Question

Without using division method show that `sqrt(7)` is an irrational numbers.

Answer 1

Let `sqrt(7)` be a raised number.
∴ `sqrt(7) = "a"/"b"`
⇒ 7 = `"a"^2/"b"^2`
⇒ a² = 7b²
Since a² is divisible by 7, a is also divisible by 7.       ....(I)
Let a = 7c
⇒ a² = 49c²
⇒ 7b² = 49c²
⇒ b² = 7c²
Since b² is divisible by 7, b is also divisible by 7.   ....(II)
From (I) and (II), we get a and b both divisible by 7.
i.e., a and b have a common factor 7.
This contradicts our assumption that `"a"/"b"` is rational.
i.e. a and b do not have any common factor other than unity (1).
⇒ `"a"/"b"` is not rational
⇒ `sqrt(7)` is not rational, i.e.`sqrt(7)` is irrational.