# Prove sqrt(6) is irrational

1. ### Klion

14
Alright, heading says it all. This is a nice problem heh.. I can see how to prove sqrt(5) is irrational. I think this method works up to the points where the fact 5 is a prime is used, (ie prime lemma) on 5 which doesn't work so well on 6! hehe

Was thinking of maybe using product of primes somehow but.. hmm dunno

Anyway for sqrt(5) went like this (proof by contradiction)

Prove sqrt(5) is not rational.

Suppose sqrt(5) = a/b, where a,b E Z+ (suppose sqrt(5) is rational)
we'll also assume gcd(a,b) = 1 (otherwise just divide a,b by gcd)
sqrt(5) = a/b <=> 5 = a^2/b^2 or 5*b^2=a^2
so 5|a^2, but 5 is a prime so 5|a Ea' 5*a'=a so 5b^2=(5a)^2 or 5b^2=5^2*a^2 or b^2 = 5*a'^2 so 5|b^2 by prime lemma 5|b 5|a ^ b|b so 5|gcd(a,b)
so 5|1 which is nonsense therefore our initial assumption is incorrect and 5 is in fact irrational.

-Kli

2. ### KLscilevothma

317
Suppose sqrt(6) is rational

gcd(a,b) = 1

$$\sqrt{6} = \frac{a}{b}$$

$$6b^2 = a^2$$

Now there are 4 possibilities.

1) both a and b are even.
It cannot be true since gcd(a,b)=1

2) a is odd and b is even
It cannot be true too. If a is odd, then left hand side will be even while right hand side will be odd.

3) both a and b are odd.
It cannot be true too and the reason is the same as that in (2)

4) a is even and b is odd.

Let a = 2m and b = 2n + 1

$$6b^2 = a^2$$

$$6*{(2n+1)}^2 = {2m}^2$$

$$12m^2 + 12m +3 = 2n^2$$

left hand side is odd while right hand side is even

Therefore sqrt(6) is irrational

3. ### HallsofIvy

41,055
Staff Emeritus
"Let a = 2m and b =2n + 1
6b2= a2

6(2n+1)2= 2m2"

No, it should be 6(2n+1)2= (2m)2

"12m2+ 12m+ 3= 2n2"

Now this completely baffles me. Even assuming you accidently switched m and n, it doesn't follow.
From 6(2n+1)2= (2m)2 you get
6(4n2+ 4n+ 1)= 4m2 or
24n2+ 24n+ 6= 4m2 and both sides are clearly even.

4. ### KLscilevothma

317
How about dividing both sides by 2, then we get
12n2+ 12n+ 3= 2m2
where left hand side is odd while right hand side is even.

If we don't divide both sides by 2, left hand side cannot be divisible by 4 while right hand side can.

5. ### mathman

6,706
How about the following (for any n not a square of an integer).

sqrt(n)=a/b, where a and b are integers with no common factors, and b >1.

n=a2/b2

The numerator and denominator still have no common factors, so n is not an integer.

6. ### phoenixthoth

any n in what? is n any nonnegative rational number, then? if it's any nonnegative integer n not the square of an integer, it seems odd to end by saying "n is not an integer."

7. ### mathman

6,706
Sorry for the confusion. I assumed we were discussing integers in general. My discussion is about (positive) integers which are not squares of integers. In n is a square of an integer, then b=1.

8. ### phoenixthoth

then wouldn't "n is not an integer" prove it?

9. ### uart

2,776
Hi phoenixthoth, Mathman is using proof by contradiction . In the first line of his proof what he means is : Assume that n is a positive integer that is not a perfect square.

He goes on to show that assuming sqrt(n) to be rational, but non integer (see b>1), leads directly to a contradiction of the assumption that n was an integer. Hence the result that the square root of all integers, other than the perfects squares, are irrational. It's a perfectly valid proof.

Last edited: Nov 28, 2003
10. ### phoenixthoth

it seems fishy because it was easier to prove it in general for nonsquare nonnegative integers n than for n=6 and it's usually the case that it's easier to do it for n=6 and harder in general. i'm at some point if i feel like it going to check where the assumption that n is not a square was used and if this proof applies to square integers as well. i suppose that's what enables one to say b>1.

11. ### phoenixthoth

and 4 = 2*2 both of which are prime

7,521
Staff Emeritus
What has 4 = 2 X 2 got to do with that? Some squares are squares of primes and some aren't, like 6 X 6 = 36. How does that affect the proof?

13. ### kx21

0
...

(Repost of 'disappeared' Post with Refinement)

Proof;-

1) Each SQRT(Prime Number) is irrational &

2) SQRT(Prime Number 1) * SQRT(Prime Number 2) is irrational if Prime Number 1 is not equal to Prime Number 2.

3) SQRT(6) = SQRT(2) * SQRT(3)

4) 2 & 3 are two different Prime Numbers.

i.e.

5) SQRT(6) is irrational.

This completes the proof...

kx21

https://www.physicsforums.com/member.php?s=&action=getinfo&find=lastposter&forumid=80

Last edited by a moderator: Dec 1, 2003
14. ### phoenixthoth

this was in reference to a deleted post.

15. ### phoenixthoth

assumption #1 is as clear as conclusion #5. some people would not take #1 without proof.

selfadjoint, this post in its original form was about six words long and the point in bringing up 4=2x2 is that it should have been stated that the two primes must be different; it was stated in the second form. this argument would have to be messaged before it would work for all n. the main thing there would be to prove/say a nonsquare has not all even powers in its prime factorization.

Last edited: Dec 1, 2003
16. ### uart

2,776
The problem is that you never really establish the truth of either point 1 or point 2 anywhere in the proof kx21

I still think that the best proof is as follows.

Let $$\sqrt{n} = \frac{a}{b}$$

Then $$a^2 = b^2 n$$ <- Eqn1

From the fundamental therom of arithmetic (that is the uniqueness of prime factorization) it is clear that any integer squared contains only even powered prime factors.

Thus the LHS of Eqn1 can contain only even powered prime factors whereas the RHS of Eqn1 will contain odd powered prime factors whenever n contains any odd powered prime factors.

So LHS = RHS is possible if (and only if) n contains only even powered prime factors. This is of course equivalent to the requirement that n be a perfect square.

In other words, either n is a perfect square or $$\sqrt{n}$$ is irrational.

Last edited: Dec 1, 2003
17. ### kx21

0
*** a 'M' Generalization with a bonus Link Phi, e & i... ***

Theorem- The irrationality of SQRT(N)

SQRT(N) is irrational if

N = p1*p2...*pk, where

pi i =1 to k are different Prime Numbers.

For instances,

1) Is SQRT(6) irrational?

...

It's irrational as 6 = 2 * 3

[?]

2) Is SQRT(30) irrational?

A simple Number Test,

with a Bonus Link about Phi (Golden Ratio), e, pi for all:-

http://superstringtheory.com/forum/extraboard/messages12/376.html

Happy exploring & have a wonderful memory & time...

kx21

https://www.physicsforums.com/member.php?s=&action=getinfo&find=lastposter&forumid=80

Last edited by a moderator: Dec 1, 2003
18. ### uart

2,776
While correct that theorem is only "half baked" as there are very many integers n for which sqrt(n) is irrational but yet n cannot be expressed as such a simple prime product as stated by the therom (that is, with all prime factors to unity power only).

Why not do it properly as per either mathmans or my own post, and establish irrationality for sqrt of all integers which are not perfect squares ?

Also, if you're going to just quote a therom that has already established primeness of certain square roots (as opposed to establishing it from something more fundamental like the uniqueness of prime factorization) then why pick such a "half baked" theorem in the first place? Why not use a better theorem to begin with.

Theorem : Either n is a perfect square or sqrt(n) is irrational.

Then all you need to do is to say. "Since 6 in not a perfect then by the above theorem then the sqrt of 6 is irrational." QED.

Really, if you are going to be lame then at least do it properly. ;-)

19. ### kx21

0
...

Let me think...

Are there any 'Topological Defects' hidden in your Theorem.

Eureka!!!

n = -1.

Proof:-

Given SQRT(-1)= i * i

This Completes the proof...

[?]

kx21
https://www.physicsforums.com/member.php?s=&action=getinfo&find=lastposter&forumid=80

Last edited by a moderator: Dec 1, 2003
20. ### phoenixthoth

it was clarified earlier that n is a nonnegative nonsquare integer. i is irrational if irrational means not rational. but irrational usually refers to R\Q. technically, this doesn't violate an or statement though i think what was mean by adding the word "either" was exclusive or, Xor.