Prove that root 5 - root 3 is an irrational number

We can prove this by "Proof by contradiction", and we can find a contradiction through arithmetic basis.









We can prove by contradiction based on the fact that the square root of 15 is irrational. We've made our assumption that we can write √5 - √3 in fraction form. By making √15 the subject, we would have contradicted our original assumption.




Now we've hit our jackpot. Since we can write √15 in a rational form, we've contradicted ourselves. This implies that our original assumption was wrong, which was that we can write √5 - √3 in fraction form. This further implicates that √5 - √3 cannot be rewritten in simplified fraction form, which means that √5 - √3 is irrational.

Thus, our proof is complete.