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.