\begin{definition}[Definition:Power (Algebra)/Real Number/Definition 3/Binary Expansion]
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:
{{begin-eqn}}
{{eqn | l = \psi_1
      | r = n + \sum_{j \mathop = 1}^k \frac {d_1} {2^k} = n + \frac {d_1} 2 + \cdots + \frac {d_k} {2^k}
}}
{{eqn | l = \psi_2
      | r = \psi_1 + \dfrac 1 {2^k}
}}
{{end-eqn}}
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 $x = 1$:
:$x^r = 1$
\end{definition}