Question

prove that 1/√11 is an irrational number .​

Answer 1

Answer:

So that 1/√11 can be written as p/q, where p, q are coprime integers and q ≠ 0. Here it creates a contradiction that, the LHS p/q is rational while the RHS √11 is irrational. ... Hence our earlier assumption is contradicted and reached the conclusion that √11 is irrational.

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Answer 2

Let us consider 1/

11

be a rational number, then

1/

11

=p/q, where ‘p’ and ‘q’ are integers, $$q \neq 0$$ and p, q have no common factors (except 1).

So,

1/11=p

2

/q

2

q

2

=11p

2

…. (1)

As we know, ‘11’ divides 11p

2

, so ‘11’ divides q

2

as well. Hence, ‘11’ is prime.

So 11 divides q

Now, let q=11k, where ‘k’ is an integer

Square on both sides, we get

q

2

=121k

2

11p

2

=121k

2

[Since, q

2

=11p

2

, from equation (1)]

p

2

=11k

2

As we know, ‘11’ divides 11k

2

, so ‘11’ divides p

2

as well. But ‘11’ is prime.

So 11 divides p

Thus, p and q have a common factor of 11. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).

We can say that 1/

11

is

not a rational number.

1/ 11 is an irrational number.

Hence proved.

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