So, I'm trying to prove that the square root of 3 is irrational.
2. Attempt at a Solution
2x can be any even number and 2x+1 can be any odd number. Since an irrational number is any number that can't be expressed as a ratio of two integers, I just have to show that the ratio of any two integers squared, either odd/even, even/odd, or odd/odd will not equal 3.
I think I've proved that any odd/even or even/odd cannot be the square root of three but I'm having trouble with odd/odd.
For odd/even and even/odd:
(2x+1)/(2x)=sqrt(3) ==> (2x+1)=sqrt(3)(2x)
square both sides:
4x^2+4x+1=12x^2 ==>even+1 does not equal an even.
However if I apply the same method with two odd numbers say (2x+1)/(2y+1)=sqrt(3). get the following:
I'm not sure where to go from there...Thanks in advance for the help