.g., pown(−3,5) is −243. • powr treats 00 as NaN (Not-a-Number – undefined). The value is also NaN for cases like powr(−3,2) where the base is less than zero. The value is defined by epower×log(base). Programming languages Most programming language with a power function are implemented using the IEEE pow function and therefore evaluate 00 as 1. The later C[28] and C++ standards describe this as the normative behaviour. The Java standard[29] mandates this behavior. The .NET Framework method System.Math.Pow also treats 00 as 1.[30] Mathematics software • Sage simplifies b0 to 1, even if no constraints are placed on b.[31] It takes 00 to be 1, but does not simplify 0x for other x. • Maple simplifies b0 to 1 even if no constraints are placed on b, and evaluates 00 to 1. Maple 16 simplifies 0x to 0. • Macsyma also simplifies b0 to 1 even if no constraints are placed on b, but issues an error for 00. For x>0, it simplifies 0x to 0. {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=

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Limits of powers

The section zero to the power of zero gives a number of examples of limits which 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 ask at what points this function does have a limit.

More precisely, consider the function f(x,y) = xy defined on D = {(x,y) ∈ R2 : x > 0}. Then D can be viewed as a subset of Template:Overline2 (that is, the set of all pairs (x,y) with x,y belonging to the extended real number line Template:Overline = [−∞, +∞], endowed with the product topology), which will contain the points at which the function f has a limit.

In fact, f has a limit at all accumulation points of D, except for (0,0), (+∞,0), (1,+∞) and (1,−∞).[35] 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.

Under this definition by continuity, we obtain:

• x+∞ = +∞ and x−∞ = 0, when 1 < x ≤ +∞.
• x+∞ = 0 and x−∞ = +∞, when 0 ≤ x < 1.
• 0y = 0 and (+∞)y = +∞, when 0 < y ≤ +∞.
• 0y = +∞ and (+∞)y = 0, when −∞ ≤ y < 0.

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

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

Efficient computation with integer exponents

The simplest method of computing bn requires n − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2100, note that 100 = 64 + 32 + 4. Compute the following in order:

1. 22 = 4
2. (22)2 = 24 = 16
3. (24)2 = 28 = 256
4. (28)2 = 216 = 65,536
5. (216)2 = 232 = 4,294,967,296
6. (232)2 = 264 = 18,446,744,073,709,551,616
7. 264 232 24 = 2100 = 1,267,650,600,228,229,401,496,703,205,376

This series of steps only requires 8 multiplication operations instead of 99 (since the last product above takes 2 multiplications).

In general, the number of multiplication operations required to compute bn can be reduced to Θ(log n) by using exponentiation by squaring or (more generally) addition-chain exponentiation. Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for bn is a difficult problem for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available.[36]

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 f 3(x) may mean f(f(f(x))); in particular, f −1(x) usually denotes the inverse function of f. Iterated functions are of interest in the study of fractals and dynamical systems. Babbage was the first to study the problem of finding a functional square root f 1/2(x).

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 = csc x. A similar convention applies to logarithms, where log2x usually means (log x)2, not log log x.

Generalizations

In linear algebra

In linear algebra[37] A0 is defined as I for every square matrix A, and An is the product ${\displaystyle A\cdots A}$ (n factors). Moreover, if A is an invertible matrix, then A-n is defined as (A−1)n.

In abstract algebra

Exponentiation for integer exponents can be defined for quite general structures in abstract algebra.

Let X be a set with a power-associative binary operation which is written multiplicatively. Then xn is defined for any element x of X and any nonzero natural number n as the product of n copies of x, which is recursively defined by

{\displaystyle {\begin{aligned}x^{1}&=x\\x^{n}&=x^{n-1}x\quad {\hbox{for }}n>1\end{aligned}}}

One has the following properties

{\displaystyle {\begin{aligned}(x^{i}x^{j})x^{k}&=x^{i}(x^{j}x^{k})\quad {\text{(power-associative property)}}\\x^{m+n}&=x^{m}x^{n}\\(x^{m})^{n}&=x^{mn}\end{aligned}}}

If the operation has a two-sided identity element 1 (often denoted by e), then x0 is defined to be equal to 1 for any x.

{\displaystyle {\begin{aligned}x1&=1x=x\quad {\text{(two-sided identity)}}\\x^{0}&=1\end{aligned}}}{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B=

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If the operation also has two-sided inverses, and multiplication is associative then the magma is a group. The inverse of x can be denoted by x−1 and follows all the usual rules for exponents.

{\displaystyle {\begin{aligned}xx^{-1}&=x^{-1}x=1\quad {\text{(two-sided inverse)}}\\(xy)z&=x(yz)\quad {\text{(associative)}}\\x^{-n}&=\left(x^{-1}\right)^{n}\\x^{m-n}&=x^{m}x^{-n}\end{aligned}}}

If the multiplication operation is commutative (as for instance in abelian groups), then the following holds:

${\displaystyle (xy)^{n}=x^{n}y^{n}}$

If the binary operation is written additively, as it often is for abelian groups, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition". Thus, each of the laws of exponentiation above has an analogue among laws of multiplication.

When 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, xn 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.

{{safesubst:#invoke:anchor|main}}Over sets

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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.

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

${\displaystyle \bigoplus _{i\in {\mathbb {N} }}V_{i}}$

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 VN, 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.

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:

${\displaystyle S^{N}\equiv \{f\colon N\to S\}}$

This fits in with the exponentiation of cardinal numbers, in the sense that |SN| = |S||N|, where |X| is the cardinality of X. When "2" is defined as {0, 1}, we have |2X| = 2|X|, where 2X, usually denoted by P(X), is the power set of X; each subset Y of X corresponds uniquely to a function on X taking the value 1 for x ∈ Y and 0 for x ∉ Y.

In category theory

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

Of cardinal and ordinal numbers

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In set theory, there are exponential operations for cardinal and ordinal numbers.

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.[10] If κ and λ are finite, then this agrees with the ordinary arithmetic exponential operation. For example, the set of 3-tuples of elements from a 2-element set has cardinality 8 = 23. In cardinal arithmetic, κ0 is always 1 (even if κ is an infinite cardinal or zero).

Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.

Repeated exponentiation

Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation. This sequence of operations is expressed by the Ackermann function and Knuth's up-arrow notation. Just as exponentiation grows faster than multiplication, which is faster growing than addition, tetration is faster growing than exponentiation. Evaluated at (3,3), the functions addition, multiplication, exponentiation, tetration yield 6, 9, 27, and 7,625,597,484,987 (=333=327=33) respectively.

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:

Many programming languages lack syntactic support for exponentiation, but provide library functions.

In Bash, C, C++, C#, Java, JavaScript, Perl, PHP, Python and Ruby, the symbol ^ represents bitwise XOR. In Pascal, it represents indirection. In OCaml and Standard ML, it represents string concatenation.

History of the notation

The term power was used by the Greek mathematician Euclid for the square of a line.[38] Archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10.[39] In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms mal for a square and kab for a cube, which later Islamic mathematicians represented in mathematical notation as m and k, respectively, by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī.[40]

In the late 16th century, Jost Bürgi used Roman Numerals for exponents.[41]

Early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie; there, the notation is introduced in Book I.[2]

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. Samuel Jeake introduced the term indices in 1696.[38] In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth).[42] Biquadrate has been used to refer to the fourth power as well.

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

Another historical synonym, involution,[43] is now rare and should not be confused with its more common meaning.

List of whole-number exponentials

n n2 n3 n4 n5 n6 n7 n8 n9 n10
2 4 8 16 32 64 128 256 512 1,024
3 9 27 81 243 729 2,187 6,561 19,683 59,049
4 16 64 256 1,024 4,096 16,384 65,536 262,144 1,048,576
5 25 125 625 3,125 15,625 78,125 390,625 1,953,125 9,765,625
6 36 216 1,296 7,776 46,656 279,936 1,679,616 10,077,696 60,466,176
7 49 343 2,401 16,807 117,649 823,543 5,764,801 40,353,607 282,475,249
8 64 512 4,096 32,768 262,144 2,097,152 16,777,216 134,217,728 1,073,741,824
9 81 729 6,561 59,049 531,441 4,782,969 43,046,721 387,420,489 3,486,784,401
10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000 10,000,000,000

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References

1. See:
• Earliest Known Uses of Some of the Words of Mathematics
• Michael Stifel, Arithmetica integra (Nuremberg ("Norimberga"), (Germany): Johannes Petreius, 1544), Liber III (Book 3), Caput III (Chapter 3): De Algorithmo numerorum Cossicorum. (On algorithms of algebra.), page 236. Stifel was trying to conveniently represent the terms of geometric progressions. He devised a cumbersome notation for doing that. On page 236, he presented the notation for the first eight terms of a geometric progression (using 1 as a base) and then he wrote: "Quemadmodum autem hic vides, quemlibet terminum progressionis cossicæ, suum habere exponentem in suo ordine (ut 1ze habet 1. 1ʓ habet 2 &c.) sic quilibet numerus cossicus, servat exponentem suæ denominationis implicite, qui ei serviat & utilis sit, potissimus in multiplicatione & divisione, ut paulo inferius dicam." (However, you see how each term of the progression has its exponent in its order (as 1ze has a 1, 1ʓ has a 2, etc.), so each number is implicitly subject to the exponent of its denomination, which [in turn] is subject to it and is useful mainly in multiplication and division, as I will mention just below.) [Note: Most of Stifel's cumbersome symbols were taken from Christoff Rudolff, who in turn took them from Leonardo Fibonacci's Liber Abaci (1202), where they served as shorthand symbols for the Latin words res/radix (x), census/zensus (x2), and cubus (x3).]
2. René Descartes, Discourse de la Méthode … (Leiden, (Netherlands): Jan Maire, 1637), appended book: La Géométrie, book one, page 299. From page 299: " … Et aa, ou a2, pour multiplier a par soy mesme; Et a3, pour le multiplier encore une fois par a, & ainsi a l'infini ; … " ( … and aa, or a2, in order to multiply a by itself; and a3, in order to multiply it once more by a, and thus to infinity ; … )
3. {{#invoke:citation/CS1|citation |CitationClass=book }}
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7. This definition of a principal root of unity can be found in:
• {{#invoke:citation/CS1|citation
|CitationClass=book }} Online resource
• {{#invoke:citation/CS1|citation
|CitationClass=book }} Defined on page 351, available on Google books.
8. Complex number to a complex power may be real at Cut The Knot gives some references to ii
9. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
10. N. Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5.
11. "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".{{#invoke:citation/CS1|citation |CitationClass=book }}
12. {{#invoke:citation/CS1|citation |CitationClass=book }}
13. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
14. sci.math FAQ: What is 0^0?
15. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
16. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
17. "Since ln(0) does not exist, 0z is undefined. For Re(z) > 0, we define it arbitrarily as 0." George F. Carrier, Max Krook and Carl E. Pearson, Functions of a Complex Variable: Theory and Technique , 2005, p. 15
18. "For z=0, w≠0, we define 0w = 0, while 00 is not defined." Mario Gonzalez, Classical Complex Analysis, Chapman & Hall, 1991, p. 56.
19. "... Let's start at x = 0. Here xx is undefined." Mark D. Meyerson, The xx Spindle, Mathematics Magazine 69, no. 3 (June 1996), 198-206.
20. Augustin-Louis Cauchy, Cours d'Analyse de l'École Royale Polytechnique (1821). In his Oeuvres Complètes, series 2, volume 3.
21. Guillaume Libri, Note sur les valeurs de la fonction 00x, Journal für die reine und angewandte Mathematik 6 (1830), 67–72.
22. Guillaume Libri, Mémoire sur les fonctions discontinues, Journal für die reine und angewandte Mathematik 10 (1833), 303–316.
23. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
24. Donald E. Knuth, Two notes on notation, Amer. Math. Monthly 99 no. 5 (May 1992), 403–422 (arXiv:math/9205211 [math.HO]).
25. Examples include Edwards and Penny (1994). Calculus, 4th ed, Prentice-Hall, p. 466, and Keedy, Bittinger, and Smith (1982). Algebra Two. Addison-Wesley, p. 32.
26. Donald C. Benson, The Moment of Proof : Mathematical Epiphanies. New York Oxford University Press (UK), 1999. ISBN 978-0-19-511721-9
27. {{#invoke:citation/CS1|citation |CitationClass=book }}
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29. Template:Cite web
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33. Template:Cite web
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35. N. Bourbaki, Topologie générale, V.4.2.
36. Template:Cite doi
37. Chapter 1, Elementary Linear Algebra, 8E, Howard Anton
38. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
39. For further analysis see The Sand Reckoner.
40. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
41. Cajori, Florian (2007). A History of Mathematical Notations; Vol I. Cosimo Classics. Pg 344 ISBN 1602066841
42. Template:Cite web
43. 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.