Hello there!

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.

Hope it helps.

goodluck!

Hello,

For the given question, there are two ways in which you can proceed to prove:

1. 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.
2. 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.

Hope it helps.