Question

State true or false.
$\sqrt{2}$ and $\sqrt{3}$ are irrational numbers.

• A. True
• B. False

Answer 1

The correct option is A True

Let us assume that the square root of the prime number $p$ is rational. Hence we can write $\sqrt{p}=\frac{a}{b}$ where a and b are coprime numbers.
Then $p=\frac{a^2}{b^2}$ and so $p{b}^2=a^2$.
Hence, $p$ divides $a^2$,so $p$ divides $a$.
Substitute $a$ by $pk$. Find out that $p$ divides $b$.
Hence, this is a contradiction as they should be relatively prime i.e., H.C.F.(a,b)=1.