List of top Mathematics Questions on Irrational Numbers

Given that \(\sqrt{5}\) is an irrational number, prove that \(2 + 3\sqrt{5}\) is an irrational number.
Prove that \(6 - 4\sqrt{5}\) is an irrational number, given that \(\sqrt{5}\) is an irrational number.
Prove that \(6 - 4\sqrt{5}\) is an irrational number, given that \(\sqrt{5}\) is an irrational number.
For real numbers \(x\) and \(y\), we define \(x R y\) iff \(x - y + \sqrt{5}\) is an irrational number. Then, relation \(R\) is:

State whether the following statements are true or false. Justify your answers. 

(i) Every irrational number is a real number. 

(ii) Every point on the number line is of the form √m , where m is a natural number. 

(iii) Every real number is an irrational number

Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Show how √5 can be represented on the number line.