Sqrt2 Irrational

Date: 05/17/99 at 12:53:00
From: G.Arjun
Subject: Real Numbers

I would like to know the answer to the question, How do you prove that 
the square root of 2 (or other roots of other real numbers) is 
irrational? I am not able to understand the argument.

Date: 05/18/99 at 15:13:35
From: Doctor Floor
Subject: Re: Real Numbers

Hello, G. Arjun,

Thanks for your question!

To prove that sqrt(2) [sqrt means square root] is an irrational number, 
we have to show that it cannot be written as a/b, where a and b are 
integers. 

Such a proof has to be done indirectly: 

1. We assume that sqrt(2) could be written as a/b for integer a and b;
2. We show that this leads to a contradiction. So something impossible 
   has to be derived from our assumption.

The fact that we derive someting impossible from the fact that 
sqrt(2) = a/b shows that sqrt(2)=a/b must be false, and we have 
proven the theorem.

Let's assume that sqrt(2) could be written as a/b for integer a and b. 
We can derive:

  sqrt(2) = a/b [sqare both sides]
        2 = a^2/b^2 [multiply by b^2; ^2 means squared]
    2*b^2 = a^2
      a^2 = 2*b^2

Now we have two integers in the equation, 2*b^2 and a^2. We can factor 
these integers into primes, and find:

  a^2   = 2^m * {product of odd primes}
  2*b^2 = 2^n * {product of odd primes}

Because a^2 is a square, m has to be even.
Because b^2 is a square, and "2*" brings an additional factor 2, 
n has to be odd.

So m and n are not equal.

But if a^2 = 2*b^2, then the factorisation into primes of these two 
has to be equal. So, m and n should be equal.

We have found a contradiction. And our assumption, that sqrt(2) = a/b 
for integers a and b, cannot hold. This proves that sqrt(2) is 
irrational.

I hope this helps you to understand.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/17/99 at 14:42:20
From: Wayne
Subject: Square root of two

Can you prove to me how the square root of two is irrational?

Date: 05/17/99 at 16:20:17
From: Doctor Rob
Subject: Re: Square root of two

There are a couple of ways to do that. Both are proofs by 
contradiction; that is, they assume that sqrt(2) is rational, and 
derive an impossible conclusion from that assumption. Here is one:

Assume sqrt(2) = a/b, reduced to lowest terms (that is, a and b have 
no common factor besides 1). Then 2 = (a/b)^2 = a^2/b^2, so 
2*b^2 = a^2. That means that a^2 is even, which implies that a is 
even. Then write a = 2*c, a^2 = 4*c^2 = 2*b^2, so 2*c^2 = b^2. That 
means that b^2 is even, which implies that b is even. This means 
that a and b have the common factor 2, which is a contradiction. 
Thus no such fraction a/b can exist, and sqrt(2) is irrational.

Here's another:

Assume sqrt(2) = a/b, with a and b positive integers, and least
denominator among all such fractions. Then

                        2*b^2 = a^2,
  2*b^2 - a*b = a^2 - a*b,            4*b^2 > 2*b^2 = a^2 > b^2,
  b*(2*b-a) = a*(a-b),                2*b > a > b,
  (2*b-a)/(a-b) = a/b = sqrt(2),      b > a - b > 0  and  2*b - a > 0.

Now (2*b-a)/(a-b) is a fraction equal to sqrt(2) with positive integer
numerator and denominator, but smaller denominator than a/b has.
This is a contradiction. Thus no such fraction a/b can exist, and
sqrt(2) is irrational.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/