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 (230)
Orphanage
Unclass'd
Unproven (519)
Corrections (17)
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: Very high
[parent] alternative proof that $\sqrt{2}$ is irrational (Proof)

Following is a proof that $ \sqrt{2}$ is irrational.

The polynomial $ x^2-2$ is irreducible over $ \mathbb{Z}$ by Eisenstein's criterion with $ p=2$. Thus, $ x^2-2$ is irreducible over $ \mathbb{Q}$ by Gauss's lemma. Therefore, $ x^2-2$ does not have any roots in $ \mathbb{Q}$. Since $ \sqrt{2}$ is a root of $ x^2-2$, it must be irrational.

This method generalizes to show that any number of the form $ \sqrt[r]{n}$ is not rational, where $ r \in \mathbb{Z}$ with $ r>1$ and $ n \in \mathbb{Z}$ such that there exists a prime $ p$ dividing $ n$ with $ p^2$ not dividing $ n$.



"alternative proof that $\sqrt{2}$ is irrational" is owned by Wkbj79.
(view preamble)

View style:

See Also: irrational, Eisenstein criterion, Gauss's lemma II


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

Cross-references: prime, rational, number, roots, Eisenstein's criterion, irreducible, polynomial, irrational, proof
There are 2 references to this entry.

This is version 5 of alternative proof that $\sqrt{2}$ is irrational, born on 2007-04-14, modified 2007-08-23.
Object id is 9183, canonical name is AlternativeProofThatSqrt2IsIrrational.
Accessed 1114 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)
 12E05 (Field theory and polynomials :: General field theory :: Polynomials )
 13A05 (Commutative rings and algebras :: General commutative ring theory :: Divisibility)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy
Imaginary irrational number? by pahio on 2007-04-15 03:49:36
Hi, are there more people thinking that non-real complex numbers (in Continental Europe: imaginary numbers) are irrational -- because they are not rational? I think such a terminology is exceptional.
Jussi
[ reply | up ]
Interact
post | correct | update request | add example | add (any)