Question

Prove root 3 is irrational number


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Answer 1

Answer:

Steps to prove:

Let √3 be rational number

√3=p/q where p and q are integers where q!=0 and p and q have no common factors (except 1)

Squaring both the sides

3=p^2/q^2

p^2=3q^2 (1)

As 3 divides 3q^2 so 3 divides p^2 but 3 is prime.

3 divides p

Let p=3k where k is an integer

Substituting the value of p in(1)

(3k)^2=3q^2

9k^2=3q^2

q^2=3k^2

As 3 divides 3k^2 so 3 divides q^2 but 3 is prime.

3 divides q

Thus p and q have a common factor 3. But this contradicts the fact that p and q have no common factors (except 1)

Hence, √3 is not a rational number.It is an irrational number.