where is the reciprocal of .
The case of arbitrary exponents is somewhat more complicated. A possible strategy is to define roots, then rational powers, and then extend by continuity. Our approach is different. For and , we define the set of all reals that one would want to be smaller than , and then define the latter as the least upper bound of this set. To be more precise, let and define
We then define to be the least upper bound of . For we define
The exponential operation possesses a number of important properties (http://planetmath.org/PropertiesOfTheExponential), some of which characterize it up to uniqueness.
It is also possible to define the exponential operation in terms of the exponential function and the natural logarithm. Since these concepts require the context of differential theory, it seems preferable to give a basic definition that relies only on the foundational property of the reals.
|Date of creation||2013-03-22 12:29:59|
|Last modified on||2013-03-22 12:29:59|
|Last modified by||rmilson (146)|