## Number Systems

## Chapter 1

## Exercise 1.2

**EX 1.2 QUESTION 1.**

**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.**

**Solution:**

(i) True, because of the collection of all rational numbers and all irrational numbers is real numbers.

(ii) False, because negative numbers cannot be the square root.

(iii) False, because every real number cannot be an irrational number.

**EX 1.2 QUESTION 2.**

**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.**

**Solution:**

The square roots of all positive integers are not irrational.

For example,

√4 = 2 is rational.

√9 = 3 is rational.

**EX 1.2 QUESTION 3.**

**Show how √5 can be represented on the number line.**

**Solution:**

To represent the √5 on the number line,

take AB = 2 units. make a perpendicular BC at b such that BC = 1 unit.

Applying Pythagoras theorem,

AB^{2}+BC^{2} = CA^{2}

2^{2}+1^{2} = CA^{2} = 5

⇒ CA = √5

now taking A as a center and AC is the radius , mark an arc on AB, which intersect at √5.