Proving a number is irrational by contradiction

Hi

I have this worked example to prove is an irrational number via contradiction. I will put it down here now. But my question is how do I do this with as I get stuck half way through this when we come to talking about even numbers.

Step 1: Suppose that is rational.

Step 2: This means that its possible to find where a and b have only a common factor of 1 such that:

, b not equal to 0.

Squaring both sides:

Step 3:

This means that must be even and thus must be even.

i.e.

Step 4:

This means that b^2 and b must be even.

Step 5: So both a and b are even. This means that a and b have a common factor or 2. This contradicts the original hypothesis and so the hypothesis must be false. This means that is not able to be expressed in the form a/b and is therefore an irrational number.

The end.

So that is the example proof. Here is what I have done so far with the example.

Step 1: Suppose that is rational.

Step 2: This means that its possible to find where a and b have only a common factor of 1 such that:

, b not equal to 0.

Squaring both sides:

Step 3:

This condition is only met when a is odd and b is odd. I just did this with a little trial and error. So a is odd. Thus a = 2k + 1

Step 4:

......

Ok this is where I come unstuck. Does anyone have any suggestions?

David.

Thankyou both for your help