prove that root five is irrational
Answers
Given: √5
We need to prove that √5 is irrational
Proof:
Let us assume that √5 is a rational number.
Sp it t 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
√5 is an irrational number
Ur answer :-
Let us assume that √5 be a rational number.
then we can write it in the form of p/q where q≠0
Also p and q are co-prime numbers.
√5 = p/q
squaring on both sides we get,
5 = p²/q²
5q² = p²
here p is divisible by 5
→ p = 5k where k is positive integer
Again squaring on both sides
p² = 25k²
substituting 5q² = p²
→ 5q² = 25k²
→ q² = 5k²
q is divisible by 5
From this we can say that p and q have a common factor 5
→ It contradicts to our assumption p and q are co-prime.
→√5 is an irrational number.
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