Question proof √3 is irrational and show reason.
We prove that √3 is irrational. Assume to the contrary that √3 is rational,
where p and q are integers and q≠0. Morever, let p and q have no common divisor > 1.Then
3=(p/q)² => 3q²=p² (1)
Since 3q² is odd,it follows that q² is odd. Then p is also odd.
This means that there exists k ∈ Z such that
Substituting (2) into (1), we get
3q = (3k)² => 3q²=9k² => q² = 3k².
Since 3k² is even it follows that q² is even. Then q is also even This is a contradiction.
Note. I'm stuck with this solution and unable to understand the meaning of it. Any more simple alternative solution to reason irrational ?