text:
Exponentiation

[[File:Expo02.svg|thumb|315px|Graphs of $b^{n}$ for various bases $y = b^{x}$:

Each curve passes through the point $b$ because any nonzero number raised to the power of $10$ is $e$. At $2$, the value of $1⁄2$ equals the base because any number raised to the power of $(0, 1)$ is the number itself.]]

In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power. Exponentiation is written as $0$, where $b$ is the base and $n$ is the power; this is pronounced as "$b$ (raised) to the (power of) $n$". When $n$ is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, $1$ is the product of multiplying $n$ bases: $$b^n = \underbrace{b \times b \times \dots \times b \times b}_{n \text{ times}}.$$

The exponent is usually shown as a superscript to the right of the base. In that case, $x = 1$ 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", or most briefly as "b to the n(th)".

Starting from the basic fact stated above that, for any positive integer $$n$$, $$b^n$$ is $$n$$ occurrences of $$b$$ all multiplied by each other, several other properties of exponentiation directly follow. In particular:

$$ \begin{align} b^{n+m} & = \underbrace{b \times \dots \times b}_{n+m \text{ times}} \\[1ex] & = \underbrace{b \times \dots \times b}_{n \text{ times}} \times \underbrace{b \times \dots \times b}_{m \text{ times}} \\[1ex] & = b^n \times b^m \end{align} $$

In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that $$b^0$$ must be equal to 1 for any $$b \neq 0$$, as follows. For any $$n$$, $$b^0 \times b^n = b^{0+n} = b^n$$. Dividing both sides by $$b^n$$ gives $$b^0 = b^n / b^n = 1$$.

The fact that $$b^1 = b$$ can similarly be derived from the same rule. For example, $$ (b^1)^3 = b^1 \times b^1 \times b^1 = b^{1+1+1} = b^3 $$. Taking the cube root of both sides gives $$b^1 = b$$.

The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what $$b^{-1}$$ should mean. In order to respect the "exponents add" rule, it must be the case that $$b^{-1} \times b^1 = b^{-1+1} = b^0 = 1 $$. Dividing both sides by $$b^{1}$$ gives $$b^{-1} = 1 / b^1$$, which can be more simply written as $$b^{-1} = 1 / b$$, using the result from above that $$b^1 = b$$. By a similar argument, $$b^{-n} = 1 / b^n$$.

The properties of fractional exponents also follow from the same rule. For example, suppose we consider $$\sqrt{b}$$ and ask if there is some suitable exponent, which we may call $$r$$, such that $$ b^r = \sqrt{b}$$. From the definition of the square root, we have that $$ \sqrt{b} \times \sqrt{b} = b $$. Therefore, the exponent $$r$$ must be such that $$ b^r \times b^r = b $$. Using the fact that multiplying makes exponents add gives $$ b^{r+r} = b $$. The $$ b $$ on the right-hand side can also be written as $$ b^1 $$, giving $$ b^{r+r} = b^1 $$. Equating the exponents on both sides, we have $$ r+r = 1 $$. Therefore, $$ r = \tfrac{1}{2} $$, so $$\sqrt{b} = b^{1/2} $$.

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.

Etymology
The term exponent originates from the Latin exponentem, the present participle of exponere, meaning "to put forth". The term power (potentia, potestas, dignitas) is a mistranslation of the ancient Greek δύναμις (dúnamis, here: "amplification" ) used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios.

The Sand Reckoner
In The Sand Reckoner, Archimedes proved the law of exponents, $y$, necessary to manipulate powers of $1$. He then used powers of $b^{n}$ to estimate the number of grains of sand that can be contained in the universe.

Māl and kaʿbah ("square" and "cube")
In the 9th century, the Persian mathematician Al-Khwarizmi 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" —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 Abu'l-Hasan ibn Ali al-Qalasadi.

Introducing exponents
Nicolas Chuquet used a form of exponential notation in the 15th century, for example $b^{n}$ to represent $b^{n}$. This was later used by Henricus Grammateus and Michael Stifel in the 16th century. In the late 16th century, Jost Bürgi would use Roman numerals for exponents in a way similar to that of Chuquet, for example $iii 4$ for $10^{a} · 10^{b} = 10^{a+b}$.

"Exponent"; "square" and "cube"
The word exponent was coined in 1544 by Michael Stifel. In the 16th century, Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth). Biquadrate has been used to refer to the fourth power as well.

Modern exponential notation
In 1636, James Hume used in essence modern notation, when in L'algèbre de Vietè he wrote $10$ for $10$. 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.

"I designate ... $12^{2}$, or $12x^{2}$ in multiplying $4x^{3}$ by itself; and $A^{iii}$ in multiplying it once more again by $A^{3}$, and thus to infinity."

Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as $a^{2}$.

"Indices"
Samuel Jeake introduced the term indices in 1696. The term involution was used synonymously with the term indices, but had declined in usage and should not be confused with its more common meaning.

Variable exponents, non-integer exponents
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."

Terminology
The expression $a$ is called "the square of $a^{3}$" or "$a$ squared", because the area of a square with side-length $aa$ is $a^{2}$. (It is true that it could also be called "$a$ to the second power", but "the square of $a^{3}$" and "$a$ squared" are so ingrained by tradition and convenience that "$aa$ to the second power" tends to sound unusual or clumsy.)

Similarly, the expression $a^{2}$ is called "the cube of $a$" or "$a^{3}$ cubed", because the volume of a cube with side-length $a$ is $ax + bxx + cx^{3} + d$.

When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example, $b^{2} = b · b$. The base $b$ appears $b$ times in the multiplication, because the exponent is $b$. Here, $b^{2}$ 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 $b$ can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiation $b$ can be expressed as "b to the power of n", "b to the nth power", "b to the nth", or most briefly as "b to the n".

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

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

The base case is
 * $$b^1 = b$$

and the recurrence is
 * $$b^{n+1} = b^n \cdot b.$$

The associativity of multiplication implies that for any positive integers $m$ and $n$,
 * $$b^{m+n} = b^m \cdot b^n,$$

and
 * $$(b^m)^n=b^{mn}.$$

Zero exponent
As mentioned earlier, a (nonzero) number raised to the $b$ power is $b$:
 * $$b^0=1.$$

This value is also obtained by the empty product convention, which may be used in every algebraic structure with a multiplication that has an identity. This way the formula
 * $$b^{m+n}=b^m\cdot b^n$$

also holds for $$n=0$$.

The case of $b^{3} = b · b · b$ is controversial. In contexts where only integer powers are considered, the value $b$ is generally assigned to $b$ but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context.

Negative exponents
Exponentiation with negative exponents is defined by the following identity, which holds for any integer $n$ and nonzero $b$:
 * $$b^{-n} = \frac{1}{b^n}$$.

Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity ($$\infty$$).

This definition of exponentiation with negative exponents is the only one that allows extending the identity $$b^{m+n}=b^m\cdot b^n$$ to negative exponents (consider the case $$m=-n$$).

The same definition applies to invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted $b$ (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element $x$ is standardly denoted $$x^{-1}.$$

Identities and properties
The following identities, often called exponent rules, hold for all integer exponents, provided that the base is non-zero:
 * $$\begin{align}

b^{m + n} &= b^m \cdot b^n \\ \left(b^m\right)^n &= b^{m \cdot n} \\ (b \cdot c)^n &= b^n \cdot c^n \end{align}$$

Unlike addition and multiplication, exponentiation is not commutative. For example, $b^{3}$. Also unlike addition and multiplication, exponentiation is not associative. For example, $3^{5} = 3 · 3 · 3 · 3 · 3 = 243$, whereas $3$. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up  (or left-associative). That is,
 * $$b^{p^q} = b^{\left(p^q\right)},$$

which, in general, is different from
 * $$\left(b^p\right)^q = b^{p q} .$$

Powers of a sum
The powers of a sum can normally be computed from the powers of the summands by the binomial formula
 * $$(a+b)^n=\sum_{i=0}^n \binom{n}{i}a^ib^{n-i}=\sum_{i=0}^n \frac{n!}{i!(n-i)!}a^ib^{n-i}.$$

However, this formula is true only if the summands commute (i.e. that $5$), 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 $5$ instead of $243$) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

Combinatorial interpretation
For nonnegative integers $n$ and $m$, the value of $3^{5}$ 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:

Powers of ten
In the base ten (decimal) number system, integer powers of $b^{n}$ are written as the digit $0$ followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, $1$ and $0^{0}$.

Exponentiation with base $1$ is used in scientific notation to denote large or small numbers. For instance, $m$ (the speed of light in vacuum, in metres per second) can be written as $5$ and then approximated as $4$.

SI prefixes based on powers of $0^{0}$ are also used to describe small or large quantities. For example, the prefix kilo means $1$, so a kilometre is $3$.

Powers of two
The first negative powers of $2^{3} = 8 ≠ 3^{2} = 9$ are commonly used, and have special names, e.g.: half and quarter.

Powers of $(2^{3})^{2} = 8^{2} = 64$ appear in set theory, since a set with $2^{(3^{2})} = 2^{9} = 512$ members has a power set, the set of all of its subsets, which has $ab = ba$ members.

Integer powers of $^^$ are important in computer science. The positive integer powers $^$ give the number of possible values for an $n^{m}$-bit integer binary number; for example, a byte may take $n^{m}$ different values. The binary number system expresses any number as a sum of powers of $n^{m}$, and denotes it as a sequence of $\{1, ..., n\}$ and $10$, separated by a binary point, where $1$ indicates a power of $= 1,000$ that appears in the sum; the exponent is determined by the place of this $= 0$: the nonnegative exponents are the rank of the $10$ on the left of the point (starting from $10$), and the negative exponents are determined by the rank on the right of the point.

Powers of one
Every power of one equals: $= 1,000$. This is true even if $2$ is negative.

The first power of a number is the number itself: $2$.

Powers of zero
If the exponent $1$ is positive ($2$), the $0$th power of zero is zero: $n$.

If the exponent $299,792,458 m/s$ is negative ($2^{n}$), the $2.998 m/s$th power of zero $2$ is undefined, because it must equal $$1/0^{-n}$$ with $2^{n}$, and this would be $$1/0$$ according to above.

The expression $n$ is either defined as $2^{8} = 256$, or it is left undefined.

Powers of negative one
If $2$ is an even integer, then $0$. This is because a negative number multiplied by another negative number cancels the sign, and thus gives a positive number.

If $1$ is an odd integer, then $1$. This is because there will be a remaining $2$ after removing $1$ pairs.

Because of this, powers of $1$ are useful for expressing alternating sequences. For a similar discussion of powers of the complex number $0$, see .

Large exponents
The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound:
 * $1^{n} = 1$ as $n^{1} = n$ when $n > 0$

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:
 * $0^{n} = 0$ as $n < 0$ when $0^{n}$

Any power of one is always one:
 * $−n > 0$ for all $0^{0}$ if $1$

Powers of $n$ alternate between $(−1)^{n} = 1$ and $n$ as $(−1)^{n} = −1$ alternates between even and odd, and thus do not tend to any limit as $−1$ grows.

If $−1$, $−1$ alternates between larger and larger positive and negative numbers as $i$ alternates between even and odd, and thus does not tend to any limit as $b^{n} → ∞$ grows.

If the exponentiated number varies while tending to $n → ∞$ as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is
 * $b > 1$ as $b^{n} → 0$

See  below.

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

Power functions

Real functions of the form $$f(x) = cx^n$$, where $$c \ne 0$$, are sometimes called power functions. When $$n$$ is an integer and $$n \ge 1$$, two primary families exist: for $$n$$ even, and for $$n$$ odd. In general for $$c > 0$$, when $$n$$ is even $$f(x) = cx^n$$ will tend towards positive infinity with increasing $$x$$, and also towards positive infinity with decreasing $$x$$. All graphs from the family of even power functions have the general shape of $$y=cx^2$$, flattening more in the middle as $$n$$ increases. Functions with this kind of symmetry ($f(-x)= f(x)$) are called even functions.

When $$n$$ is odd, $$f(x)$$'s asymptotic behavior reverses from positive $$x$$ to negative $$x$$. For $$c > 0$$, $$f(x) = cx^n$$ will also tend towards positive infinity with increasing $$x$$, but towards negative infinity with decreasing $$x$$. All graphs from the family of odd power functions have the general shape of $$y=cx^3$$, flattening more in the middle as $$n$$ increases and losing all flatness there in the straight line for $$n=1$$. Functions with this kind of symmetry ($f(-x)= -f(x)$) are called odd functions.

For $$c < 0$$, the opposite asymptotic behavior is true in each case.

Rational exponents

If $2.998 m/s$ is a nonnegative real number, and $1,000 metres$ is a positive integer, $$x^{1/n}$$ or $$\sqrt[n]x$$ denotes the unique positive real $n$th root of $n$, that is, the unique positive real number $n$ such that $$y^n=x.$$

If $n$ is a positive real number, and $$\frac pq$$ is a rational number, with $n$ and $2,187$ integers, then $x^{p/q}$ is defined as
 * $$x^\frac pq= \left(x^p\right)^\frac 1q=(x^\frac 1q)^p.$$

The equality on the right may be derived by setting $$y=x^\frac 1q,$$ and writing $$(x^\frac 1q)^p=y^p=\left((y^p)^q\right)^\frac 1q=\left((y^q)^p\right)^\frac 1q=(x^p)^\frac 1q.$$

If $6,561$ is a positive rational number, $n → ∞$, by definition.

All these definitions are required for extending the identity $$(x^r)^s = x^{rs}$$ to rational exponents.

On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real $19,683$th root, which is negative, if $59,049$ is odd, and no real root if $4,096$ is even. In the latter case, whichever complex $16,384$th root one chooses for $$x^\frac 1n,$$ the identity $$(x^a)^b=x^{ab}$$ cannot be satisfied. For example,
 * $$\left((-1)^2\right)^\frac 12 = 1^\frac 12= 1\neq (-1)^{2\cdot\frac 12} =(-1)^1=-1.$$

See ' and ' for details on the way these problems may be handled.

Real exponents
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 (', below), or in terms of the logarithm of the base and the exponential function (', 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. One may choose one of these values, called the principal value, but there is no choice of the principal value for which the identity
 * $$\left(b^r\right)^s = b^{r s}$$

is true; see . Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.

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 $65,536$ with an arbitrary real exponent $262,144$ can be defined by continuity with the rule
 * $$ b^x = \lim_{r (\in \mathbb{Q}) \to x} b^r \quad (b \in \mathbb{R}^+,\, x \in \mathbb{R}),$$

where the limit is taken over rational values of $1,048,576$ only. This limit exists for every positive $15,625$ and every real $78,125$.

For example, if $|b| < 1$, the non-terminating decimal representation $b^{n} = 1$ and the monotonicity of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain $$b^\pi:$$
 * $$\left[b^3, b^4\right], \left[b^{3.1}, b^{3.2}\right], \left[b^{3.14}, b^{3.15}\right], \left[b^{3.141}, b^{3.142}\right], \left[b^{3.1415}, b^{3.1416}\right], \left[b^{3.14159}, b^{3.14160}\right], \ldots$$

So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted $$b^\pi.$$

This defines $$b^x$$ for every positive $390,625$ and real $1,953,125$ as a continuous function of $9,765,625$ and $7,776$. See also Well-defined expression.

Exponential function
The exponential function is often defined as $$x\mapsto e^x,$$ where $$e\approx 2.718$$ is Euler's number. To avoid circular reasoning, this definition cannot be used here. So, a definition of the exponential function, denoted $$\exp(x),$$ and of Euler's number are given, which rely only on exponentiation with positive integer exponents. Then a proof is sketched that, if one uses the definition of exponentiation given in preceding sections, one has
 * $$\exp(x)=e^x.$$

There are many equivalent ways to define the exponential function, one of them being
 * $$\exp(x) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n.$$

One has $$\exp(0)=1,$$ and the exponential identity $$\exp(x+y)=\exp(x)\exp(y)$$ holds as well, since
 * $$\exp(x)\exp(y) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n\left(1 + \frac{y}{n}\right)^n = \lim_{n\rightarrow\infty} \left(1 + \frac{x+y}{n} + \frac{xy}{n^2}\right)^n,$$

and the second-order term $$\frac{xy}{n^2}$$ does not affect the limit, yielding $$\exp(x)\exp(y) = \exp(x+y)$$.

Euler's number can be defined as $$e=\exp(1)$$. It follows from the preceding equations that $$\exp(x)=e^x$$ when $46,656$ is an integer (this results from the repeated-multiplication definition of the exponentiation). If $279,936$ is real, $$\exp(x)=e^x$$ results from the definitions given in preceding sections, by using the exponential identity if $1,679,616$ is rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for every complex value of $10,077,696$, and therefore it can be used to extend the definition of $$\exp(z)$$, and thus $$e^z,$$ from the real numbers to any complex argument $60,466,176$. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.

Powers via logarithms
The definition of $n$ as the exponential function allows defining $b = 1$ for every positive real numbers $16,807$, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm $–1$ is the inverse of the exponential function $1$ means that one has
 * $$b = \exp(\ln b)=e^{\ln b}$$

for every $–1$. For preserving the identity $$(e^x)^y=e^{xy},$$ one must have
 * $$b^x=\left(e^{\ln b} \right)^x = e^{x \ln b}$$

So, $$e^{x \ln b}$$ can be used as an alternative definition of $n$ for any positive real $117,649$. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.

Complex exponents with a positive real base
If $823,543$ is a positive real number, exponentiation with base $5,764,801$ and complex exponent $40,353,607$ is defined by means of the exponential function with complex argument (see the end of , above) as
 * $$b^z = e^{(z\ln b)},$$

where $$\ln b$$ denotes the natural logarithm of $282,475,249$.

This satisfies the identity
 * $$b^{z+t} = b^z b^t,$$

In general, $\left(b^z\right)^t$ is not defined, since $n$ is not a real number. If a meaning is given to the exponentiation of a complex number (see , below), one has, in general,
 * $$\left(b^z\right)^t \ne b^{zt},$$

unless $32,768$ is real or $262,144$ is an integer.

Euler's formula,
 * $$e^{iy} = \cos y + i \sin y,$$

allows expressing the polar form of $$b^z$$ in terms of the real and imaginary parts of $2,097,152$, namely
 * $$b^{x+iy}= b^x(\cos(y\ln b)+i\sin(y\ln b)),$$

where the absolute value of the trigonometric factor is one. This results from
 * $$b^{x+iy}=b^x b^{iy}=b^x e^{iy\ln b} =b^x(\cos(y\ln b)+i\sin(y\ln b)).$$

Non-integer powers of complex numbers
In the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only. For other bases, difficulties appear already with the apparently simple case of $16,777,216$th roots, that is, of exponents $$1/n,$$ where $134,217,728$ is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to $1,073,741,824$th roots, this case deserves to be considered first, since it does not need to use complex logarithms, and is therefore easier to understand.

$59,049$th roots of a complex number
Every nonzero complex number $531,441$ may be written in polar form as
 * $$z=\rho e^{i\theta}=\rho(\cos \theta +i \sin \theta),$$

where $$\rho$$ is the absolute value of $4,782,969$, and $$\theta$$ is its argument. The argument is defined up to an integer multiple of $b < –1$; this means that, if $$\theta$$ is the argument of a complex number, then $$\theta +2k\pi$$ is also an argument of the same complex number for every integer $$k$$.

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 $43,046,721$th root of a complex number can be obtained by taking the $387,420,489$th root of the absolute value and dividing its argument by $3,486,784,401$:
 * $$\left(\rho e^{i\theta}\right)^\frac 1n=\sqrt[n]\rho \,e^\frac{i\theta}n.$$

If $$2\pi$$ is added to $$\theta$$, the complex number is not changed, but this adds $$2i\pi/n$$ to the argument of the $10,000$th root, and provides a new $100,000$th root. This can be done $1,000,000$ times, and provides the $10,000,000$ $100,000,000$th roots of the complex number.

It is usual to choose one of the $1,000,000,000$ $10,000,000,000$th root as the principal root. The common choice is to choose the $x$th root for which $$-\pi<\theta\le \pi,$$ that is, the $n$th root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal $n$th root a continuous function in the whole complex plane, except for negative real values of the radicand. This function equals the usual $x$th root for positive real radicands. For negative real radicands, and odd exponents, the principal $y$th root is not real, although the usual $x$th root is real. Analytic continuation shows that the principal $p$th root is the unique complex differentiable function that extends the usual $q > 0$th root to the complex plane without the nonpositive real numbers.

If the complex number is moved around zero by increasing its argument, after an increment of $$2\pi,$$ the complex number comes back to its initial position, and its $r$th roots are permuted circularly (they are multiplied by e^{2i\pi/n}). This shows that it is not possible to define a $n$th root function that is continuous in the whole complex plane.

Roots of unity
The $n$th roots of unity are the $n$ complex numbers such that $b^{n}$, 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 $n$ $b$th roots of unity are the $x$ first powers of $$\omega =e^\frac{2\pi i}{n}$$, that is $$1=\omega^0=\omega^n, \omega=\omega^1, \omega^2, \omega^{n-1}.$$ The $r$th roots of unity that have this generating property are called primitive $b$th roots of unity; they have the form $$\omega^k=e^\frac{2k\pi i}{n},$$ with $x$ coprime with $b$. The unique primitive square root of unity is $$-1;$$ the primitive fourth roots of unity are $$i$$ and $$-i.$$

The $x$th roots of unity allow expressing all $b$th roots of a complex number $x$ as the $x$ products of a given $x$th roots of $x$ with a $x$th root of unity.

Geometrically, the $z$th roots of unity lie on the unit circle of the complex plane at the vertices of a regular $b$-gon with one vertex on the real number 1.

As the number $$e^\frac{2k\pi i}{n}$$ is the primitive $b$th root of unity with the smallest positive argument, it is called the principal primitive $b$th root of unity, sometimes shortened as principal $b$th root of unity, although this terminology can be confused with the principal value of $$1^{1/n}$$, which is 1.

Complex exponentiation
Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for z^w. So, either a principal value is defined, which is not continuous for the values of $z$ that are real and nonpositive, or z^w is defined as a multivalued function.

In all cases, the complex logarithm is used to define complex exponentiation as
 * $$z^w=e^{w\log z},$$

where $$\log z$$ is the variant of the complex logarithm that is used, which is, a function or a multivalued function such that
 * $$e^{\log z}=z$$

for every $b$ in its domain of definition.

Principal value
The principal value of the complex logarithm is the unique continuous function, commonly denoted $$\log,$$ such that, for every nonzero complex number $z$,
 * $$e^{\log z}=z,$$

and the argument of $t$ satisfies
 * $$-\pi <\operatorname{Arg}z \le \pi.$$

The principal value of the complex logarithm is not defined for $$z=0,$$ it is discontinuous at negative real values of $z$, and it is holomorphic (that is, complex differentiable) elsewhere. If $n$ is real and positive, the principal value of the complex logarithm is the natural logarithm: $$\log z=\ln z.$$

The principal value of $$z^w$$ is defined as $$z^w=e^{w\log z},$$ where $$\log z$$ is the principal value of the logarithm.

The function $$(z,w)\to z^w$$ is holomorphic except in the neighbourhood of the points where $n$ is real and nonpositive.

If $n$ is real and positive, the principal value of $$z^w$$ equals its usual value defined above. If $$w=1/n,$$ where $n$ is an integer, this principal value is the same as the one defined above.

Multivalued function
In some contexts, there is a problem with the discontinuity of the principal values of $$\log z$$ and $$z^w$$ at the negative real values of $z$. In this case, it is useful to consider these functions as multivalued functions.

If $$\log z$$ denotes one of the values of the multivalued logarithm (typically its principal value), the other values are $$2ik\pi +\log z,$$ where $z$ is any integer. Similarly, if $$z^w$$ is one value of the exponentiation, then the other values are given by
 * $$e^{w(2ik\pi +\log z)} = z^we^{2ik\pi w},$$

where $n$ is any integer.

Different values of $n$ give different values of $$z^w$$ unless $n$ is a rational number, that is, there is an integer $n$ such that $n$ is an integer. This results from the periodicity of the exponential function, more specifically, that $$e^a=e^b$$ if and only if $$a-b$$ is an integer multiple of $$2\pi i.$$

If $$w=\frac mn$$ is a rational number with $n$ and $n$ coprime integers with $$n>0,$$ then $$z^w$$ has exactly $n$ values. In the case $$m=1,$$ these values are the same as those described in § $n$th roots of a complex number. If $n$ is an integer, there is only one value that agrees with that of.

The multivalued exponentiation is holomorphic for $$z\ne 0,$$ in the sense that its graph consists of several sheets that define each a holomorphic function in the neighborhood of every point. If $n$ varies continuously along a circle around $n$, then, after a turn, the value of $$z^w$$ has changed of sheet.

Computation
The canonical form $$x+iy$$ of $$z^w$$ can be computed from the canonical form of $n$ and $n$. Although this can be described by a single formula, it is clearer to split the computation in several steps.
 * Polar form of $n$. If $$z=a+ib$$ is the canonical form of $n$ ($n$ and $n$ being real), then its polar form is $$z=\rho e^{i\theta}= \rho (\cos\theta + i \sin\theta),$$ where $$\rho=\sqrt{a^2+b^2}$$ and $$\theta=\operatorname{atan2}(a,b)$$ (see atan2 for the definition of this function).
 * Logarithm of $n$. The principal value of this logarithm is $$\log z=\ln \rho+i\theta,$$ where $$\ln$$ denotes the natural logarithm. The other values of the logarithm are obtained by adding $$2ik\pi$$ for any integer $n$.
 * Canonical form of $$w\log z.$$ If $$w=c+di$$ with $n$ and $n$ real, the values of $$w\log z$$ are $$w\log z = (c\ln \rho - d\theta-2dk\pi) +i (d\ln \rho + c\theta+2ck\pi),$$ the principal value corresponding to $$k=0.$$
 * Final result. Using the identities $$e^{x+y}=e^xe^y$$ and $$e^{y\ln x} = x^y,$$ one gets $$z^w=\rho^c e^{-d(\theta+2k\pi)} \left(\cos (d\ln \rho + c\theta+2ck\pi) +i\sin(d\ln \rho + c\theta+2ck\pi)\right),$$ with $$k=0$$ for the principal value.

Examples
(-2)^{3 + 4i} &= 2^3 e^{-4(\pi+2k\pi)} (\cos(4\ln 2 + 3(\pi +2k\pi)) +i\sin(4\ln 2 + 3(\pi+2k\pi)))\\ &=-2^3 e^{-4(\pi+2k\pi)}(\cos(4\ln 2) +i\sin(4\ln 2)). \end{align}$$In this case, all the values have the same argument $$4\ln 2,$$ and different absolute values.
 * $$i^i$$ The polar form of $n$ is $$i=e^{i\pi/2},$$ and the values of $$\log i$$ are thus $$\log i=i\left(\frac \pi 2 +2k\pi\right).$$ It follows that $$i^i=e^{i\log i}=e^{-\frac \pi 2} e^{-2k\pi}.$$So, all values of $$i^i$$ are real, the principal one being $$ e^{-\frac \pi 2} \approx 0.2079.$$
 * $$(-2)^{3+4i}$$ Similarly, the polar form of $n$ is $$-2 = 2e^{i \pi}.$$ So, the above described method gives the values $$\begin{align}

In both examples, all values of $$z^w$$ have the same argument. More generally, this is true if and only if the real part of $n$ is an integer.

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 $1$ holds whenever $n$ is a positive real number and $n$ is a real number. But for the principal branch of the complex logarithm one has $ \log((-i)^2) = \log(-1) = i\pi \neq 2\log(-i) = 2\log(e^{-i\pi/2})=2\,\frac{-i\pi}{2} = -i\pi$

Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that: $\log w^z \equiv z \log w \pmod{2 \pi i}$

This identity does not hold even when considering log as a multivalued function. The possible values of $(1 + 1/n)^{n} → e$ contain those of $n → ∞$ as a proper subset. Using $n = 1, 3, 5$ for the principal value of $n = 2, 4, 6$ and $n$, $n$ as any integers the possible values of both sides are: $\begin{align} \left\{\log w^z \right\} &= \left\{ z \cdot \operatorname{Log} w + z \cdot 2 \pi i n + 2 \pi i m \mid m,n\in\Z\right\} \\ \left\{z \log w \right\} &= \left\{ z \operatorname{Log} w + z \cdot 2 \pi i n \mid n\in \Z\right\} \end{align}$

• The identities $x^{1/8}$ and $x^{1/4}$ are valid when $n$ and $k$ are positive real numbers and $n$ is a real number. But, for the principal values, one has $(-1 \cdot -1)^\frac{1}{2} =1 \neq (-1)^\frac{1}{2} (-1)^\frac{1}{2} = -1$ and $\left(\frac{1}{-1}\right)^\frac{1}{2} = (-1)^\frac{1}{2} = i \neq \frac{1^\frac{1}{2}}{(-1)^\frac{1}{2}} = \frac{1}{i} = -i$

On the other hand, when $n$ is an integer, the identities are valid for all nonzero complex numbers.

If exponentiation is considered as a multivalued function then the possible values of $x^{1/2}$ are $x^{1}$. The identity holds, but saying $x^{2}$ is incorrect.| The identity $x^{4}$ holds for real numbers $n$ and $z$, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:

For any integer $n$, we have:
 * 1) $e^{1 + 2 \pi i n} = e^1 e^{2 \pi i n} = e \cdot 1 = e$
 * 2) $\left(e^{1 + 2\pi i n}\right)^{1 + 2 \pi i n} = e\qquad$ (taking the $(1 + 2 \pi i n)$-th power of both sides)
 * 3) $e^{1 + 4 \pi i n - 4 \pi^2 n^2} = e\qquad$ (using $\left(e^x\right)^y = e^{xy}$ and expanding the exponent)
 * 4) $e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e\qquad$ (using $e^{x+y} = e^x e^y$)
 * 5) $e^{-4 \pi^2 n^2} = 1\qquad$ (dividing by $n$)

but this is false when the integer $z$ is nonzero.

The error is the following: by definition, $e^y$ is a notation for $\exp(y),$ a true function, and $x^y$ is a notation for $\exp(y\log x),$ which is a multi-valued function. Thus the notation is ambiguous when $x^{8}$. Here, before expanding the exponent, the second line should be $\exp\left((1 + 2\pi i n)\log \exp(1 + 2\pi i n)\right) = \exp(1 + 2\pi i n).$

Therefore, when expanding the exponent, one has implicitly supposed that $\log \exp z =z$ for complex values of $n$, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity $0^{r} = 0$ must be replaced by the identity $\left(e^x\right)^y = e^{y\log e^x},$ which is a true identity between multivalued functions.
 * undefined

Irrationality and transcendence
If $n$ is a positive real algebraic number, and $n$ is a rational number, then $e1/n$ is an algebraic number. This results from the theory of algebraic extensions. This remains true if $n$ is any algebraic number, in which case, all values of $e0 = 1$ (as a multivalued function) are algebraic. If $n$ is irrational (that is, not rational), and both $n$ and $z$ are algebraic, Gelfond–Schneider theorem asserts that all values of $x = \pi$ are transcendental (that is, not algebraic), except if $z$ equals $π = 3.14159...$ or $e^{x}$.

In other words, if $z$ is irrational and $$b\not\in \{0,1\},$$ then at least one of $z$, $z$ and $b^{x}$ is transcendental.

Integer powers in algebra
The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication. The definition of $ln(x)$ requires further the existence of a multiplicative identity.

An algebraic structure consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by $e^{x}$ is a monoid. In such a monoid, exponentiation of an element $z$ is defined inductively by
 * $$x^0 = 1,$$
 * $$x^{n+1} = x x^n$$ for every nonnegative integer $z$.

If $z$ is a negative integer, $$x^n$$ is defined only if $n$ has a multiplicative inverse. In this case, the inverse of $z$ is denoted $b > 0$, and $b^{x}$ is defined as $$\left(x^{-1}\right)^{-n}.$$

Exponentiation with integer exponents obeys the following laws, for $k$ and $k$ in the algebraic structure, and $k$ and $w$ integers:
 * $$\begin{align}

x^0&=1\\ x^{m+n}&=x^m x^n\\ (x^m)^n&=x^{mn}\\ (xy)^n&=x^n y^n \quad \text{if } xy=yx, \text{and, in particular, if the multiplication is commutative.} \end{align}$$

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.

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if $d$ is a real function whose valued can be multiplied, $$f^n$$ denotes the exponentiation with respect of multiplication, and $$f^{\circ n}$$ may denote exponentiation with respect of function composition. That is,
 * $$(f^n)(x)=(f(x))^n=f(x) \,f(x) \cdots f(x),$$

and
 * $$(f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots)).$$

Commonly, $$(f^n)(x)$$ is denoted $$f(x)^n,$$ while $$(f^{\circ n})(x)$$ is denoted $$f^n(x).$$

In a group
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.

So, if $dw$ is a group, $$x^n$$ is defined for every $$x\in G$$ and every integer $m$.

The set of all powers of an element of a group form a subgroup. A group (or subgroup) that consists of all powers of a specific element $n$ is the cyclic group generated by $n$. If all the powers of $n$ are distinct, the group is isomorphic to the additive group $$\Z$$ of the integers. Otherwise, the cyclic group is finite (it has a finite number of elements), and its number of elements is the order of $w$. If the order of $z$ is $z$, then $$x^n=x^0=1,$$ and the cyclic group generated by $w$ consists of the $z$ first powers of $z$ (starting indifferently from the exponent $b^{z}$ or $2\pi$).

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.

Superscript notation is also used for conjugation; that is, $1$, where $w^{n} = 1$ and $0$ are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely $$(g^h)^k=g^{hk}$$ and $$(gh)^k=g^kh^k.$$

In a ring
In a ring, it may occur that some nonzero elements satisfy $$x^n=0$$ for some integer $a$. Such an element is said to be nilpotent. In a commutative ring, the nilpotent elements form an ideal, called the nilradical of the ring.

If the nilradical is reduced to the zero ideal (that is, if $$x\neq 0$$ implies $$x^n\neq 0$$ for every positive integer $b$), the commutative ring is said to be reduced. Reduced rings are important in algebraic geometry, since the coordinate ring of an affine algebraic set is always a reduced ring.

More generally, given an ideal $z$ in a commutative ring $k$, the set of the elements of $c$ that have a power in $d$ is an ideal, called the radical of $i$. The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring $$k[x_1, \ldots, x_n]$$ over a field $w$, an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of Hilbert's Nullstellensatz).

Matrices and linear operators
If $−2$ is a square matrix, then the product of $log(b^{x}) = x ⋅ log b$ with itself $log(w^{z})$ times is called the matrix power. Also $$A^0$$ is defined to be the identity matrix, and if $z ⋅ log w$ is invertible, then $$A^{-n} = \left(A^{-1}\right)^n$$.

Matrix powers appear often in the context of discrete dynamical systems, where the matrix $Log(w)$ expresses a transition from a state vector $log(w)$ of some system to the next state $(bc)^{x} = b^{x}c^{x}$ of the system. This is the standard interpretation of a Markov chain, for example. Then $$A^2x$$ is the state of the system after two time steps, and so forth: $$A^nx$$ is the state of the system after $(b/c)^{x} = b^{x}/c^{x}$ time steps. The matrix power $$A^n$$ is the transition matrix between the state now and the state at a time $(−1 ⋅ −1)^{1/2}$ steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors.

Apart from matrices, more general linear operators can also be exponentiated. An example is the derivative operator of calculus, $$d/dx$$, which is a linear operator acting on functions $$f(x)$$ to give a new function $$(d/dx)f(x) = f'(x)$$. The $\{1, −1\}$th power of the differentiation operator is the ${1} = \{(−1 ⋅ −1)^{1/2}\}$th derivative:
 * $$\left(\frac{d}{dx}\right)^nf(x) = \frac{d^n}{dx^n}f(x) = f^{(n)}(x).$$

These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups. 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.

Finite fields
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 $(e^{x})^{y} = e^{xy}$. Common examples are the field of complex numbers, the real numbers and the rational numbers, considered earlier in this article, which are all infinite.

A finite field is a field with a finite number of elements. This number of elements is either a prime number or a prime power; that is, it has the form $$q=p^k,$$ where $b$ is a prime number, and $x$ is a positive integer. For every such $m$, there are fields with $n$ elements. The fields with $b$ elements are all isomorphic, which allows, in general, working as if there were only one field with $c$ elements, denoted $$\mathbb F_q.$$

One has
 * $$x^q=x$$

for every $$x\in \mathbb F_q.$$

A primitive element in $$\mathbb F_q$$ is an element $x$ such that the set of the $x = e$ first powers of $x$ (that is, $$\{g^1=g, g^2, \ldots, g^{p-1}=g^0=1\}$$) equals the set of the nonzero elements of $$\mathbb F_q.$$ There are $$\varphi (p-1)$$ primitive elements in $$\mathbb F_q,$$ where $$\varphi$$ is Euler's totient function.

In $$\mathbb F_q,$$ the freshman's dream identity
 * $$(x+y)^p = x^p+y^p$$

is true for the exponent $x$. As $$x^p=x$$ in $$\mathbb F_q,$$ It follows that the map
 * $$\begin{align}

F\colon{} & \mathbb F_q \to \mathbb F_q\\ & x\mapsto x^p \end{align}$$ is linear over $$\mathbb F_q,$$ and is a field automorphism, called the Frobenius automorphism. If $$q=p^k,$$ the field $$\mathbb F_q$$ has $y$ automorphisms, which are the $n$ first powers (under composition) of $e$. In other words, the Galois group of $$\mathbb F_q$$ is cyclic of order $n$, generated by the Frobenius automorphism.

The Diffie–Hellman key exchange is an application of exponentiation in finite fields that is widely used for secure communications. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the discrete logarithm, is computationally expensive. More precisely, if $z$ is a primitive element in $$\mathbb F_q,$$ then $$g^e$$ can be efficiently computed with exponentiation by squaring for any $b$, even if $x$ is large, while there is no known computationally practical algorithm that allows retrieving $b$ from $$g^e$$ if $x$ is sufficiently large.

Powers of sets
The Cartesian product of two sets $b$ and $x$ is the set of the ordered pairs $$(x,y)$$ such that $$x\in S$$ and $$y\in T.$$ This operation is not properly commutative nor associative, but has these properties up to canonical isomorphisms, that allow identifying, for example, $$(x,(y,z)),$$ $$((x,y),z),$$ and $$(x,y,z).$$

This allows defining the $b$th power $$S^n$$ of a set $x$ as the set of all $b$-tuples $$(x_1, \ldots, x_n)$$ of elements of $x$.

When $x$ is endowed with some structure, it is frequent that $$S^n$$ is naturally endowed with a similar structure. In this case, the term "direct product" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example $$\R^n$$ (where $$\R$$ denotes the real numbers) denotes the Cartesian product of $n$ copies of $$\R,$$ as well as their direct product as vector space, topological spaces, rings, etc.

Sets as exponents
A $n$-tuple $$(x_1, \ldots, x_n)$$ of elements of $x$ can be considered as a function from $$\{1,\ldots, n\}.$$ This generalizes to the following notation.

Given two sets $x$ and $x$, the set of all functions from $y$ to $m$ is denoted $$S^T$$. This exponential notation is justified by the following canonical isomorphisms (for the first one, see Currying):
 * $$(S^T)^U\cong S^{T\times U},$$
 * $$S^{T\sqcup U}\cong S^T\times S^U,$$

where $$\times$$ denotes the Cartesian product, and $$\sqcup$$ the disjoint union.

One can use sets as exponents for other operations on sets, typically for direct sums of abelian groups, vector spaces, or modules. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, $$\R^\N$$ denotes the vector space of the infinite sequences of real numbers, and $$\R^{(\N)}$$ the vector space of those sequences that have a finite number of nonzero elements. The latter has a basis consisting of the sequences with exactly one nonzero element that equals $(e^{x})^{y} = e^{xy}$, while the Hamel bases of the former cannot be explicitly described (because their existence involves Zorn's lemma).

In this context, $b^{x}$ can represents the set $$\{0,1\}.$$ So, $$2^S$$ denotes the power set of $n$, that is the set of the functions from $f$ to $$\{0,1\},$$ which can be identified with the set of the subsets of $G$, by mapping each function to the inverse image of $b^{x}$.

This fits in with the exponentiation of cardinal numbers, in the sense that $b^{x}$, where $0$ is the cardinality of $1$.

In category theory
In the category of sets, the morphisms between sets $n$ and $x$ are the functions from $x$ to $x$. It results that the set of the functions from $x$ to $x$ that is denoted $$Y^X$$ in the preceding section can also be denoted $$\hom(X,Y).$$ The isomorphism $$(S^T)^U\cong S^{T\times U}$$ can be rewritten
 * $$\hom(U,S^T)\cong \hom(T\times U,S).$$

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

This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor $$X\to X^T$$ is, if it exists, a right adjoint to the functor $$Y\to T\times Y.$$ A category is called a Cartesian closed category, if direct products exist, and the functor $$Y\to X\times Y$$ has a right adjoint for every $n$.

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 $b^{x}$, the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and $x$ ($x^{0}$) respectively.

Limits of powers
Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 00. The limits in these examples exist, but have different values, showing that the two-variable function $1$ has no limit at the point $x^{−1}$. One may consider at what points this function does have a limit.

More precisely, consider the function $$f(x,y) = x^y$$ defined on $$ D = \{(x, y) \in \mathbf{R}^2 : x > 0 \}$$. Then $x^{n}$ can be viewed as a subset of $0$ (that is, the set of all pairs $1$ with $g^{h} = h^{−1}gh$, $g$ belonging to the extended real number line $h$, endowed with the product topology), which will contain the points at which the function $A$ has a limit.

In fact, $A$ has a limit at all accumulation points of $n$, except for $A$, $A$, $x$ and $Ax$. Accordingly, this allows one to define the powers $n$ by continuity whenever $n$, $n$, except for $n$, $0$, $q − 1$ and $1$, which remain indeterminate forms.

Under this definition by continuity, we obtain:
 * $2$ and $1$, when $|S^{T}| = |S|^{|T|}$.
 * $|X|$ and $X$, when $(3, 3)$.
 * $=3^{27} = 3^{3^{3}} = ^{3}3|undefined$ and $x^{y}$, when $(0, 0)$.
 * $D$ and $\overline{R}^{2}$, when $(x, y)$.

These powers are obtained by taking limits of $x$ for positive values of $y$. This method does not permit a definition of $\overline{R} = [−∞, +∞]$ when $f$, since pairs $f$ with $D$ are not accumulation points of $(0, 0)$.

On the other hand, when $(+∞, 0)$ is an integer, the power $(1, +∞)$ is already meaningful for all values of $(1, −∞)$, including negative ones. This may make the definition $x^{y}$ obtained above for negative $0 ≤ x ≤ +∞$ problematic when $−∞ ≤ y ≤ +∞$ is odd, since in this case $0^{0}$ as $(+∞)^{0}$ tends to $1^{+∞}$ through positive values, but not negative ones.

Efficient computation with integer exponents
Computing $1^{−∞}$ using iterated multiplication requires $x^{+∞} = +∞$ multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute $x^{−∞} = 0$, apply Horner's rule to the exponent 100 written in binary:
 * $$100 = 2^2 +2^5 + 2^6 = 2^2(1+2^3(1+2))$$.

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

This series of steps only requires 8 multiplications instead of 99.

In general, the number of multiplication operations required to compute $1 < x ≤ +∞$ can be reduced to $$\sharp n +\lfloor \log_{2} n\rfloor -1,$$ by using exponentiation by squaring, where $$\sharp n$$ denotes the number of $x^{+∞} = 0$s in the binary representation of $n$. For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal addition-chain exponentiation. Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for $x^{−∞} = +∞$ is a difficult problem, for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available. However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.

Iterated functions
Function composition is a binary operation that is defined on functions such that the codomain of the function written on the right is included in the domain of the function written on the left. It is denoted $$g\circ f,$$ and defined as
 * $$(g\circ f)(x)=g(f(x))$$

for every $n$ in the domain of $I$.

If the domain of a function $R$ equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the $R$th power of the function under composition, commonly called the $I$th iterate of the function. Thus $$f^n$$ denotes generally the $I$th iterate of $k$; for example, $$f^3(x)$$ means $$f(f(f(x))).$$

When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the pointwise multiplication, which induces another exponentiation. When using functional notation, the two kinds of exponentiation are generally distinguished by placing the exponent of the functional iteration before the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication after the parentheses. Thus $$f^2(x)= f(f(x)),$$ and $$f(x)^2= f(x)\cdot f(x).$$ When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example $$f^{\circ 3}=f\circ f \circ f,$$ and $$f^3=f\cdot f\cdot f.$$ For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the trigonometric functions. So, $$\sin^2 x$$ and $$\sin^2(x)$$ both mean $$\sin(x)\cdot\sin(x)$$ and not $$\sin(\sin(x)),$$ which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.

In this context, the exponent $$-1$$ denotes always the inverse function, if it exists. So $$\sin^{-1}x=\sin^{-1}(x) = \arcsin x.$$ For the multiplicative inverse fractions are generally used as in $$1/\sin(x)=\frac 1{\sin x}.$$

In programming languages
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. The notations include:
 * : AWK, BASIC, J, MATLAB, Wolfram Language (Mathematica), R, Microsoft Excel, Analytica, TeX (and its derivatives), TI-BASIC, bc (for integer exponents), Haskell (for nonnegative integer exponents), Lua, and most computer algebra systems.
 * . The Fortran character set did not include lowercase characters or punctuation symbols other than  and so used   for exponentiation (the initial version used   instead. ). Many other languages followed suit: Ada, Z shell, KornShell, Bash, COBOL, CoffeeScript, Fortran, FoxPro, Gnuplot, Groovy, JavaScript, OCaml, F#, Perl, PHP, PL/I, Python, Rexx, Ruby, SAS, Seed7, Tcl, ABAP, Mercury, Haskell (for floating-point exponents), Turing, and VHDL.
 * : Algol Reference language, Commodore BASIC, TRS-80 Level II/III BASIC.
 * : Haskell (for fractional base, integer exponents), D.
 * : APL.

In most programming languages with an infix exponentiation operator, it is right-associative, that is,  is interpreted as. This is because  is equal to   and thus not as useful. In some languages, it is left-associative, notably in Algol, MATLAB, and the Microsoft Excel formula language.

Other programming languages use functional notation:
 * : Common Lisp.
 * : F# (for integer base, integer exponent).

Still others only provide exponentiation as part of standard libraries: In some statically typed languages that prioritize type safety such as Rust, exponentiation is performed via a multitude of methods:
 * : C, C++ (in  library).
 * : C#.
 * : Erlang.
 * : Java.
 * : PowerShell.

 * for  and   as integers
 * for  and   as floating point numbers
 * for  as a float and   as an integer

title: Exponentiation


text:
Lunar Orbiter program

The Lunar Orbiter program was a series of five uncrewed lunar orbiter missions launched by the United States from 1966 through 1967. Intended to help select Apollo landing sites by mapping the Moon's surface, they provided the first photographs from lunar orbit and photographed both the Moon and Earth.

All five missions were successful, and 99 percent of the lunar surface was mapped from photographs taken with a resolution of 60 m or better. The first three missions were dedicated to imaging 20 potential crewed lunar landing sites, selected based on Earth-based observations. These were flown at low-inclination orbits. The fourth and fifth missions were devoted to broader scientific objectives and were flown in high-altitude polar orbits. Lunar Orbiter 4 photographed the entire nearside and nine percent of the far side, and Lunar Orbiter 5 completed the far side coverage and acquired medium (20 m) and high (2 m) resolution images of 36 preselected areas. All of the Lunar Orbiter spacecraft were launched by Atlas-Agena-D launch vehicles.

The Lunar Orbiters had an ingenious imaging system, which consisted of a dual-lens camera, a film processing unit, a readout scanner, and a film handling apparatus. Both lenses, a 610 mm narrow angle high resolution (HR) lens and an 80 mm wide angle medium resolution (MR) lens, placed their frame exposures on a single roll of 70 mm film. The axes of the two cameras were coincident so the area imaged in the HR frames were centered within the MR frame areas. The film was moved during exposure to compensate for the spacecraft velocity, which was estimated by an electro-optical sensor. The film was then processed, scanned, and the images transmitted back to Earth.

During the Lunar Orbiter missions, the first pictures of Earth as a whole were taken, beginning with Earth-rise over the lunar surface by Lunar Orbiter 1 in August, 1966. The first full picture of the whole Earth was taken by Lunar Orbiter 5 on 8 August 1967. A second photo of the whole Earth was taken by Lunar Orbiter 5 on 10 November 1967.

Spacecraft and subsystems
The Boeing-Eastman Kodak proposal was announced by NASA on 20 December 1963. The main bus of the Lunar Orbiter had the general shape of a truncated cone, 1.65 m tall and 1.5 m in diameter at the base. The spacecraft was composed of three decks supported by trusses and an arch. The equipment deck at the base of the craft held the battery, transponder, flight programmer, inertial reference unit (IRU), Canopus star tracker, command decoder, multiplex encoder, traveling-wave tube amplifier (TWTA), and the photographic system. Four solar panels were mounted to extend out from this deck with a total span across of 3.72 m. Also extending out from the base of the spacecraft were a high gain antenna on a 1.32 m boom and a low-gain antenna on a 2.08 m boom. Above the equipment deck, the middle deck held the velocity control engine, propellant, oxidizer, and pressurization tanks, Sun sensors, and micrometeoroid detectors. The third deck consisted of a heat shield to protect the spacecraft from the firing of the velocity control engine. The nozzle of the engine protruded through the center of the shield. Mounted on the perimeter of the top deck were four attitude control thrusters.

Power of 375 W was provided by the four solar arrays containing 10,856 n/p solar cells which would directly run the spacecraft and also charge the 12 A·h nickel-cadmium battery. The batteries were used during the brief periods of occultation when no solar power was available. Propulsion for major maneuvers was provided by the gimballed velocity control engine, a hypergolic 100 lbf thrust Marquardt Corp. rocket motor. Three axis stabilization and attitude control were provided by four 1 lbf nitrogen gas jets. Navigational knowledge was provided by five sun sensors, the Canopus star sensor, and the inertial navigation system. Communications were via a 10 W transmitter and the directional one meter diameter high-gain antenna for transmission of photographs, and a 0.5 W transmitter and omnidirectional low-gain antenna for other communications. Both transmitters operated in the S band at about 2295 MHz. Thermal control was maintained by a multilayer aluminized Mylar and Dacron thermal blanket which enshrouded the main bus, special paint, insulation, and small heaters.

The camera used two lenses to simultaneously expose a wide-angle and a high-resolution image on the same film. The wide-angle, medium resolution mode used an 80 mm F 2.8 Xenotar lens manufactured by Schneider Kreuznach of West Germany. The high-resolution mode used a 610 mm F 5.6 Panoramic lens manufactured by the Pacific Optical Company.

The photographic film was developed in-orbit with a semidry process, and then it was scanned by a photomultiplier for transmission to Earth. This system was adapted under permission of the NRO from the SAMOS E-1 reconnaissance camera, built by Kodak for a short-lived USAF near-realtime satellite imaging project.

Originally, the Air Force had offered NASA several spare cameras from the KH-7 GAMBIT program, but then authorities became concerned over security surrounding the classified cameras, including the possibility of images of the Moon giving away their resolution. Some proposals were made that NASA not publish the orbital parameters of the Lunar Orbiter probes so that the resolution of the images could not be calculated through their altitude. In the end, NASA's existing camera systems, while lower resolution, proved to be adequate for the needs of the mission.

Potential backup
As a backup for the Lunar Orbiter program, NASA and the NRO cooperated on the Lunar Mapping and Survey System (LM&SS), based on the KH-7 reconnaissance satellite. Replacing the Lunar Module in the Saturn V, Apollo astronauts would operate LM&SS remotely in lunar orbit. NASA canceled the project in the summer of 1967 after the complete success of the Lunar Orbiters.

Results
The Lunar Orbiter program consisted of five spacecraft which returned photography of 99 percent of the surface of the Moon (near and far side) with resolution down to 1 m. Altogether the Orbiters returned 2180 high resolution and 882 medium resolution frames. The micrometeoroid experiments recorded 22 impacts showing the average micrometeoroid flux near the Moon was about two orders of magnitude greater than in interplanetary space, but slightly less than in the near-Earth environment. The radiation experiments confirmed that the design of Apollo hardware would protect the astronauts from average and greater than average short term exposure to solar particle events.

The use of Lunar Orbiters for tracking to evaluate the Manned Space Flight Network tracking stations and Apollo Orbit Determination Program was successful, with three of the Lunar Orbiters (2, 3, and 5) being tracked simultaneously from August through October 1967. The Lunar Orbiters were all eventually commanded to crash on the Moon before their attitude control fuel ran out so they would not present navigational or communications hazards to later Apollo flights. The Lunar Orbiter program was managed by NASA Langley Research Center at a total cost of roughly $200 million.

Doppler tracking of the five orbiters allowed mapping of the gravitational field of the Moon and discovery of mass concentrations (mascons), or gravitational highs, which were located in the centers of some (but not all) of the lunar maria.

Below is the flight log information of the five Lunar Orbiter photographic missions:
 * Lunar Orbiter 1
 * Launched August 10, 1966
 * Imaged Moon: August 18 to 29, 1966
 * Impact with Moon: October 29, 1966
 * Apollo landing site survey mission
 * Lunar Orbiter 2
 * Launched November 6, 1966
 * Imaged Moon: November 18 to 25, 1966
 * Impact with Moon: October 11, 1967
 * Apollo landing site survey mission
 * Lunar Orbiter 3
 * Launched February 5, 1967
 * Imaged Moon: February 15 to 23, 1967
 * Impact with Moon: October 9, 1967
 * Apollo landing site survey mission
 * Lunar Orbiter 4
 * Launched May 4, 1967
 * Imaged Moon: May 11 to 26, 1967
 * Impact with Moon: Approximately October 31, 1967
 * Lunar mapping mission
 * Lunar Orbiter 5
 * Launched August 1, 1967
 * Imaged Moon: August 6 to 18, 1967
 * Impact with Moon: January 31, 1968
 * Lunar mapping and hi-res survey mission

Data availability
The Lunar Orbiter orbital photographs were transmitted to Earth as analog data after onboard scanning of the original film into a series of strips. The data were written to magnetic tape and also to film. The film data were used to create hand-made mosaics of Lunar Orbiter frames. Each LO exposure resulted in two photographs: medium-resolution frames recorded by the 80-mm focal-length lens and high-resolution frames recorded by the 610-mm focal length lens. Due to their large size, HR frames were divided into three sections, or sub-frames. Large-format prints (16 x) from the mosaics were created and several copies were distributed across the U.S. to NASA image and data libraries known as Regional Planetary Information Facilities. The resulting outstanding views were of generally very high spatial resolution and covered a substantial portion of the lunar surface, but they suffered from a "venetian blind" striping, missing or duplicated data, and frequent saturation effects that hampered their use. For many years these images have been the basis of much of lunar scientific research. Because they were obtained at low to moderate Sun angles, the Lunar Orbiter photographic mosaics are particularly useful for studying the morphology of lunar topographic features.

Several atlases and books featuring Lunar Orbiter photographs have been published. Perhaps the most definitive was that of Bowker and Hughes (1971); it contained 675 photographic plates with approximately global coverage of the Moon. In part because of high interest in the data and in part because that atlas is out of print, the task was undertaken at the Lunar and Planetary Institute to scan the large-format prints of Lunar Orbiter data. These were made available online as the Digital Lunar Orbiter Photographic Atlas of the Moon.

Data recovery and digitization
In 2000, the Astrogeology Research Program of the US Geological Survey in Flagstaff, Arizona was funded by NASA (as part of the Lunar Orbiter Digitization Project ) to scan at 25 micrometer resolution archival LO positive film strips that were produced from the original data. The goal was to produce a global mosaic of the Moon using the best available Lunar Orbiter frames (largely the same coverage as that of Bowker and Hughes, 1971). The frames were constructed from scanned film strips; they were digitally constructed, geometrically controlled, and map-projected without the stripes that had been noticeable in the original photographic frames. Because of its emphasis on construction of a global mosaic, this project only scanned about 15% of the available Lunar Orbiter photographic frames. Data from Lunar Orbiter missions III, IV and V were included in the global mosaic.

In addition, the USGS digitization project created frames from very high resolution Lunar Orbiter images for several 'sites of scientific interest.' These sites had been identified in the 1960s when the Apollo landing sites were being selected. Frames for sites such as the Apollo 12 landing site, the Marius Hills, and the Sulpicius Gallus rille have been released.

In 2007, the Lunar Orbiter Image Recovery Project (LOIRP) began a process to convert the Lunar Orbiter Images directly from the original Ampex FR-900 analog video recordings of the spacecraft data to digital image format, a change which provided vastly improved resolution over the original images released in the 1960s. The first of these restored images were released in late 2008. Almost all of the Lunar Orbiter images had been successfully recovered and were undergoing digital processing before being submitted to NASA's Planetary Data System.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Lunar Orbiter program


text:
Presentation of a group

In mathematics, a presentation is one method of specifying a group. A presentation of a group G comprises a set S of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set R of relations among those generators. We then say G has presentation

 * $$\langle S \mid R\rangle.$$

Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.

As a simple example, the cyclic group of order n has the presentation

 * $$\langle a \mid a^n = 1\rangle,$$

where 1 is the group identity. This may be written equivalently as

 * $$\langle a \mid a^n\rangle,$$

thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that do include an equals sign.

Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group.

A closely related but different concept is that of an absolute presentation of a group.

Background
A free group on a set S is a group where each element can be uniquely described as a finite length product of the form:

 * $$s_1^{a_1} s_2^{a_2} \cdots s_n^{a_n}$$

where the si are elements of S, adjacent si are distinct, and ai are non-zero integers (but n may be zero). In less formal terms, the group consists of words in the generators and their inverses, subject only to canceling a generator with an adjacent occurrence of its inverse.

If G is any group, and S is a generating subset of G, then every element of G is also of the above form; but in general, these products will not uniquely describe an element of G.

For example, the dihedral group D8 of order sixteen can be generated by a rotation, r, of order 8; and a flip, f, of order 2; and certainly any element of D8 is a product of rs and fs.

However, we have, for example, $rfr = f^{−1}$, $r^{7} = r^{−1}$, etc., so such products are not unique in D8. Each such product equivalence can be expressed as an equality to the identity, such as

 * $rfrf = 1$, or
 * $r^{8} = 1$, or

Informally, we can consider these products on the left hand side as being elements of the free group $f ^{2} = 1$, and let $F = ⟨r, f⟩$. That is, we let R be the subgroup generated by the strings rfrf, r8, f 2, each of which is also equivalent to 1 when considered as products in D8.

If we then let N be the subgroup of F generated by all conjugates x−1Rx of R, then it follows by definition that every element of N is a finite product x1−1r1x1 ... xm−1rm xm of members of such conjugates. It follows that each element of N, when considered as a product in D8, will also evaluate to 1; and thus that N is a normal subgroup of F. Thus D8 is isomorphic to the quotient group $R = ⟨rfrf, r^{8}, f ^{2}⟩$. We then say that D8 has presentation

 * $$\langle r, f \mid r^8 = 1, f^2 = 1, (rf)^2 = 1\rangle.$$

Here the set of generators is $F/N$, and the set of relations is $S = {r, f }$. We often see R abbreviated, giving the presentation

 * $$\langle r, f \mid r^8 = f^2 = (rf)^2 = 1\rangle.$$

An even shorter form drops the equality and identity signs, to list just the set of relators, which is $R = {r ^{8} = 1, f ^{2} = 1, (rf )^{2} = 1}$. Doing this gives the presentation

 * $$\langle r, f \mid r^8, f^2, (rf)^2 \rangle.$$

All three presentations are equivalent.

Notation
Although the notation ${r ^{8}, f ^{2}, (rf )^{2}} |undefined$ used in this article for a presentation is now the most common, earlier writers used different variations on the same format. Such notations include the following:

Definition
Let S be a set and let FS be the free group on S. Let R be a set of words on S, so R naturally gives a subset of $$F_S$$. To form a group with presentation $$\langle S \mid R\rangle$$, take the quotient of $$F_S$$ by the smallest normal subgroup that contains each element of R. (This subgroup is called the normal closure N of R in $$F_S$$.) The group $$\langle S \mid R\rangle$$ is then defined as the quotient group
 * $$\langle S \mid R \rangle = F_S / N.$$

The elements of S are called the generators of $$\langle S \mid R\rangle$$ and the elements of R are called the relators. A group G is said to have the presentation $$\langle S \mid R\rangle$$ if G is isomorphic to $$\langle S \mid R\rangle$$.

It is a common practice to write relators in the form $$x = y$$ where x and y are words on S. What this means is that $$y^{-1}x\in R$$. This has the intuitive meaning that the images of x and y are supposed to be equal in the quotient group. Thus, for example, rn in the list of relators is equivalent with $$r^n=1$$.

For a finite group G, it is possible to build a presentation of G from the group multiplication table, as follows. Take S to be the set elements $$g_i$$ of G and R to be all words of the form $$g_ig_jg_k^{-1}$$, where $$g_ig_j=g_k$$ is an entry in the multiplication table.

Alternate definition
The definition of group presentation may alternatively be recast in terms of equivalence classes of words on the alphabet $$S \cup S^{-1}$$. In this perspective, we declare two words to be equivalent if it is possible to get from one to the other by a sequence of moves, where each move consists of adding or removing a consecutive pair $$x x^{-1}$$ or $$x^{-1} x$$ for some $x$ in $S$, or by adding or removing a consecutive copy of a relator. The group elements are the equivalence classes, and the group operation is concatenation.

This point of view is particularly common in the field of combinatorial group theory.

Finitely presented groups
A presentation is said to be finitely generated if S is finite and finitely related if R is finite. If both are finite it is said to be a finite presentation. A group is finitely generated (respectively finitely related, ) if it has a presentation that is finitely generated (respectively finitely related, a finite presentation). A group which has a finite presentation with a single relation is called a one-relator group.

Recursively presented groups
If S is indexed by a set I consisting of all the natural numbers N or a finite subset of them, then it is easy to set up a simple one to one coding (or Gödel numbering) f : FS → N from the free group on S to the natural numbers, such that we can find algorithms that, given f(w), calculate w, and vice versa. We can then call a subset U of FS recursive (respectively recursively enumerable) if f(U) is recursive (respectively recursively enumerable). If S is indexed as above and R recursively enumerable, then the presentation is a recursive presentation and the corresponding group is recursively presented. This usage may seem odd, but it is possible to prove that if a group has a presentation with R recursively enumerable then it has another one with R recursive.

Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. However a theorem of Graham Higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group. From this we can deduce that there are (up to isomorphism) only countably many finitely generated recursively presented groups. Bernhard Neumann has shown that there are uncountably many non-isomorphic two generator groups. Therefore, there are finitely generated groups that cannot be recursively presented.

History
One of the earliest presentations of a group by generators and relations was given by the Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus – a presentation of the icosahedral group. The first systematic study was given by Walther von Dyck, student of Felix Klein, in the early 1880s, laying the foundations for combinatorial group theory.

Examples
The following table lists some examples of presentations for commonly studied groups. Note that in each case there are many other presentations that are possible. The presentation listed is not necessarily the most efficient one possible.

An example of a finitely generated group that is not finitely presented is the wreath product $$\mathbf{Z} \wr \mathbf{Z}$$ of the group of integers with itself.

Some theorems
"Theorem. Every group has a presentation."

To see this, given a group G, consider the free group FG on G. By the universal property of free groups, there exists a unique group homomorphism $⟨S$ whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in FG, therefore is equal to its normal closure, so $⟨S$. Since the identity map is surjective, φ is also surjective, so by the First Isomorphism Theorem, $(S | R)$. This presentation may be highly inefficient if both G and K are much larger than necessary.

"Corollary. Every finite group has a finite presentation."

One may take the elements of the group for generators and the Cayley table for relations.

Novikov–Boone theorem
The negative solution to the word problem for groups states that there is a finite presentation ${S; R}$  for which there is no algorithm which, given two words u, v, decides whether u and v describe the same element in the group. This was shown by Pyotr Novikov in 1955 and a different proof was obtained by William Boone in 1958.

Constructions
Suppose G has presentation $⟨S; R⟩$ and H has presentation $φ : F_{G} → G$ with S and T being disjoint. Then
 * the free product $⟨G | K⟩ = F_{G}/K$ has presentation $⟨G | K⟩ ≅ im(φ) = G$ and
 * the direct product $⟨S | R⟩$ has presentation $⟨S | R⟩$, where [S, T] means that every element from S commutes with every element from T (cf. commutator).

Deficiency
The deficiency of a finite presentation $⟨T | Q⟩$ is just $G ∗ H$ and the deficiency of a finitely presented group G, denoted def(G), is the maximum of the deficiency over all presentations of G. The deficiency of a finite group is non-positive. The Schur multiplicator of a finite group G can be generated by −def(G) generators, and G is efficient if this number is required.

Geometric group theory
A presentation of a group determines a geometry, in the sense of geometric group theory: one has the Cayley graph, which has a metric, called the word metric. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups.

Further, some properties of this graph (the coarse geometry) are intrinsic, meaning independent of choice of generators.

title: Presentation of a group


text:
Zulu traditional religion

Zulu traditional religion consists of the beliefs and spiritual practices of the Zulu people of southern Africa. It contains numerous deities commonly associated with animals or general classes of natural phenomena. Unkulunkulu is known to be the Supreme Creator.

The Creator and the ancestors
Similar to some other Bantu religions, Adherents of Zulu traditional religion believe in honoring ancestors (Amadlozi). Unkulunkulu is the highest god and is the creator of humanity.

UmkhuluwoMkhulu ("the greatest one") was created in Uhlanga, a huge swamp of reeds, before he came to Earth. Individuals base their morality or behavior ethics on his judgment or “apprehension of the Ultimate Reality.”

  Unkulunkulu is sometimes conflated with the sky god Umvelinqangi (meaning "he who was in the very beginning"), the god of thunder, earthquake whose other name is Unsondo, and is the son of Unkulunkulu, the Father, and Nomkhubulwane, the Mother.

The word nomkhubulwane means the one who shapeshifts into any form of an animal. Another name given for the supreme being Umkhuluwomkhulu is uSomandla, the ultimate source of all existence. European settlers used the word Unkulunkulu in order to try to explain their belief in the God of the bible to the people of Zululand.

According to Irvin Hexham (1981), "there is no evidence of belief in a heavenly deity or sky god in Zulu religion before the advent of Europeans". However, other scholars such as Eileen Jensen Krige, Isaac Schapera, Axel-Ivar Berglund (1976), Hammond-Took, and John Mbiti disagree with Hexham's analysis. They argue that the "lord of heaven" or Zulu sky god has always existed in the traditional Zulu belief system, a deity who they argue is greater than the "archetypal ancestor and creator, Unkulunkulu".

Other deities

 * Nomhoyi/Mamlambo, the goddess of rivers
 * Nomkhubulwane/Mbaba Mwana Waresa, sometimes called the Zulu Demeter, who is a goddess of the rainbow, agriculture, rain and beer (which she invented)
 * Inkosazana, another fertility goddess
 * uNgungi, the deity of the blacksmiths
 * iNyanga the Moon goddess is associated with healers who are called IziNyanga, the word nyanga is a Zulu word for the Moon
 * Sonzwaphi the deity of healing
 * Ukhulukhulwana (or UkhuluKhukwan) a star being ancestor who came from the stars and found the ancient Zulus living like animals and without laws. He taught them to build huts and taught them the high laws of isiNtu. The word unkulunkulu is suspected to be a corruption of the word umkhuluwomkhulu.

title: Zulu traditional religion


text:
Akan

Akan may refer to:

People and languages

 * Akan people, an ethnic group in Ghana and Côte d'Ivoire
 * Akan languages, a language group within the wider Central Tano languages
 * Kwa languages, a language group which includes Akan
 * Central Tano languages, a language group which includes Akan

Places

 * Akan (Ghana parliament constituency)
 * Akan District, Hokkaido, Japan
 * Akan, Hokkaido, a town in Akan District, Hokkaido
 * Akan National Park
 * Akan Volcanic Complex, a volcano in Hokkaidō, Japan
 * Lake Akan, a lake in Hokkaidō, Japan
 * Akan River, a river in Hokkaidō, Japan
 * Akan, Wisconsin, a town in the United States

Other uses

 * Akan religion, traditional beliefs and religious practices of the Akan people
 * Akan (surname), a surname
 * Akan names, names of Ghana origin
 * Akan (biblical figure), a person mentioned in the Book of Genesis
 * Akan (Maya god), a deity in Maya religion (identified with the god A')
 * Akan (あかん), a Japanese Kansai dialect phrase meaning "No way"

title: Akan