Question

Prove that $\dfrac{1}{{\sqrt {11} }}$ is an irrational number.

Answer 1

Hint: In mathematics, the irrational numbers are all the real numbers which are not rational numbers.
That is, irrational numbers cannot be expressed as the ratio of two integers. The symbol Q′ represents the set of irrational numbers and is read as Q prime.

Complete step-by-step answer:
To prove that $\dfrac{1}{{\sqrt {11} }}$ is irrational. Firstly we take $\dfrac{1}{{\sqrt {11} }}$ and then rationalize it. After that, we consider that it is a rational number and proceed with the proof. This proof leads to a contradiction which proves it irrational.
Now $\dfrac{1}{{\sqrt {11} }}$ = $\dfrac{1}{{\sqrt {11} }} \times \dfrac{{\sqrt {11} }}{{\sqrt {11} }} = \dfrac{{\sqrt {11} }}{{11}}$
If possible let $\dfrac{{\sqrt {11} }}{{11}}$ be a rational number. So there exists p,q which are integers such that
$\dfrac{{\sqrt {11} }}{{11}} = \dfrac{p}{q}$ $q \ne 0$ and p and q are co-prime.
  $
   \Rightarrow \sqrt {11} = \dfrac{p}{q} \times 11 \\
  \sqrt {11} = \dfrac{{11p}}{q} \\
$
 Now 11p and q are integers
$ \Rightarrow \dfrac{{11p}}{q}$ is a rational number.
$ \Rightarrow \sqrt {11} $ is rational. Which is not true a $\sqrt {11} $ irrational number.
Claim:- $\sqrt {11} $ is an irrational number.
If possible let $\sqrt {11} $is a rational number. Then there exists a and b which are integers such that
$\sqrt {11} = \dfrac{a}{b}$ $b \ne 0$, a and b are co-prime
$\Rightarrow$ $11 = \dfrac{{{a^2}}}{{{b^2}}}$ (squaring both sides)
$ \Rightarrow {a^2} = 11{b^2}............(i)$
$\Rightarrow$ 11 divides ${a^2}$
$ \Rightarrow 11$ divides $a$
Let a = 11c
Put a in (i)
${(11c)^2} = 11{b^2}$
$121{c^2} = 11{b^2}$
$11{c^2} = {b^2}$
$ \Rightarrow {b^2} = 11{c^2}$
$\Rightarrow$ 11 divides ${b^2}$
$\Rightarrow$ 11 divides b
SO we see that 11 divides both a and b. which is a contradiction because a and b are co-prime
So our supposition is wrong.
∴ $\sqrt {11} $ is an irrational number. This proves claims.
So in the main proof our supposition is wrong.
∴ $\dfrac{{\sqrt {11} }}{{11}}$ is an irrational number.
$\Rightarrow$ $\dfrac{1}{{\sqrt {11} }}$ is irrational number

Note: The product and quotient of a rational and irrational number are irrational. We cannot express irrational numbers as a ratio of two integers. When an irrational number is written with a decimal point, the numbers after the decimal point continue infinitely with no repeatable pattern.
For example 1.567183906 (not repeating) - irrational number
When irrational numbers are encountered by a computer program, they must be estimated.