Prove that √5 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
Hence proved
Let √5 be a rational number
Therefore, √5= p/q [ p and q are in their least terms i.e., HCF of (p,q)=1 and q ≠ 0
On squaring both sides, we get
p²= 5q² ...(1)
Clearly, 5 is a factor of 5q²
⇒ 5 is a factor of p² [since, 5q²=p²]
⇒ 5 is a factor of p
Let p =5 m for all m ( where m is a positive integer)
Squaring both sides, we get
p²= 25 m² ...(2)
From (1) and (2), we get
5q² = 25m² ⇒ q²= 5m²
Clearly, 5 is a factor of 5m²
⇒ 5 is a factor of q² [since, q² = 5m²]
⇒ 5 is a factor of q
Thus, we see that both p and q have common factor 5 which is a contradiction that H.C.F. of (p,q)= 1
Therefore, Our supposition is wrong
Hence √5 is not a rational number i.e., irrational number.