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if  is irrational then  is irrational
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(Theorem)
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Proof. 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. 
Note that the converse is not true. For example, $\sqrt{2}$ is irrational and $\left(\sqrt{2}\right)^2=2$ is rational.
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"if  is irrational then  is irrational" is owned by Wkbj79. [ full author list (3) | owner history (2) ]
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Cross-references: converse, rational number, rational, contrapositive, irrational, integer, real number
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This is version 10 of if  is irrational then  is irrational, born on 2004-04-18, modified 2008-04-30.
Object id is 5779, canonical name is IfAnIsIrrationalThenAIsIrrational.
Accessed 3109 times total.
Classification:
| AMS MSC: | 11J72 (Number theory :: Diophantine approximation, transcendental number theory :: Irrationality; linear independence over a field) | | | 11J82 (Number theory :: Diophantine approximation, transcendental number theory :: Measures of irrationality and of transcendence) |
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Pending Errata and Addenda
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