if an is irrational then a is irrational
Theorem.
If a be a real number and n is an integer such that an is irrational, then a is irrational.
Proof.
We show this by way of contrapositive. In other words, we show that, if a is rational, then an is rational.
Let a be rational. Then there exist integers b and c with c≠0 such that a=bc. Thus, an=bncn, which is a rational number.
∎
Note that the converse is not true. For example, √2 is irrational and (√2)2=2 is rational.
Title | if an is irrational then a is irrational |
---|---|
Canonical name | IfAnIsIrrationalThenaIsIrrational |
Date of creation | 2013-03-22 14:18:50 |
Last modified on | 2013-03-22 14:18:50 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 13 |
Author | Wkbj79 (1863) |
Entry type | Theorem |
Classification | msc 11J82 |
Classification | msc 11J72 |