Question

4 Prove that 1/√11
is an irrational number.​

Answer 1

Step-by-step explanation:

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.

Answer 2

Answer:

Is not an rational number