# Number System

## Exercise 1.1 Part 1

Question 1: Is zero a rational number? Can you write it in the form of p/q, where p and q are integers and q ≠ 0?

**Answer:** We know that, raional number is a number which can be written in the form of p/q, where q is not equal to zero. Thus,

(a) Yes, zero is a rational number, as it can be written in the form of p/q.

(b) Since, 0 (zero) is an integer thus, zero can be written in the form of p/q. Example : 0/1, 0/2, 0/3, etc. where 0 (zero) is an integer.

Question 2: Find six rational numbers between 3 and 4.

**Answer:**First method to find rational number between two number:

A rational number between two numbers can be found by calculating average of two given numbers.Thus, six rational numbers between given two numbers i.e. 3 and 4 can be calculated as follows:

(1) First rational number between 3 and 4 can be calculated by finding average between them.

`(3+4)/(2)=7/2`

(2) As 7/2 is a rational number between 3 and 4, consequently, rational number between 3 and 7/2 will be one of the rational number between 3 and 4, which can be found by calcuating average between 3 and 7/2

`((7/2)+3)/2=((7+6)/2)/2`

`=(13/2)/2=13/(2xx2)=13/4`

(3) Again as as 7/2 is a rational number between 3 and 4, consequently, rational number between 4 and 7/2 will be the rational number between 3 and 4, which can be found by average between 4 and 7/2

`(4+(7/2))/2=((8+7)/2)/2`

`=(15/2)/(2)=15/(2xx2)=15/4`

(4) Similarly, rational number between 3 and 13/4 will be the rational number between 3 and 4, since 13/4 is a rational number between 3 and 4. Thus next rational number between 3 and 4 can be found by calculating average between 3 and 13/4

`(3+13/4)/2=((12+13)/4)/2`

`=(25/4)/2=25/(4xx2)=25/8`

(5) Similarly, rational number between 4 and 13/4 will be the rational number between 3 and 4, which can be found by calculating average between 4 and 13/4

`(4+13/4)/2=((16+13)/4)/2`

`=(29/4)/2=29/(4xx2)=29/8`

(6) Similarly next rational number between 3 and 4 can be found by calculating average between 3 and 15/4, since 15/4 is a rational number between 3 and 4 which we calculated in the step (3).

`(3+15/4)/2=((12+15)/4)/2`

`=(27/4)/2=27/(4xx2)=27/8`

Thus, six rational numbers between 3 and 4 are 7/2, 13/4, 15/4, 25/8, 29/8 and 27/8.

One can find as many rational number between given any two numbers by calcuating average as above.

Second method to find rational number between two number:

Step – 1: 3 and 4 can also be written as 3/1 and 4/1

Step -2: Since we have to find six rational numbers between 3 and 4, so multiply the numerator and denominator by 6+1 i.e. 7

This will come as follows:

`3=3/1=(3xx7)/(1xx7)=21/7` and

`4=4/1=(4xx7)/(1xx7)=28/7`

Step – 3: Now, we have to find six rational numbers between `21/7` and `28/7`

So, `(22/7),(23/7),(24/7),(25/7),(26/7) and (27/7)` are the six rational numbers between 3 and 4.