Prove that √3 is an irrational number.

1.	Prove that √3 is an irrational number.
Answer» Let us assume that √3 is a rational number.Then, as we know a rational number should be in the form of p/q where p and q are co- prime number.So,√3 = p/q { where p and q are co- prime}√3q = pNow, by squaring both the side we get,(√3q)² = p²3q² = p² ........ ( i )So,if 3 is the factor of p²then, 3 is also a factor of p ..... ( ii )=> Let p = 3m { where m is any integer }squaring both sidesp² = (3m)²p² = 9m² putting the value of p² in equation ( i )3q² = p² 3q² = 9m²q² = 3m²So,if 3 is factor of q²q is divisible by 3From this we can say p and q have a common factor 3It contradicts our assumption p and q are co primes Therefore root 3 is an irrational number

Discussion