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 "square root of m", where m is a natural number. (iii) Every real number is an irrational number.

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 $\sqrt{m}$, where $m$ is a natural number.

(iii) Every real number is an irrational number.

(i) True. An irrational number is a real number that cannot be expressed as a simple fraction. This means irrational numbers are considered to be part of the real number system but they cannot be exactly written as a ratio of two integers. The real number system contains both rational numbers (numbers that can be expressed as the quotient of two integers) and irrational numbers.

(ii) False. Not every point on the number line can be represented as $\sqrt{m}$, where $m$ is a natural number. For example, rational numbers and negative numbers cannot be represented in this form. While $\sqrt{m}$, for natural numbers $m$, does represent some points on the number line (specifically, the points corresponding to the square roots of natural numbers), there are many more points on the number line that do not fit this description, including all the rational numbers that are not perfect squares, all negative numbers, and other irrational numbers that are not square roots of natural numbers (like $\pi$ or $e$).

(iii) False. Not every real number is an irrational number. Real numbers include both rational numbers (which can be expressed as the quotient of two integers, including integers themselves as the special case where the denominator is 1) and irrational numbers (which cannot be expressed as the ratio of two integers). Rational numbers, such as 1/2, 2, or -5, are examples of real numbers that are not irrational.