Question

show that 5 - 2√3 is an irrational number?​

Answer 1

ANSWER:

5-2√3 is an irrational number.

TO PROVE:

• 5-2√3 is an IRRATIONAL number.

SOLUTION:

Let 5-2√3 is a rational number which can be expressed in the form of p/q where p and q have no common factor other than one.

$\implies \: 5 - 2 \sqrt{3} = \frac{p}{q} \\ \implies \: 5 - \frac{p}{q} = 2 \sqrt{3} \\ \implies \: \frac{5q - p}{q} = 2 \sqrt{3} \\ \implies \: \frac{5q - p}{2q} = \sqrt{3} \\ \implies \: \sqrt{3} = \frac{5q - p}{2q}$

This shows that √3 is a Rational number But we know that √3 is an Irrational number. Thus our supposition was wrong so 5-2√3 is an Irrational number.

NOTE:

• The root of all prime number is an Irrational number.
• The above method of proving irrational number is called contradiction method.
• In contradiction method we first suppose the Wrong but at last we prove that our supposition is wrong.