Exponentiation
The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define
Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.
The term exponent originates from the Latin exponentem, the present participle of exponere, meaning "to put forth".
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"[9]—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.
In the late 16th century, Jost Bürgi would use Roman numerals for exponents in a way similar to that of Chuquet, for example iii4 for 4x3.
In 1636, James Hume used in essence modern notation, when in L'algèbre de Viète he wrote Aiii for A3.
Thus they would write polynomials, for example, as ax + bxx + cx3 + d. Samuel Jeake introduced the term indices in 1696.
Earlier Leonardo Torres Quevedo contributed Essays on Automation (1914) which had suggested the floating-point representation of numbers.
Eventually educators and engineers adopted scientific notation of numbers, consistent with common reference to order of magnitude in a ratio scale.
[19] For instance, in 1961 the School Mathematics Study Group developed the notation in connection with units used in the metric system.
Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up[27][28][29] (or left-associative).
It follows that in computer algebra, many algorithms involving integer exponents must be changed when the exponentiation bases do not commute.
For example, the prefix kilo means 103 = 1000, so a kilometre is 1000 m. The first negative powers of 2 have special names:
For example, See § Real exponents and § Non-integer powers of complex numbers for details on the way these problems may be handled.
Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function ex means that one has for every b > 0.
This results from 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 nth roots, that is, of exponents
Although the general theory of exponentiation with non-integer exponents applies to nth roots, this case deserves to be considered first, since it does not need to use complex logarithms, and is therefore easier to understand.
The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments.
This makes the principal nth root a continuous function in the whole complex plane, except for negative real values of the radicand.
the complex number comes back to its initial position, and its nth roots are permuted circularly (they are multiplied by
Geometrically, the nth roots of unity lie on the unit circle of the complex plane at the vertices of a regular n-gon with one vertex on the real number 1.
[34][35][36] 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
in the sense that its graph consists of several sheets that define each a holomorphic function in the neighborhood of every point.
[42] 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.
The latter has a basis consisting of the sequences with exactly one nonzero element that equals 1, while the Hamel bases of the former cannot be explicitly described (because their existence involves Zorn's lemma).
Then D can be viewed as a subset of R2 (that is, the set of all pairs (x, y) with x, y belonging to the extended real number line R = [−∞, +∞], endowed with the product topology), which will contain the points at which the function f has a limit.
This method does not permit a definition of xy when x < 0, since pairs (x, y) with x < 0 are not accumulation points of D. On the other hand, when n is an integer, the power xn is already meaningful for all values of x, including negative ones.
Programming languages generally express exponentiation either as an infix operator or as a function application, as they do not support superscripts.
[49] The notations include: In most programming languages with an infix exponentiation operator, it is right-associative, that is, a^b^c is interpreted as a^(b^c).
