The Degree with the Irrational Exponent

Let′s suppose `a` is irrational number. What is the sense of record `a^a` , where `a` - positive number? Let′s consider three events: `a=1` , `a>1` , `0<a<1` .

  1. If `a=0` , then we suppose `1^a=1` .
  2. Suppose `a>1`. Let′s take any rational number `r_1<a` and any rational number `r_2>a`. Thereat `r_1<r_2` and `a^(r_1)<a^(r_2)` . In this case `a^a` means the number, that is between `a^(r_1)` and `a^(r_2)` for any rational numbers `r_1` and `r_2` such, that `r_1<a` and `r_2>a`. It is proved, that there is such number and singular for any `a>1` and for any irrational number `a`.
  3. Suppose `0<a<1`. Let′s take any rational number `r_1<a` and any rational number `r_2>a`. Thereat `r_1<r_2` and `a^(r_1)>a^(r_2)` . In this case `a^a` means such number, that is between `a^(r_2)` and `a^(r_1)` for any rational numbers `r_1` and `r_2`, satisfying the inequality `r_1<a<r_2`. It is proved, that there is such number and singular for any number `a` with the interval `(0,1)` and any irrational number `a`.