Question

Prove 1/√11 is an irrational number. Please provide proper steps ​

Answer 1

Step-by-step explanation:

So that 1/√11 can be written as p/q, where p, q are coprime integers and q ≠ 0.

Thus,

\displaystyle \frac{p}{q}=\frac{1}{\sqrt{11}}

q

p

=

11

1

Taking the reciprocals...

\begin{gathered}\displaystyle \frac{1}{\frac{p}{q}}=\frac{1}{\frac{1}{\sqrt{11}}} \\ \\ \\ \\ \Rightarrow \ \frac{q}{p}=\sqrt{11}\end{gathered}

q

p

1

=

11

1

1

⇒

p

q

=

11

Here it creates a contradiction that, the LHS p/q is rational while the RHS √11 is irrational. Here it seems that √11 can be written in fractional form.

Hence our earlier assumption is contradicted and reached the conclusion that √11 is irrational.

Answer 2

Answer:

Thus, 11 is a common factor of a and b both. But it contradicts the fact that a and b have no common factor other than 1. So, our supposition is wrong. Hence, √11 is irrational.

Step-by-step explanation:

If √11 is irrational, how can you prove it?

By the method of contradiction..

Let √11 be rational , then there should exist √11=p/q ,where p & q are coprime and q≠0(by the definition of rational number). So,

√11= p/q

On squaring both side, we get,

11= p²/q² or,

11q² = p². …………….eqñ (i)

Since , 11q² = p² so ,11 divides p² & 11 divides p

Let 11 divides p for some integer c ,

so ,

p= 11c

On putting this value in eqñ(i) we get,

11q²= 121p²

or, q²= 11p²

So, 11 divides q² for p²

Therefore 11 divides q.

So we get 11 as a common factor of p & q but we assumpt that p & q are coprime so it contradicts our statement. Our supposition is wrong and √11 is irrational.

Hope it will help you ⚘☘⚘☘⚘