prove that √3 is an irrational number it’s very urgent don’t spam please​

2 thoughts on “prove that √3 is an irrational number it’s very urgent don’t spam please​”

1. Answer:

   √3 cannot be express in p/q forms except with 1

   Step-by-step explanation:

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

   Then, there exist positive integers a and b such that

   3=ba where, a and b, are co-prime i.e. their HCF is 1

   Now,

   3=ba

   ⇒3=b2a2

   ⇒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.

   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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