Question

prove that √5 is irrational​

Answer 1

Answer:

You can start by finding out the square root of 5 by division method and keep calculating for up-to 6 or more places of decimal (Eg: 2.23606798). Then at the end you need to mention: Since the given answer upto 8 places of decimal ( as shown in brackets) is non-recurring and non-repetitive, therefore it must be an irrational number. The second approach would be to prove it by contradictory method. For this you need to assume that root 5 is rational at first.

So it can be expressed in the form p/q where p,q are co-prime integers and q≠0.

⇒√5=p/q On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number.

This contradicts our given assumption. So, root 5 is an irrational number