Question

Given that √2 is irrational, prove that (11-5√2 ) is an irrational number.

Answer 1

ANSWER:

• (11-5√2) is an Irrational number.

GIVEN:

• √2 is an irrational number.

TO PROVE:

• 11-5√2 is an irrational number.

SOLUTION:

Let (11-5√2) be a rational number which can be expressed in the form of p/q where p and q have no other common factor than 1.

$\implies \: 11 - 5 \sqrt{2} = \dfrac{p}{q} \\ \implies \: 11 - \dfrac{p}{q} = 5 \sqrt{2} \\ \implies \: \dfrac{11q - p}{q} = 5 \sqrt{2} \\ \implies \: \frac{11q - p}{5q} = \sqrt{2}$

Here :

• (11q-p)/5q is rational but √2 is an Irrational number.
• Thus our contradiction is wrong.
• So (11-5√2) is an Irrational number.

NOTE:

• This method of proving an Irrational number is called contradiction method.
• in this method we first contradict a fact then we prove that our supposition was wrong.