Question

prove that root 3 is irrational​

Answer 1

Answer:

yes it is irrational number

Answer 2

To Prove :–

• √3 is an irrational.

Proof :–

Suppose √3 be a rational number.

So, let $\dfrac{a}{b}$ = √3 (a & b have no common factor other than 1)

i.e. a & b are co-primes.

Now, squaring both sides

⟹ $\dfrac{{a}^{2}}{{b}^{2}}$ = 3 –––––– (1)

⟹ $\dfrac{{a}^{2}}{3} = {b}^{2}$

• 3 divides ${a}^{2}$

• 3 divides a –––––– (2)

Again,

⟹ a² = 3b² [from equation (1)]

Put, a = 3c [c is another integer]

⟹ (3c)² = 3b²

⟹ 9c² = 3b²

⟹bc² = $\dfrac{{\cancel{3}b}^{2}}{\cancel9}$

⟹ c² = $\dfrac{{b}^{2}}{3}$

• 3 divides b²

• 3 divides b –––––– (3)

From (2) and (3),

3 divides a & b.

Which contradicts our assumption therefore, √3 is not a rational number.

Hence,

√3 is an irrational.