Exponentiation Totally Explained

Everything about Power Mathematics totally explained

Exponentiation is a mathematical operation, written aⁿ, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication:
ť

a^n = underbrace, we've |2

^X| = 2^|X|, where 2^X, usually denoted by PX, is the power set of X; each subset Y of X corresponds uniquely to a function on X taking the value 1 for x ∈ Y and 0 for x ∉ Y.

Exponentiation in category theory

In a Cartesian closed category, the exponential operation can be used to raise an arbitrary object to the power of another object. This generalizes the Cartesian product in the category of sets.

Exponentiation of cardinal and ordinal numbers

In set theory, there are exponential operations for cardinal and ordinal numbers.
If κ and λ are cardinal numbers, the expression κ^λ represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ. If κ and λ are finite then this agrees with the ordinary exponential operation. For example, the set of 3-tuples of elements from a 2-element set has cardinality 8.
Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process. In the ordinal numbers, exponentiation is defined by transfinite induction. For ordinals α and β, the exponential α^β is the supremum of the ordinal product α^γα over all γ < β.

Repeated exponentiation

Just as exponentiation of natural numbers is motivated by repeated multiplication, it's possible to define an operation based on repeated exponentiation; this operation is sometimes called tetration. Iterating tetration leads to another operation, and so on. This sequence of operations is captured by the Ackermann function and Knuth's up-arrow notation.

Exponentiation in programming languages

The superscript notation x^y is convenient in handwriting but inconvenient for typewriters and computer terminals that align the baselines of all characters on each line. Many programming languages have alternate ways of expressing exponentiation that don't use superscripts:

x ↑ y: Algol, Commodore BASIC

x ^ y: BASIC, J, Matlab, R, Microsoft Excel, TeX (and its derivatives), Haskell (for integer exponents), and most computer algebra systems

x ** y: Ada, Bash, Fortran, FoxPro, Perl, Python, Ruby, SAS, ABAP, Haskell (for floating-point exponents), Turing

x * y: APL

Power(x, y): Microsoft Excel, Delphi/Pascal (declared in "Math"-unit)

pow(x, y): C, C++, PHP

Math.pow(x, y): Java, JavaScript, Modula-3

Math.Pow(x, y): C#

(expt x y): Common Lisp, Scheme In Bash, C, C++, C#, Java, JavaScript, PHP, Python and Ruby, the symbol ^ represents bitwise XOR. In Pascal, it represents indirection.

History of the notation

The term power was used by Euclid for the square of a line. Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel. Samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), surfolide (fifth), zenzicube (sixth), second surfolide (seventh) and Zenzizenzizenzic (eighth). Biquadrate has been used to refer to the fourth power as well.
Another historical synonym, involution, is now rare and shouldn't be confused with its more common meaning.

Further Information

Get more info on 'Power Mathematics'.

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