## NCERT Class VI Mathematics Chapter 6 Integers

National Council of Educational Research and Training (NCERT) Book for Class VI

Subject: Mathematics

Chapter: Chapter 6 – Integers

Class VI NCERT Mathematics Text Book Chapter 6 Integers is given below.

**6.1 Introduction**

Sunita’s mother has 8 bananas. Sunita has to go for a picnic with her.

friends. She wants to carry 10 bananas with herCan her mother give 10 bananas to her? She does not have enough, so she borrows 2 bananas from her neighbour to be returned later. After giving 10 bananas to Sunita, how many bananas are left with her mother? Can we say that she has zero bananas? She has no bananas with her, but has to return two to her neighbour. So when she gets some more bananas, say 6, she will return 2 and be left with 4 only.

Ronald goes to the market to purchase a pen. He has only Rs 12 with him but the pen costs Rs 15. The shopkeeper writes Rs 3 as due amount from him. He writes Rs 3 in his diary to remember Ronald’s debit. But how would he remember whether Rs 3 has to be given or has to be taken from Ronald? Can

he express this debit by some colour or sign? Ruchika and Salma are playing a game using a number strip which is marked from 0 to 25 at equal intervals.

To begin with, both of them placed a coloured token at the zero mark. Two coloured dice are placed in a bag and are taken out by them one by one. If the die is red in colour, the token is moved forward as per the number shown on throwing this die. If it is blue, the token is moved backward as per the number

shown when this die is thrown. The dice are put back into the bag after each move so that both of them have equal chance of getting either die. The one who reaches the 25th mark first is the winner. They play the game. Ruchika gets the red die and gets four on the die after throwing it. She, thus, moves the token to mark four on the strip. Salma also happens to take out the red die and wins 3 points and, thus, moves her token to number 3.

In the second attempt, Ruchika secures three points with the red die and Salma gets 4 points but with the blue die. Where do you think both of them should place their token after the second attempt?

Ruchika moves forward and reaches 4 + 3 i.e. the 7th mark.

Whereas Salma placed her token at zero position. But Ruchika objected saying she should be behind zero. Salma agreed. But there is nothing behind zero. What can they do?

Salma and Ruchika then extended the strip on the other side. They used a blue strip on the other side.

Now, Salma suggested that she is one mark behind zero, so it can be marked as blue one. If the token is at blue one, then the position behind blue one is blue two. Similarly, blue three is behind blue two. In this way they decided to move backward. Another day while playing they could not find blue paper, so Ruchika said, let us use a sign on the other side as we are moving in opposite

direction. So you see we need to use a sign going for numbers less than zero. The sign that is used is the placement of a minus sign attached to the number. This indicates that numbers with a negative sign are less than zero. These are called negative numbers.

**Do This**

**(Who is where?)**

Suppose David and Mohan have started walking from zero position in opposite directions. Let the steps to the right of zero be represented by ‘+’ sign and to the left of zero represented by ‘–’ sign. If Mohan moves 5 steps to the right of zero it can be represented as +5 and if David moves 5 steps to

the left of zero it can be represented as – 5. Now represent the following positions with + or – sign :

(a) 8 steps to the left of zero. (b) 7 steps to the right of zero.

(c) 11 steps to the right of zero. (d) 6 steps to the left of zero.

**Do This**

**(Who follows me?)**

We have seen from the previous examples that a movement to the right is made if the number by which we have to move is positive. If a movement of only 1 is made we get the successor of the number.

Write the succeeding number of the following :

A movement to the left is made if the number by which the token has to move is negative.

If a movement of only 1 is made to the left, we get the predecessor of a number.

**6.1.1 Tag me with a sign**

We have seen that some numbers carry a minus sign. For example, if we want to show Ronald’s due amount to the shopkeeper we would write it as – 3.

**6.2 Integers**

The first numbers to be discovered were natural numbers i.e. 1, 2, 3, 4,… If we include zero to the collection of natural numbers, we get a new collection of numbers known as whole numbers i.e. 0, 1, 2, 3, 4,… You have studied these numbers in the earlier chapter. Now we find that there are negative numbers too. If we put the whole numbers and the negative numbers together, the new collection of numbers will look like 0, 1, 2, 3, 4, 5,…, –1, – 2, – 3, –4, –5, … and this collection of numbers is known as Integers. In this collection, 1, 2, 3, … are said to be positive integers and – 1, – 2, – 3,…. are said to be negative integers.

Let us understand this by the following figures. Let us suppose that the figures represent the collection of numbers written against them.

Then the collection of integers can be understood by the following diagram in which all the earlier collections are included :

Draw a line and mark some points at equal distance on it as shown in the figure. Mark a point as zero on it. Points to the right of zero are positive integers and are marked + 1, + 2, + 3, etc. or simply 1, 2, 3 etc. Points to the left of zero are negative integers and are marked – 1, – 2, – 3 etc.

In order to mark – 6 on this line, we move 6 points to the left of zero. (Fig 6.1)

Let us once again observe the integers which are represented on the

number line.

We know that 7 > 4 and from the number line shown above, we observe that 7 is to the right of 4 (Fig 6.3).

Similarly, 4 > 0 and 4 is to the right of 0. Now, since 0 is to the right of –3 so, 0 > – 3. Again, – 3 is to the right of – 8 so, – 3 > – 8.

Thus, we see that on a number line the number increases as we move to the right and decreases as we move to the left.

Therefore, – 3 < – 2, – 2 < – 1, – 1 < 0, 0 < 1, 1 < 2, 2 < 3 so on.

Hence, the collection of integers can be written as…, –5, –4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5…

**Example 1** : By looking at the number line, answer the following questions : Which integers lie between – 8 and – 2? Which is the largest integer and the smallest integer among them?

**Solution :** Integers between – 8 and – 2 are – 7, – 6, – 5, – 4, – 3. The integer – 3 is the largest and – 7 is the smallest.

**If, I am not at zero what happens when I move?**

Let us consider the earlier game being played by Salma and Ruchika.

Suppose Ruchika’s token is at 2. At the next turn she gets a red die which after throwing gives a number 3. It means she will move 3 places to the right of 2. Thus, she comes to 5.

By looking at the number line, answer the following question :

**Example 2** : (a) One button is kept at – 3. In which direction and how many steps should we move to reach at – 9?

(b) Which number will we reach if we move 4 steps to the right of – 6.

**Solution** : (a) We have to move six steps to the left of – 3.

(b) We reach – 2 when we move 4 steps to the right of – 6.

**EXERCISE 6.1**

(b) Is point G a negative integer or a positive integer?

(c) Write integers for points B and E.

(d) Which point marked on this number line has the least value?

(e) Arrange all the points in decreasing order of value.

5. Following is the list of temperatures of five places in India on a particular day of the year.

Plot the name of the city against its temperature.

(c) Which is the coolest place?

(d) Write the names of the places where temperatures are above 10°C.

6. In each of the following pairs, which number is to the right of the other on the number line?

(a) 2, 9 (b) – 3, – 8 (c) 0, – 1

(d) – 11, 10 (e) – 6, 6 (f) 1, – 100

7. Write all the integers between the given pairs (write them in the increasing order.)

(a) 0 and – 7 (b) – 4 and 4

(c) – 8 and – 15 (d) – 30 and – 23

8. (a) Write four negative integers greater than – 20.

(b) Write four negative integers less than – 10.

9. For the following statements, write True (T) or False (F). If the statement is false, correct the statement.

(a) – 8 is to the right of – 10 on a number line.

(b) – 100 is to the right of – 50 on a number line.

(c) Smallest negative integer is – 1.

(d) – 26 is greater than – 25.

10. Draw a number line and answer the following :

(a) Which number will we reach if we move 4 numbers to the right of – 2.

(b) Which number will we reach if we move 5 numbers to the left of 1.

(c) If we are at – 8 on the number line, in which direction should we move to reach – 13?

(d) If we are at – 6 on the number line, in which direction should we move to reach – 1?

**6.3 Additon of Integers**

**Do This**

**(Going up and down)**

In Mohan’s house, there are stairs for going up to the terrace and for going down to the godown.

Let us consider the number of stairs going up to the terrace as positive integer, the number of stairs going down to the godown as negative integer, and the number representing ground level as zero.

Do the following and write down the answer as integer :

(a) Go 6 steps up from the ground floor.

(b) Go 4 steps down from the ground floor.

(c) Go 5 steps up from the ground floor and then go 3 steps up further from there.

(d) Go 6 steps down from the ground floor and then go down further 2 steps from there.

(e) Go down 5 steps from the ground floor and then move up 12 steps from there.

(f) Go 8 steps down from the ground floor and then go up 5 steps from there.

(g) Go 7 steps up from the ground floor and then 10 steps down from there. Ameena wrote them as follows :

(a) + 6 (b) – 4 (c) (+5) + (+ 3) = + 8 (d) (– 6) + (–2) = – 4

(e) (– 5) + (+12) = + 7 (f) (– 8) + (+5) = – 3 (g) (+7) + (–10) = 17

She has made some mistakes. Can you check her answers and correct those that are wrong?

(i) Let us add 3 and 5 on number line.

On the number line, we first move 3 steps to the left of 0 reaching – 3, then we move 5 steps to the left of – 3 and reach – 8. (Fig 6.5)

Thus, (– 3) + (– 5) = – 8.

We observe that when we add two positive integers, their sum is a positive integer. When we add two negative integers, their sum is a negative integer.

(iii) Suppose we wish to find the sum of (+ 5) and (– 3) on the number line. First we move to the right of 0 by 5 steps reaching 5. Then we move 3 steps to the left of 5 reaching 2. (Fig 6.6)

Thus, (+ 5) + (– 3) = 2

(iv) Similarly, let us find the sum of (– 5) and (+ 3) on the number line.

First we move 5 steps to the left of 0 reaching – 5 and then from this

point we move 3 steps to the right. We reach the point – 2.

Thus, (– 5) + (+3) = – 2. (Fig 6.7)

(b)We want to know an integer which is 5 less than 3; so we start from 3 and move to the left by 5 steps and obtain –2 as shown below :

Therefore, 5 less than 3 is –2. (Fig 6.10)

**Example 4 :** Find the sum of (– 9) + (+ 4) + (– 6) + (+ 3)

**Solution** : We can rearrange the numbers so that the positive integers and the negative integers are grouped together. We have

(– 9) + (+ 4) + (– 6) + (+ 3) = (– 9) + (– 6) + (+ 4) + (+ 3) = (– 15) + (+ 7) = – 8

**Example 5** : Find the value of (30) + (– 23) + (– 63) + (+ 55)

**Solution** : (30) + (+ 55) + (– 23) + (– 63) = 85 + (– 86) = – 1

**Example 6** : Find the sum of (– 10), (92), (84) and (– 15)

**Solution** : (– 10) + (92) + (84) + (– 15) = (– 10) + (– 15) + 92 + 84

= (– 25) + 176 = 151

**EXERCISE 6.2**

1. Using the number line write the integer which is :

(a) 3 more than 5

(b) 5 more than –5

(c) 6 less than 2

(d) 3 less than –2

2. Use number line and add the following integers :

(a) 9 + (– 6)

(b) 5 + (– 11)

(c) (– 1) + (– 7)

(d) (– 5) + 10

(e) (– 1) + (– 2) + (– 3)

(f) (– 2) + 8 + (– 4)

3. Add without using number line :

(a) 11 + (– 7) (b) (– 13) + (+ 18)

(c) (– 10) + (+ 19) (d) (– 250) + (+ 150)

(e) (– 380) + (– 270) (f) (– 217) + (– 100)

4. Find the sum of :

(a) 137 and – 354 (b) – 52 and 52

(c) – 312, 39 and 192 (d) – 50, – 200 and 300

5. Find the sum :

(a) (– 7) + (– 9) + 4 + 16

(b) (37) + (– 2) + (– 65) + (– 8)

**6.4 Subtraction of Integers with the help of a Number Line**

We have added positive integers on a number line. For example, consider 6+2. We start from 6 and go 2 steps to the right side. We reach at 8. So, 6 + 2 = 8. (Fig 6.11)

We also saw that to add 6 and (–2) on a number line we can start from 6 and then move 2 steps to the left of 6. We reach at 4. So, we have, 6 + (–2) = 4. (Fig 6.12)

Thus, we find that, to add a positive integer we move towards the right on a number line and for adding a negative integer we move towards left. We have also seen that while using a number line for whole numbers, for subtracting 2 from 6, we would move towards left. (Fig 6.13)

Fig 6.13

i.e. 6 – 2 = 4

What would we do for 6 – (–2)? Would we move towards the left on the number line or towards the right?

If we move to the left then we reach 4.

Then we have to say 6 – (–2) = 4. This is not true because we know 6 – 2 = 4 and 6 – 2 ≠ 6 – (–2).

So, we have to move towards the right. (Fig 6.14)

i.e. 6 – (–2) = 8

This also means that when we subtract a negative integer we get a greater integer. Consider it in another way. We know that additive inverse of (–2) is 2. Thus, it appears that adding the additive inverse of –2 to 6 is the same as subtracting (–2) from 6.

We write 6 – (–2) = 6 + 2.

Let us now find the value of –5 – (–4) using a number line. We can say that this is the same as –5 + (4), as the additive inverse of –4 is 4. We move 4 steps to the right on the number line starting from –5. (Fig 6.15)

We reach at –1.

i.e. –5 + 4 = –1. Thus, –5 – (–4) = –1.

**Example 7 :** Find the value of – 8 – (–10) using number line

**Solution** : – 8 – (– 10) is equal to – 8 + 10 as additive inverse of –10 is 10.

On the number line, from – 8 we will move 10 steps towards right. (Fig 6.16)

We reach at 2. Thus, –8 – (–10) = 2

Hence, to subtract an integer from another integer it is enough to add the additive inverse of the integer that is being subtracted, to the other integer.

**Example 8** : Subtract (– 4) from (– 10)

**Solution** : (– 10) – (– 4) = (– 10) + (additive inverse of – 4) = –10 + 4 = – 6

**Example 9 :** Subtract (+ 3) from (– 3)

**Solution** : (– 3) – (+ 3) = (– 3) + (additive inverse of + 3) = (– 3) + (– 3) = – 6

**EXERCISE 6.3**

1. Find

(a) 35 – (20) (b) 72 – (90)

(c) (– 15) – (– 18) (d) (–20) – (13)

(e) 23 – (– 12) (f) (–32) – (– 40)

2. Fill in the blanks with >, < or = sign.

(a) (– 3) + (– 6) ______ (– 3) – (– 6)

(b) (– 21) – (– 10) _____ (– 31) + (– 11)

(c) 45 – (– 11) ______ 57 + (– 4)

(d) (– 25) – (– 42) _____ (– 42) – (– 25)

3. Fill in the blanks.

(a) (– 8) + _____ = 0

(b) 13 + _____ = 0

(c) 12 + (– 12) = ____

(d) (– 4) + ____ = – 12

(e) ____ – 15 = – 10

4. Find

(a) (– 7) – 8 – (– 25)

(b) (– 13) + 32 – 8 – 1

(c) (– 7) + (– 8) + (– 90)

(d) 50 – (– 40) – (– 2)

**What have we discussed?**

1. We have seen that there are times when we need to use numbers with a negative sign. This is when we want to go below zero on the number line. These are called negative numbers. Some examples of their use can be in temperature scale, water level in lake or river, level of oil in tank etc. They are also used to denote debit account or outstanding dues.

2. The collection of numbers…, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, … is called integers. So, – 1, – 2, – 3, – 4, … called negative numbers are negative integers and 1, 2, 3, 4, … called positive numbers are the positive integers.

3. We have also seen how one more than given number gives a successor and one less than given number gives predecessor.

**4. We observe that**

(a) When we have the same sign, add and put the same sign.

(i) When two positive integers are added, we get a positive integer

[e.g. (+ 3) + ( + 2) = + 5].

(ii) When two negative integers are added, we get a negative integer

[e.g. (–2) + ( – 1) = – 3].

(b) When one positive and one negative integers are added we subtract them and put the sign of the bigger integer. The bigger integer is decided by ignoring the signs of the integers [e.g. (+4) + (–3) = + 1 and (–4) + ( + 3) = – 1].

(c) The subtraction of an integer is the same as the addition of its additive inverse.

5. We have shown how addition and subtraction of integers can also be shown on a number line.

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