Proof by contrapositive Prove that if the product $ab$ is irrational, then either $a$ or $b$ (or both) must be irrational.

How do I prove this by contrapositive? What is contrapositive?
The statement you want to prove is:


  If $ab$ is irrational, then $a$ is irrational or $b$ is irrational.


The contrapositive is:


  If not($a$ is irrational or $b$ is irrational), then not($ab$ is irrational).


A more natural way to state this (using DeMorgan's Law) is:


  If both $a$ and $b$ are rational, then $ab$ is rational.


This last statement is indeed true. Since the truth of a statement and the truth of its contrapositive always agree, one can conclude the original statement is true, as well.
If $a,b \in \Bbb{Z}^+$ and $a\neq b$, then $ax^2+bx+(b-a) = 0$ has no positive integer root (solution).