Question

show the following are irrational number i) 2- root 3​

Answer 1

Given:

A number 2-√3

To Show:

It is irrational

Proof:

On the contarary let us assume that 2-√3 is rational.

So,it can be expressed in the form of p/q ,where p and q are integers and q ≠ 0 . Also,HCF of p and q is 1.

=> $\sf{\dfrac{p}{q}=2-\sqrt{3}}$

Squaring both sides.

=>$\sf{\dfrac{p^{2}}{q^{2}}=(2-\sqrt{3})^{2}}$

=>$\sf{\dfrac{p^{2}}{q^{2}}=4+3-4\sqrt{3}}$

using

• (a-b)²=a²+b²-2ab.

=>$\sf{\dfrac{p^{2}}{q^{2}}=7-4\sqrt{3}}$

=>$\sf{7-\dfrac{p^{2}}{q^{2}}=4\sqrt{3}}$

=>$\sf{\dfrac{7q^{2}-p^{2}}{4q^{2}}=\sqrt{3}}$

But √3 is irrational and 7q²-p²/4q² is rational .

Thus we arrived at a contradiction .So,our assumption was wrong .

.°. 2-√3 is a irrational number.