Question

find that √3 is a irrational number​

Answer 1

Answer:

Let us assume on the contrary that √3 is a rational number.

Then, there exist positive integers a and b such that

√3 = a/b where, a and b, are co-prime i.e. their

HCF is 1

Now,

√3 = a/b

⇒3= a²/b²

⇒3b² =a²

⇒3 divides a² [∵3 divides 3b² ]

⇒3 divides a ...(i)

⇒a=3c for some integer c

⇒a² =9c²

⇒3b² =9c² [∵a² =3b² ]

⇒b² =3c²

⇒3 divides b² [∵3 divides 3c² ]

⇒3 divides b...(ii)

From (i) and (ii), we observe that a and b have at least 3 as a common factor. But, this contradicts the fact that a and b are co-prime. This means that our assumption is not correct.

Hence, √3 is an irrational number.

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