Definition:Power (Algebra)/Real Number/Definition 3/Binary Expansion

Definition
Let $x \in \R_{>0}$ be a (strictly) positive real number.

Let $r \in \R$ be a real number.

First let $x > 1$.

Let $r$ be expressed in binary notation:
 * $r = n \cdotp d_1 d_2 d_3 \ldots$

where $d_1, d_2, d_3 \ldots$ are in $\set {0, 1}$.

For $k \in \Z_{> 0}$, let $\psi_1, \psi_2 \in \Q$ be rational numbers defined as:

Then $x^r$ is defined as the (strictly) positive real number $\xi$ defined as:
 * $\ds \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$

In this context, $x^{\psi_1}, x^{\psi_2}$ denote $x$ to the rational powers $\psi_1$ and $\psi_2$.

Next let $x < 1$.

Then $x^r$ is defined as:
 * $x^r := \paren {\dfrac 1 x}^{-r}$

Finally, when $x = 1$:


 * $x^r = 1$