Let x &gt; 0, and let α be an irrational number. Can we make sense of x α ? What about the case x &lt; 0 ?

Question

Let   x  &amp;gt;  0, and let   α be an irrational number. Can we make sense of       x          α       ? What about the case   x  &amp;lt;  0 ?

Valeria Wolfe · Accepted Answer

Assuming everything here is real, there are several ways to make sense of       x    α  .Let a1,a2,a3,… be a sequence of rational numbers converging to α. Then we define      x    α    =      lim          n      →      ∞            x                  a        n             One should, of course, make sure that any sequence converging to α gives the same limit.The one I think is most often used as a definition by mathematicians is simply      x    α    =  exp  ⁡  (  α  ⋅  ln  ⁡  x  ) where both   exp and   ln may be defined several different ways, for instance  exp  ⁡  (  a  )  =  1  +  a  +            a      2        2    +            a      3        6    +  ⋯  +            a      n              n      !        +  ⋯    ln  ⁡  x  =      ∫    1    x        1    t    d  t There are many definitions to choose from here, though.Then we have this one, which is not so common, but somewhat of a favourite of mine, if one knows a little group theory: Consider   (      R    ,  +  ), the real numbers with the operation of addition and   (            R        +    ,  ⋅  ), the positive real numbers with multiplication. As groups, these are isomorphic, and there are many different homomorphisms between them, but each one may be uniquely determined by where it sends   1  ∈  (      R    ,  +  ) (and we probably need to assume that it&#039;s &quot;nice&quot; for some suitable notion of niceness, like being continuous, or monotonic).One of these homomorphisms   f  :  (      R    ,  +  )  →  (            R        +    ,  ⋅  ) has   f  (  1  )  =  x. We then define      x    α    =  f  (  α  )

Let x > 0 , and let α be an irrational number. Can we make sens

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