NCERT Class 9 Mathematics Textbook for Blind Students made Screen Readable by Professor T K Bansal.

In your earlier classes, you have learnt about the number line and how to represent various types of numbers on it, (see Fig. 1.1).

Note : For the sake of convenience, through out this chapter, the captions of the figures are given above the figure themselves and the alt text is provided with each and every figure. I hope you will enjoy this chapter, happy learning!

Fig. 1.1 : The number line

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!

Fig. 1.2

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 natural numbers! Recall that we denote this collection by the symbol capital N.

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

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 integers, and this collections of integers is denoted by the symbol capital Z.

Are there some numbers still left on the number line? Of course! There are numbers like, ½, 3/4, − 2005/2006. If you put all such numbers also into your bag, it will now be the collection of rational numbers. The collection of rational numbers is denoted by the capital letter Q. ‘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÷q, where, p and q are integers and q ≠ 0.
(Why do we insist that q ≠ 0?)

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

For example, − 25 can be written as − 25 /1, here p = − 25 and q = 1. Therefore, the rational numbers also include the natural numbers, whole numbers and integers.

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

\[\frac{1}{2}\ =\ \frac{2}{4}\ =\ \frac{10}{20}\ =\ \frac{25}{50}\ =\ \frac{47}{94},\]

and so on. These are equivalent rational numbers (or fractions). However, when we say that p/q is a rational number, or when we represent p/q 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-prime). So, on the number line, among the infinitely many fractions equivalent to ½ , we will choose ½ 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.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.

Solution :

(i) False, because zero is a whole number but not a natural number.

(ii) True, because every integer m can be expressed in the form m/1 , and so it is a rational number.

(iii) False, because 3/5 is not an integer.

Example 2

Find five rational numbers between 1 and 2.

Solution:

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 + s) ÷ 2 lies between r and s. So, 3/2 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 5/4, 11/8, 13/8, and 7/4.

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., 1 = 6/6 and 2 = 12/6. Then you can check that 7/6, 8/6, 9/6, 10/6 and 11/6 are all rational numbers between 1 and 2. So, the five numbers are 7/6, 4/3, 3/2, 5/3 and 11/6.

Remark : Notice that in Example 2, we were asked to find five rational numbers between 1 and 2. But, we 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.

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:
1. What are the numbers that are left on the number line, called?
2. How do we recognise them? That is, how do we distinguish them from the rationals (rational numbers)?

These questions will be answered in the next section.

EXERCISE 1.1

Note : For the sake of convenience the answer to each question is given just after the question itself. The letters Q represents the question and the letter A represents the corresponding answer.

Q1. Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?

A1. Yes. 0 = 0/1 = 0/2 = 0/3 etc., denominator q can also be taken as negative integer.

Q2. Find six rational numbers between 3 and 4.

A2. There can be infinitely many rational numbers between numbers 3 and 4, one way is to take them
3 = 21/(6+1), 4 = 28 / (6 + 1). Then the six numbers are 22/7, 23/7, 24/7/ 26/7, 27/7.

Q3. Find five rational numbers between 3/5 and 4/5.

A3. 5/3 = 30/50, 4/5 = 40/ 50. Therefore, five rationals are : 31/50, 32/50, 33/50, 34/50, 35/50

Q4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.

A4.
(i) True, since the collection of whole numbers contains all the natural numbers.
(ii) False, for example − 2 is not a whole number.
(iii) False, for example 1/2 is a rational number but not a whole number.