# Introduction

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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 .

Now turn and walk all the way back, pick up zero and put it into the bag. You now have the collection of .

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 .

Are there some numbers still left on the line? Of course! There are numbers like .

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

You may recall the definition of rational numbers:

A number ‘r’ is called a rational number, if it can be written in the form **p** and **q** are integers and **q** ≠ 0. (Why do we insist that q ≠ 0?)

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

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Notice that all the numbers now in the bag can be written in the form

You also know that the rational numbers do not have a unique representation in the form

These are equivalent rational numbers (or fractions)._ However,when we say that

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

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

(i) Every whole number is a natural number.

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

**(ii) Every integer is a rational number.**

**Reason: because every integer m can be expressed in the form **

(iii) Every rational number is an integer.

**Reason: because **

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**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

So,

**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.,

Then you can check that

**Remark** : Notice that in Example 2, you were asked to find five rational numbers between 1 and 2. But, you must have realised that in fact there are infinitely many rational numbers between 1 and 2.

**In general, there are infinitely many rational numbers between any two given rational numbers.**

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Let us take a look at the number line again. Have you picked up all the numbers? 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. The amazing thing is that there are infinitely many numbers lying between any two of these gaps too!

So we are left with the following questions:

What are the numbers, that are left on the number line, called?

How do we recognise them? That is, how do we distinguish ? Let's find out!