Question

prove that 1/root 10 is an irrational number​

Answer 1

Answer:

Let us consider 1/√10 be a rational number then, 1/√10 =p/q, where ‘p’ and ‘q’ are integers, $$q \neq 0$$ and p, q have no common factors (except 1).

SO 1/10=p^2/q

q^2=10p^2------(1)

As we know, ‘10’ divides 10p;so ‘10’ divides q^2 as well. Hence, ‘10’ is prime.

So 10divides q

Now, let q=10k, where ‘k’ is an integer

Square on both sides, we get

q^2=10k^2

10p^2=100k^2 [Since, q^2=10p^2, from equation (1)]

p^2=10k^2

As we know, ‘10’ divides 10k ^2;so ‘10’ divides p^2

as well. But ‘10' is prime.

So 10 divides p

Thus, p and q have a common factor of 10. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).

We can say that 1/

√10 is not a rational number.

So it is an irrational number

Hence proved.