In the entry general associativity, the notion of the power for elements of a set having an associative binary operation “” and for positive integers as exponents (http://planetmath.org/GeneralPower) was defined as a generalisation of the operation. Then the two power laws
are . For the validity of the third well-known power law,
the law of power of product, the commutativity of the operation is needed.
Extending the power notion for zero and negative integer exponents requires the existence of http://planetmath.org/node/10539neutral and inverse elements ( and ):
The two first power laws then remain in for all integer exponents, and if the operation is commutative, also the .
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 functions. The irrational exponents are possible to introduce by utilizing the exponential function and logarithms; another way would be to define as limit of a sequence
The last step were the imaginary (non-real complex) exponents , when also the base of the power may be other than a positive real number; the one gets the so-called general power.
|Date of creation||2013-03-22 19:08:44|
|Last modified on||2013-03-22 19:08:44|
|Last modified by||pahio (2872)|
|Defines||power of product|