| Version 6 |
Version 5 |
| \begin{cor*} |
\begin{cor*} |
| Let $a$ be a complex number and $n$ be an integer such that $a^n$ is not rational. Then $a$ is not rational. |
Let $a$ be a complex number and $n$ be an integer such that $a^n$ is not rational. Then $a$ is not rational. |
| \end{cor*} |
\end{cor*} |
|
|
| \begin{proof} |
\begin{proof} |
| We show this by way of contrapositive. In other words, we show that, if $a$ is rational, then $a^n$ is rational. |
We show this by way of contrapositive. In other words, we show that, if $a$ is rational, then $a^n$ is rational. |
|
|
| Let $a$ be rational. Then there exist integers $b$ and $c$ with $c\neq 0$ such that $\displaystyle a=\frac{b}{c}$. Thus, $\displaystyle a^n=\frac{b^n}{c^n}$, which is a rational number. |
Let $a$ be rational. Then there exist integers $b$ and $c$ with $c\neq 0$ such that $\displaystyle a=\frac{b}{c}$. Thus, $\displaystyle a^n=\frac{b^n}{c^n}$, which is a rational number. |
| \end{proof} |
\end{proof} |
|
|
| Note that the converse is not true. For example, $\sqrt{2}$ is irrational and $\left(\sqrt{2}\right)^2=2$ is rational. |
Note that the converse is not true. For example, $\sqrt{2}$ is irrational and $\left(\sqrt{2}\right)^2=2$ is rational. |
