Prove that √5 is irrational
Sol. Let us assume to the contrary that √5 is an irrational number. So it can be written in the form of p/q where p and q are co-prime integers and q is not equal to zero.
√5= p/q
squaring both sides
(√5)² = (p/q)²
5= p²/q²
p²= 5 q² ,
so p² is divisible by 5 hence p is also divisible by 5. So put p = 5 m in above equation where m is some positive integer.
(5 m)² = 5q²
25 m² = 5q²
q²= 5 m² , so q² is divisible by 5 hence q is also divisible by 5. It means p and q both are divisible by 5. Hence 5 is common factor of p and q. But it contradicts that p and q are co-prime. Therefore our assumption is wrong. Hence √5 is irrational.
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