Question

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

Answer 1

(i) Every irrational number is a real number.

True. By definition, irrational numbers are numbers that cannot be expressed as the ratio of two integers, and they are part of the real number system. Thus, every irrational number is a real number.

(ii) Every point on the number line is of the form \( \sqrt{m} \), where \( m \) is a natural number.

False. Not every point on the number line is of the form \( \sqrt{m} \), where \( m \) is a natural number. While the square roots of natural numbers like \( 1, 4, 9, 16, \dots \) represent points on the number line, there are many other points that are not square roots of natural numbers, such as rational numbers like \( 1/2 \), or irrational numbers like \( \pi \) or \( e \).

(iii) Every real number is an irrational number.

False. Real numbers include both rational and irrational numbers. Rational numbers can be expressed as the ratio of two integers (e.g., \( \frac{1}{2}, 3, -4 \)) and are not irrational. Hence, not every real number is irrational.