Exponentiation


In mathematics, exponentiation, denoted, is an operation involving two numbers: the base,, and the exponent or power,. When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases:
In particular,.
The exponent is usually shown as a superscript to the right of the base as or in computer code as b^n. This binary operation is often read as " to the power "; it may also be referred to as " raised to the th power", "the th power of ", or, most briefly, " to the ".
The above definition of immediately implies several properties, in particular the multiplication rule:
That is, when multiplying a base raised to one power times the same base raised to another power, the powers add.
Exponentiation can also be extended to powers that are not positive integers. When is non-zero, the definition
is compatible with the multiplication rule:. A similar argument suggests the definition
for negative integer powers, and in particular for any nonzero number, and also the definition
for fractional powers. For example,, meaning, which is the definition of square root:.
The definition of exponentiation can be extended in a natural way to define for any positive real base and any real number exponent. More involved definitions allow complex base and exponent, as well as certain types of matrices as base or exponent.
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 is a mistranslation of the ancient Greek δύναμις used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios.
The word exponent was coined in 1544 by Michael Stifel. In the 16th century, Robert Recorde used the terms "square", "cube", "zenzizenzic", "sursolid", "zenzicube", "second sursolid", and "zenzizenzizenzic". "Biquadrate" has been used to refer to the fourth power as well.

History

In The Sand Reckoner, Archimedes proved the law of exponents,, necessary to manipulate powers of. He then used powers of to estimate the number of grains of sand that can be contained in the universe.
In the 9th century, the Persian mathematician Al-Khwarizmi used the terms مَال 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 for a cube, which later Islamic mathematicians represented in mathematical notation as the letters mīm and kāf, respectively, by the 15th century, as seen in the work of Abu'l-Hasan ibn Ali al-Qalasadi.
Nicolas Chuquet used a form of exponential notation in the 15th century, for example to represent. 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 for.
In 1636, James Hume used in essence modern notation, when in L'algèbre de Viète he wrote for. 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.
Some mathematicians used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as.
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.
In 1748, Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing:

20th century

As calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation. For example Konrad Zuse introduced floating-point arithmetic in his 1938 computer Z1. One register contained representation of leading digits, and a second contained representation of the exponent of 10. Earlier Leonardo Torres Quevedo contributed Essays on Automation which had suggested the floating-point representation of numbers. The more flexible decimal floating-point representation was introduced in 1946 with a Bell Laboratories computer. Eventually educators and engineers adopted scientific notation of numbers, consistent with common reference to order of magnitude in a ratio scale.
For instance, in 1961 the School Mathematics Study Group developed the notation in connection with units used in the metric system.
Exponents also came to be used to describe units of measurement and quantity dimensions. For instance, since force is mass times acceleration, it is measured in kg m/sec2. Using M for mass, L for length, and T for time, the expression M L T–2 is used in dimensional analysis to describe force.

Terminology

The expression is called "the square of " or " squared", because the area of a square with side-length is.
Similarly, the expression is called "the cube of " or " cubed", because the volume of a cube with side-length is.
When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example,. The base appears times in the multiplication, because the exponent is. Here, 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 can be simply read "3 to the 5th", or "3 to the 5".

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 as one has an associative multiplication:
The base case is
and the recurrence is
The associativity of multiplication implies that for any positive integers and,
and

Zero exponent

As mentioned earlier, a number raised to the power is :
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
also holds for.
The case of is controversial. In contexts where only integer powers are considered, the value is generally assigned to 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 and nonzero :
Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity.
This definition of exponentiation with negative exponents is the only one that allows extending the identity to negative exponents.
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 . In particular, in such a structure, the inverse of an invertible element is standardly denoted

Identities and properties

The following identities, often called , hold for all integer exponents, provided that the base is non-zero:
Unlike addition and multiplication, exponentiation is not commutative: for example,, but reversing the operands gives the different value. Also unlike addition and multiplication, exponentiation is not associative: for example,, whereas. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down, not bottom-up. That is,
which, in general, is different from

Powers of a sum

The powers of a sum can normally be computed from the powers of the summands by the binomial formula
However, this formula is true only if the summands commute, which is implied if they belong to a structure that is commutative. Otherwise, if and 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 for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

Combinatorial interpretation

For nonnegative integers and, the value of is the number of functions from a set of elements to a set of elements. Such functions can be represented as -tuples from an -element set. Some examples for particular values of and are given in the following table:
The possible -tuples of elements from the set
0 = 0
1 = 1
2 = 8,,,,,,,
3 = 9,,,,,,,,
4 = 4,,,
5 = 1

Particular bases

Powers of ten

In the base ten number system, integer powers of are written as the digit followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, and.
Exponentiation with base is used in scientific notation to denote large or small numbers. For instance, can be written as and then approximated as.
SI prefixes based on powers of are also used to describe small or large quantities. For example, the prefix kilo means, so a kilometre is.