Show the square root of two is irrational?

Answer 1

Suppose to the contrary that square root of two is rational. This means we can write square root 2 = a/b where a/b is a fraction that CANNOT be reduced..that part is important. Now, 4a^2=b^2 from square both sides and multiplying So we know the left side is even since it is a multiple of 4. So the right hand side must be also even since they are equal, but that implies they were not relatively prime to start with which is a contradiciton. we conclude square root of 2 is irrational. I will help to make this a little clearer, just in case some people to not fully understand the proof. We assumed a/b was in reduced form which is equivalent to saying there are NO COMMON Factors. For exampel 1/2 or 1/3 is in reduced form but 2/4 is not because they have a common factor of 2. No if b^2 is even, then b must be even (easy to prove) . This causes a problem because if b is even then a cannot be even or they would not be relatively prime (that means they have no common factors) Here is an example of a rational number so you can see how it works 8 for example 8=a/b 8b=a so 8b is even since it is 2(4b) now that means a is even. b can be odd since even x odd is even, we can write 8=8/1 and this is inreduced form Try it with 2/3...3b=2 etc.. it works! The conclusion is that square root of two is irrational There are many many other ways to prove this, some people understand some proofs more than others. I will put some more here later.