Question

26.
Prove that V3 is irrational.​

Answer 1

Explanation:

Assume that √3 is not irrational that is √3 is rational number in the form of a/b where p &q are co-prime.

Then, √3=a/b

3=a2/b2

3b2=a2 (1)

➡a2 is divisible by 3

so a is also divisible by 3

Let a=5c

Substituting a=3c in(1)

3b2=(3a)2=25a2

b2=3a2

➡b2 is divisible by 3

so,b is also divisible by 3

since a&b are not co-prime.

Therefore,it is our contradictory assumption that √3 is a rational number.

Hence √3 is irrational number

Answer 2

let √3 be rational no

√3=p/q   (where p and q are co primes)

squaring both sides

(√3)²=(p/q)²

3=p²/q²

p²=3q²

3 is a factor of p² .so 3 is a factor of p

3q²=(3r)²

q²=3r²

3 is a factor of q so 3 is a factor of q

so 3 is a common factor of both p and q. but p and q are co primes.so both the statement contradicts. so √3 is irrational