Question

prove that root 7 as irrational number​

Answer 1

Step-by-step explanation:

Let 7 be a whole number and not a perfect square. If possible let the square root of the 7 be a rational number p/q which is in simplest form. This means that 'p' and 'q' have no common factor.

Now, p/q = (root of 7)

Or, p2/q2 = 7 [Squaring both the sides]

Or, p2 = 7q2

Implies, 7 is a factor of p2

Implies, 7 is a factor of p

Let p = 7m for some natural number 'm'.

Then, p = 7m

Or, p2 = 72m2 [Squaring both the sides]

Or, 7q2 = 49m2 [Since, p2 = nq2]

Or, q2 = 7m2

Implies, 7 is a factor of 'q2'

Implies, 7 is a factor of q.

But, 7 is a factor of 'p' and 7 is a factor of 'q'. This means that, 7 is a factor of both 'p' and 'q'. This contradicts the assumption that 'p' and 'q' have no common factor. This means that our supposition is wrong.

Hence, (root of 7) cannot be a rational number.

Answer 2

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