Hii,

Assume that √5 makes sense. After that, we can write it as a/b, where b ≠ 0 and a and b are whole numbers without a common factor.

Thus, √5 = a/b
Both sides are squared:
5 = a² / b² ⇒ a² = 5b²

This indicates that since a² is divisible by 5, so is a.
Assume that a = 5k.

Consequently, a² = 25k² ⇒ 25k² = 5b² ⇒ b² = 5k²
Consequently, b is likewise divisible by 5.

This indicates that 5 is a factor that both a and b share, which is contradictory.

√5 is therefore irrational.