Question

EN
Prove that V6 is irrational.​

Answer 1

Answer:

Suppose we consider, √6 is a rational number .

Then we can express it in the form of a/b

∴√6 = a/b, where a and b are positive integer and they are co-prime, i.e.HCF(a,b) = 1

∴√6 = a/b

=>b√6 = a

=>(b√6)² = a² [squaring both sides]

=>6b² = a²………..(1)

Here, a² is divided by 6

∴ a is also divided by 6. [we know that if p divides a², then p divides a]

∴ 6|a

=>a = 6c [c∈ℤ]

=>a² = (6c)²

=>6b² = 36c² [from (1)]

=>b² = 6c²

Here, b² is divided by 6,

∴ b is also divided by 6.

∴ 6|a and 6|b

we observe that a and b have at least 6 as a common factor. But this contradicts that “a and b are co-prime.”

It means that our consideration of “√6 is a rational number” is not true.

Hence, √6 is an irrational number.