Introduction

In your earlier classes, you have learnt about the number line and how to represent various types of numbers on it.

Just imagine you start from zero and go on walking along this number line in the positive direction. As far as your eyes can see, there are numbers, numbers and numbers.

Now suppose you start walking along the number line, and collecting some of the numbers. Get a bag ready to store them!

You might begin with picking up only natural numbers like 1, 2, 3, and so on. You know that this list goes on for ever.

(Why is this true?) So, now your bag contains infinitely many naturalwhole numbers!

Recall that we denote this collection by the symbol .

Now turn and walk all the way back, pick up zero and put it into the bag. You now have the collection of wholenatural numbers which is denoted by the symbol .

Now, stretching in front of you are many, many negative integers. Put all the negative integers into your bag. What is your new collection? Recall that it is the collection of all integersfractions, and it is denoted by the symbol ZQ.

Are there some numbers still left on the line? YesNo

Of course! There are numbers like 13,34, or even −20052006. If you put all such numbers also into the bag, it will now be the collection of rationaldecimals numbers.

The collection of rational numbers is denoted by QR.

‘Rational’ comes from the word ‘ratio’, and Q comes from the word ‘quotient’.

You may recall the definition of rational numbers:

The condition that q ≠ 0 is essential because division by zero is undefined in mathematics. If q were equal to zero, the expression pq would not represent a valid number.

In simpler terms, when q = 0, the fraction pq would be meaningless, leading to an undefined or indeterminate result. Therefore, to ensure that the expression represents a rational number, it is crucial that q is not zero.

Let's identify the different numbers and drag the numbers in below bucket components.

Instruction

Notice that all the numbers now in the bag can be written in the form pq, where p and q are integers and q ≠ 0.

For example:−25 can be written as −251 where here p = and q = .

Therefore, the rational numbers also includeexclude the natural numbers, whole numbers and integers.

You also know that the rational numbers do not have a unique representation in the form pq, where p and q are integers and q ≠ 0.

For example:

12=24=1020=2550= 4794, and so on.

These are equivalent rational numbers (or fractions).

However,when we say that pq is a rational number, or when we represent pq on the number line, we assume that q ≠ 0 and that p and q have no common factors other than 1 that is, p and q are co-primeprime.

So, on the number line, among the infinitely many fractions equivalent to 12, we will choose 12 to represent all of them.

Now, let us solve some examples about the different types of numbers, which you have studied in earlier classes.

Example 1: Are the following statements true or false? Give reasons for your answers.

(i) Every whole number is a natural number. FalseTrue

Reason: Because zero is a whole number but not a natural number.

(ii) Every integer is a rational number. TrueFalse

Reason: Because every integer m can be expressed in the form m1, and so it is a rational number.

(iii) Every rational number is an integer. FalseTrue

Reason: Because say for eg. 35 isis not a rational number but it is notis an integer.

Example 2: Find five rational numbers between 1 and 2.

We can approach this problem in at least two ways.

Solution 1 : Recall that to find a rational number between r and s, you can add r and s and divide the sum by 2, that is r+s2 lies between r and s.

So, 3252 is a number between 1 and 2. You can proceed in this manner to find four more rational numbers between 1 and 2.

These four numbers are: 5445, 118811, 138813 and 7447

Solution 2 : The other option is to find all the five rational numbers in one step. Since we want five numbers, we write 1 and 2 as rational numbers with denominator 5+1 i.e. such as 1=66 and 2=126.

Then you can check that 7667, 8668, 9669, 106610 and 116611 are all rational numbers between 1 and 2.

So, the five numbers are 76,43,32,53 and 116

Let us take a look at the number line again. Have you picked up all the numbers? NoYes

Not, yet. The fact is that there are infinitely many more numbers left on the number line! There are gaps in between the places of the numbers you picked up, and not just one or two but infinitely many.