Prove that 5 is an irrational number. Class 9

Prove that 5 is an irrational number. Class 9

Question 1.

Prove that $\sqrt{5}$ is an irrational number. Class 9

Solution:

Let us assume that $\sqrt{5}$ is a rational number.

$\sqrt{5}$ = $\frac{p}{q}$,where p and q are co-prime numbers and q ≠ 0.

Squaring both sides, we get

5 = $\frac{p^2}{q^2}$

Multiplying both sides by q², we get

5q²= p² ...(i)

Since p² = 5q², p² is divisible by 5.

∴ p is also divisible by 5.

Hence, let p = 5k,

where k is an integer.

Substituting p = 5k in equation (i), we get

5q² = (5k)²

∴ 5q² = 25k²

Dividing both sides by 5, we get

q² = 5k², which is divisible by 5

∴ q is also divisible by 5.

We have shown that both p and q are divisible by 5.

Therefore, both have a common factor 5.

But this contradicts our assumption that p and q have no common factor other than 1.

Hence, our assumption that $\sqrt{5}$ is a rational number is wrong.

∴ $\sqrt{5}$ is an irrational number.