Question

Root 3 is irrational.. True or False​

Answer 1

True, √3 is a irrational number.

Answer 2

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Yes, it is true √3 is irrational.

${\bold{\pink{\underline{\red{Exp}\purple{lana}\green{tion}\orange{:-}}}}}$

Let's prove √3 is irrational step by step explanation,

At first,

Let √3 be rational and let it's simple form be a/b

Then, a and b are integers having no common factor other than 1, and b ≠ 0.

Now,

√3 = a/b

Squaring both side

(√3)² = (a/b)²

$\bf\implies\:$3 = a²/b²

$\bf\implies\:$3b² = a²...............(i)

$\bf\implies\:$3 divides a² [∴ 3 divides 3b²]

$\bf\implies\:$3 divides a

(∴ 3 is prime and 3 divides a² $\bf\implies\:$ 3 divides a )

let a = 3c for some integer c

putting a = 3c in eq (i) ,we get

3b² = 9c² $\bf\implies\:$b² = 3c²

$\bf\implies\:$ 3 divides b² [∴3 divides 3c²]

$\bf\implies\:$ 3 divides b

(∴ 3 is prime and 3 divides b²$\bf\implies\:$ 3 divides b)

Thus,

3 is a common factor of a and b

but this contradiction the fact that a and b have no factor other than 1

This contradiction arises by assuming √3 is rational.

Hence, √3 is irrational.