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

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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 by its decimal expansion:

$r = n \cdotp d_1 d_2 d_3 \ldots$


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

\(\displaystyle \psi_1\) \(=\) \(\displaystyle n + \sum_{j \mathop = 1}^k \frac {d_1} {10^k} = n + \frac {d_1} {10} + \cdots + \frac {d_k} {10^k}\)
\(\displaystyle \psi_2\) \(=\) \(\displaystyle \psi_1 + \dfrac 1 {10^k}\)


Then $x^r$ is defined as the (strictly) positive real number $\xi$ defined as:

$\displaystyle \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$

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:

$x^r := \left({\dfrac 1 x}\right)^{-r}$


Finally, when $x = 1$:

$x^r = 1$


Binary Expansion

It is possible to use other bases than decimal:

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 $\left\{ {0, 1}\right\}$.


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

\(\displaystyle \psi_1\) \(=\) \(\displaystyle n + \sum_{j \mathop = 1}^k \frac {d_1} {2^k} = n + \frac {d_1} 2 + \cdots + \frac {d_k} {2^k}\)
\(\displaystyle \psi_2\) \(=\) \(\displaystyle \psi_1 + \dfrac 1 {2^k}\)


Then $x^r$ is defined as the (strictly) positive real number $\xi$ defined as:

$\displaystyle \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$

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:

$x^r := \left({\dfrac 1 x}\right)^{-r}$


Finally, when $x = 1$:

$x^r = 1$


Also see


Historical Note

The technique of defining the $r$th power of $x$ for real number $r$ by means of the decimal expansion of $r$, was first designed by John Wallis in $1655$, and later refined by Isaac Newton in $1669$.


Sources