Prove that the square root of an irrational number is an irrational number:
In this case you want to show that if a number is irrational, then the number is also irrational—you want to show an implication. The implication you want to show is that , where “ is irrational” and “ is irrational”. The contrapositive is , or in other words
is not irrational is not irrational
Since “not irrational” is the same as “can be written as a fraction”, you can begin with the implication “ can be written as a fraction”, and then try to show that can be written as a fraction. If you can do this, then you are done with the proof.
Write your implication as the equation . You want to show that is a fraction, so square both sides. Then you get on one side, and a fraction on the other side, so you have shown that is not irrational. This is the same as the contrapositive . You have shown that
Mathematically you can write the reasoning like this: Assume
Take the square root on both sides and get
It shows that can be written as a fraction, such that is not an irrational number. You have shown that
which is the contrapositive version of what you wanted to show, which was that if is an irrational number, must be an irrational number.