Use division method to show that √3 and √5 are irrational numbers.
Suppose for the sake of contradiction that √3 is rational
We know that rational numbers are those numbers which can be expressed in pq form ,where p and q are integers and q≠0
⟹√3=pq
Squaring on both sides
3=p2q2
⟹p2=3q2
∵p2 is a multiple of 3⟹p must be a multiple of 3
let p=3n⟹p2=9n2⟹q2=3n2
This means q is also a multiple of 3,which contradicts the fact that p and q had no common factor
Hence √3 is an irrational number
Suppose for the sake of contradiction that √5 is rationalWe know that rational numbers are those numbers which can be expressed in pq form ,where p and q are integers and q≠0
⟹√5=pq
Squaring on both sides
5=p2q2
⟹p2=5q2
∵p2 is a multiple of 5⟹p must be a multiple of 5
let p=5n⟹p2=25n2⟹q2=5n2
This means q is also a multiple of 5,which contradicts the fact that p and q had no common factor
Hence √5 is an irrational number