Question

. Prove that 3 is an irrational number.

Answer 1

Step-by-step explanation:

Let us assume to the contrary , that √3 is an rational number.

Also, let a and b are co - prime no.

√3=a/b

b√3 = a ------------ (1)

Squaring both sides we get :

3b² = a²

b²= a²/3

3 divides a² and also a .

Let a = 3c ( for some integer c )

Substituting the value of a from eq (1)

a= 3c

b√3 = 3c

Squaring both sides we get:

3b² = 9c ²

b²= 9c²/ 3

b² = 3c²

b²/3= c²

3 divides b²and also b.

But we assumed that a and b are co - prime .

We assumed wrong.

Hence, √3 is an irrational number.

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