**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.

Asked by Vishal kumar | 1 year ago | 248

**(i) Consider the irrational numbers and the real numbers separately.**

- The irrational numbers are the numbers that cannot be converted in the form \(\frac{p}{q}\), where p and q are integers and q \(\ne\) 0. (Eg: \( \sqrt{2},3\pi,..011011011...)\)
- The real number is the collection of rational numbers and irrational numbers.
- Therefore, we conclude that, every irrational number is a real number.

**(ii) Consider a number line. on a number line, we can represent negative as well as positive numbers.**

- Positive numbers are represented in the form of \( \sqrt 1, \sqrt{1.1},\sqrt{1.2}...\)
- But we cannot get a negative number after taking square root of any number. (Eg: \( \sqrt{-5}=5i\) is a complex number (which you will study in higher classes))

Therefore, we conclude that every number point on the number line is not of the form \( \sqrt m,\) where m is a natural number.

**(iii) Consider the irrational numbers and the real numbers separately.**

- Irrational numbers are the numbers that cannot be converted in the form \(\frac{p}{q},\) where p and q are integers and q \( \ne 0\).
- A real number is the collection of rational numbers (Eg: \( \frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},......)\) and irrational numbers (Eg: \( \sqrt 2, 3\pi,.011011011...)\)

So, we can conclude that every irrational number is a real number. But every real number is not an irrational number.

Therefore, every real number is not an irrational number.

Answered by Shivani Kumari | 1 year agoVisualise the representation of \( 5.3\overline{7}\) on the number line upto 5 decimal places, that is upto 5.37777.

Visualise 2.665 on the number line, using successive magnification.

Find whether the following statements are true or false:

**(i) **Every real number is either rational or irrational.

**(ii)** π is an irrational number.

**(iii)** Irrational numbers cannot be represented by points on the number line.