• Extending the power notion for zero and negative integer exponents requires the existence of http://planetmath.org/node/10539neutral and inverse elements (e and a-1):


    The two first power laws then remain in for all integer exponents, and if the operation is commutative, also the .

When the operation in question is the multiplication of real or complex numbers, the power notion may be extended for other than integer exponents.

  • One step is to introduce fractional (http://planetmath.org/FractionalNumber) exponents by using roots (http://planetmath.org/NthRoot); see the fraction power.

  • The following step would be the irrational exponents, which are in the power functionsDlmfDlmf.  The irrational exponents are possible to introduce by utilizing the exponential functionDlmfDlmfMathworld and logarithms; another way would be to define aϱ as limit of a sequenceMathworldPlanetmath


    where the limit of the rational number sequence  r1,r2,  is ϱ.  The sequence ar1,ar2, may be shown to be a Cauchy sequence.

  • The last step were the imaginary (non-real complex) exponents μ, when also the base of the power may be other than a positive real number; the one gets the so-called general power.

Title exponentiation
Canonical name Exponentiation
Date of creation 2013-03-22 19:08:44
Last modified on 2013-03-22 19:08:44
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Topic
Classification msc 20-00
Related topic ContinuityOfNaturalPower
Defines power law
Defines power of product