Prove that √5 is irrational
To prove that √5 is irrational, we need to show that it cannot be expressed as the ratio of two integers.Let us assume contrary to the statement that √5 is rational. Then we can write:√5 = a/b
where a and b are co-prime integers.
Squaring both sides,
we get:
5 = a² / b²
Multiplying both sides by b²
we get:
5b² = a²
Since a² is divisible by 5, a must be a divisible by 5.
Let a = 5k, where k is a positive integer.
Fill this value into the last equation,
5b² = (5k)²
Simplifying, we get:
b² = 5k²
This means that b² is also divisible by 5, which implies that b must also divisible with 5.
It means both a and b are divisible by 5 but a and b are coprime.
Now there is a contradiction that a and b have no common factors. So, our assumption that √5 is rational is false. Hence √5 is irrational.