Question

Prove that 2√3÷5 is irrational​

Answer 1

Answer:

To prove:

$\frac{2\sqrt{3}}{5}$ is irrational number.

Proof:

Let us first assume $\frac{2\sqrt{3}}{5}$ is a rational number.

So, it can be written as:

$\frac{2\sqrt{3}}{5}=\frac{a}{b}$ where $b\neq0$

Multiplying both sides by 5.

$5\times \frac{2\sqrt{3}}{5}=5\times \frac{a}{b}$

$2\sqrt{3}=\frac{5a}{b}$

Dividing both sides by 2.

$\frac{2\sqrt{3}}{2}=\frac{5a}{2b}$

$\sqrt{3}=\frac{5a}{2b}$

We see that $\frac{5a}{2b}$ is a rational number as it is in the $\frac{p}{q}$ form but we know $\sqrt3$ is irrational which makes the equation false making it contradicting with our assumption.

Thus our assumption is wrong.

Hence $\frac{2\sqrt{3}}{5}$ is an irrational number.