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