An important and exam oriented question from real numbers chapter in which we have given a number √2/3 and we have to show that the given number is an irrational number

Kindly give me a detailed solution of this question

RS Aggarwal, Class 10, chapter 1E, question no 21

## Let us assume that 3 is a rational number which can be expressed in the form of p/q, where p and q are integers, q≠0 and p and q are co prime that is HCF(p,q)=1.

We have,

√3=p/q

⇒√3q=p......(1)⇒3q²=p²(squaring both sides)

⇒p² is divisible by 3

⇒p is divisible by 3......(2)

Therefore, for an integer r,

p=3r

⇒√3q=3r (from(1))

⇒3q²=9r² (squaring both sides)

⇒q²=9/3r²

⇒q²=3r²

⇒q² is divisible by 3

⇒q is divisible by 3......(3)

From equations 2 and 3, we get that 3 is the common factor of p and q which contradicts that p and q are co prime. This means that our assumption was wrong.

Thus √3 is an irrational number.

Now, since multiplication of a rational number with an irrational number is an irrational number.

Hence √2/3 is an irrational number.