# Number System

## Exercise 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.

**Answer:**

(i) True

Reason: Real numbers are any number which can we think about. Thus, every irrational number is a real number.

(ii) False

Reason: A number line may have negative or positive number. Since, no negative can be the square root of a natural number, thus every point on the number line cannot be in the form of √m, where m is a natural number.

(iii) False

Reason: All numbers are real number and non terminating numbers are irrational number. For example 2, 3, 4, etc. are some example of real numbers and these are not irrational.

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.

**Answer:** No. Square roots of all positive integers are not irrational. Example 4, 9, 16, etc. are positive integers and their square roots are 2, 9 and 4 which are rational numbers.

Question 3: Show how √5 can be represented on the number line.

**Answer:**

Steps to show √5 on a number line.

**Step: 1** – Draw a number line mm’**Step: 2** – Take OA equal to one inch, i.e. one unit.

**Step: 3** – Draw a perpendicular AB equal to one inch (1 inch) on point A.**Step: 4** – Join OB. This OB will be equal to √2**Step: 5** – Draw a line BC perpendicular to OB on point B equal to OA i.e. one inch.**Step: 6** – Join OC. This OC will be equal to √3**Step: 7** – Draw a line CD equal to OA and perpendicular to OC.

**Step: 8** – Join OD. This will be √4 i.e. equal to 2.**Step: 9** – Draw ED equal to 1 inch and perpendicular to OD.

**Step: 10** – Join OE. This will be equal to √5

**Step: 11** – Cut a line segment OF equal to OE on number line. This line segment OF will be equal to √5

See how, OE is equal to √5

Here, OD = 2, DE = 1 and angle ODE = 90º

Thus, according to Pythagoras theorem.

`OE=sqrt(OD^2+DE^2)`

Or, `OE=sqrt(2^2+1^2)`

Or, `OE=sqrt(4+1)`

Or, `OE=sqrt5`