PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] if $a^n$ is irrational then ${a}$ is irrational (Theorem)
Theorem   If $ a$ be a real number and $ n$ is an integer such that $ a^n$ 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 $ 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. $ \qedsymbol$

Note that the converse is not true. For example, $ \sqrt{2}$ is irrational and $ \left(\sqrt{2}\right)^2=2$ is rational.



"if $a^n$ is irrational then ${a}$ is irrational" is owned by Wkbj79. [ full author list (3) | owner history (2) ]
(view preamble | get metadata)

View style:


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: converse, rational number, rational, contrapositive, irrational, integer, real number
There are 2 references to this entry.

This is version 10 of if $a^n$ is irrational then ${a}$ is irrational, born on 2004-04-18, modified 2008-04-30.
Object id is 5779, canonical name is IfAnIsIrrationalThenAIsIrrational.
Accessed 2693 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)
Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)