Question

$1+\sqrt{3}$ is a real rational number.

• A. True
• B. False

Answer 1

The correct option is B False

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.

$\therefore$ $\sqrt{3}$ is irrational.
$\Rightarrow$ $1+\sqrt{3}$ is an irrational number.