Textbook: Integers | Mathematics Class 6 (Maharashtra Board) PDF Download

 Page 1


12
 Count how many boys, flowers and ducks there are in the picture. We have to count 
objects in order to find out the answer to ‘How many?’. Numbers were created because 
of the need to count things in nature. We write the count of things in the form of numbers.   Dada : The numbers 1, 2, 3, 4, ... that we have used up to now for counting are called 
‘counting numbers’. They are also called natural numbers. But is it possible to 
count the stars in the sky or the grains of sand on the beach? They are innumerable 
and so are the natural numbers. Look at this list of natural numbers:
  Natural numbers : 1, 2, 3, 4, ..., 321, 322, ..., 28573, ...
Samir : We have already learnt to add and subtract these natural numbers. But when we 
subtract 5 from 5 nothing remains. The zero that we write to show that, is not 
seen in this list. 
Dada :  Of course, we cannot do without ‘zero’. The set of all natural numbers together 
with zero is the set of whole numbers. 
Whole numbers :  0, 1, 2, 3, 4, ..., 367, 368, ..., 237105, ...
Dada : We need to use some other numbers which are not there in this group.
Salma : Which are those?
Dada : Here’s an example. In Maharashtra, the temperature falls to 10 °C (10 degrees 
Celsius) or even 8 °C in winter, but not down to 0 °C. But in Kashmir, it may 
fall even below 0 °C. To show that, we need numbers that are less than zero.
3
Integers
Let’s recall.
Let’s discuss.
Page 2


12
 Count how many boys, flowers and ducks there are in the picture. We have to count 
objects in order to find out the answer to ‘How many?’. Numbers were created because 
of the need to count things in nature. We write the count of things in the form of numbers.   Dada : The numbers 1, 2, 3, 4, ... that we have used up to now for counting are called 
‘counting numbers’. They are also called natural numbers. But is it possible to 
count the stars in the sky or the grains of sand on the beach? They are innumerable 
and so are the natural numbers. Look at this list of natural numbers:
  Natural numbers : 1, 2, 3, 4, ..., 321, 322, ..., 28573, ...
Samir : We have already learnt to add and subtract these natural numbers. But when we 
subtract 5 from 5 nothing remains. The zero that we write to show that, is not 
seen in this list. 
Dada :  Of course, we cannot do without ‘zero’. The set of all natural numbers together 
with zero is the set of whole numbers. 
Whole numbers :  0, 1, 2, 3, 4, ..., 367, 368, ..., 237105, ...
Dada : We need to use some other numbers which are not there in this group.
Salma : Which are those?
Dada : Here’s an example. In Maharashtra, the temperature falls to 10 °C (10 degrees 
Celsius) or even 8 °C in winter, but not down to 0 °C. But in Kashmir, it may 
fall even below 0 °C. To show that, we need numbers that are less than zero.
3
Integers
Let’s recall.
Let’s discuss.
13
Warm water
Crushed ice and salt Crushed ice
Samir : In January, when the papers said that it was snowing in Kashmir, the temperature 
in Srinagar was - 8 °C. How do we read that?
Dada :  It is read as ‘minus eight degrees Celsius’. When we put a minus sign ( - ) before 
any number, the number obtained is less than zero. It is called a negative number. 
On a thermometer, there are increasing numbers like 1, 2, 3, ... above 0. These 
are called positive numbers. The numbers below zero are - 1, - 2, - 3, ... .
Samir :  Can we show negative numbers on the number line?
Dada : Of course! On the right of zero at distances of 1, 2, 3, .... units are the numbers 
1, 2, 3, ... . On the left of zero at 1, 2, 3, ... unit distances are the numbers           
- 1, - 2, - 3, ... . They are called negative numbers. The numbers  1, 2, 3, .... 
on the right are called positive numbers. They can be written as 1, 2, 3, ... and 
also +1, +2, +3, ... .
Salma :  On the thermometer, the positive numbers are above zero and the negative 
numbers are below it. On the number line, they are on the right and left sides 
of zero respectively. Does it mean that positive and negative numbers are on 
opposite sides of zero?
Dada : Correct!
Samir :  Then should we use positive numbers to show height above sea level and negative 
numbers for depth below sea level?
Dada  : You’re right, too! Very good!
The ‘+’ sign is generally not written before positive numbers. However, it is necessary 
to write the ‘- ’ sign of a negative number. Zero does not have any sign.
   
 Take warm water in one beaker, some crushed ice in another and a mixture of salt 
and crushed ice in a third beaker. Ask your teacher for help in measuring the temperature 
of the substance in each of the beakers using a thermometer. Note the temperatures.
Take care!
 
Try this.
Page 3


12
 Count how many boys, flowers and ducks there are in the picture. We have to count 
objects in order to find out the answer to ‘How many?’. Numbers were created because 
of the need to count things in nature. We write the count of things in the form of numbers.   Dada : The numbers 1, 2, 3, 4, ... that we have used up to now for counting are called 
‘counting numbers’. They are also called natural numbers. But is it possible to 
count the stars in the sky or the grains of sand on the beach? They are innumerable 
and so are the natural numbers. Look at this list of natural numbers:
  Natural numbers : 1, 2, 3, 4, ..., 321, 322, ..., 28573, ...
Samir : We have already learnt to add and subtract these natural numbers. But when we 
subtract 5 from 5 nothing remains. The zero that we write to show that, is not 
seen in this list. 
Dada :  Of course, we cannot do without ‘zero’. The set of all natural numbers together 
with zero is the set of whole numbers. 
Whole numbers :  0, 1, 2, 3, 4, ..., 367, 368, ..., 237105, ...
Dada : We need to use some other numbers which are not there in this group.
Salma : Which are those?
Dada : Here’s an example. In Maharashtra, the temperature falls to 10 °C (10 degrees 
Celsius) or even 8 °C in winter, but not down to 0 °C. But in Kashmir, it may 
fall even below 0 °C. To show that, we need numbers that are less than zero.
3
Integers
Let’s recall.
Let’s discuss.
13
Warm water
Crushed ice and salt Crushed ice
Samir : In January, when the papers said that it was snowing in Kashmir, the temperature 
in Srinagar was - 8 °C. How do we read that?
Dada :  It is read as ‘minus eight degrees Celsius’. When we put a minus sign ( - ) before 
any number, the number obtained is less than zero. It is called a negative number. 
On a thermometer, there are increasing numbers like 1, 2, 3, ... above 0. These 
are called positive numbers. The numbers below zero are - 1, - 2, - 3, ... .
Samir :  Can we show negative numbers on the number line?
Dada : Of course! On the right of zero at distances of 1, 2, 3, .... units are the numbers 
1, 2, 3, ... . On the left of zero at 1, 2, 3, ... unit distances are the numbers           
- 1, - 2, - 3, ... . They are called negative numbers. The numbers  1, 2, 3, .... 
on the right are called positive numbers. They can be written as 1, 2, 3, ... and 
also +1, +2, +3, ... .
Salma :  On the thermometer, the positive numbers are above zero and the negative 
numbers are below it. On the number line, they are on the right and left sides 
of zero respectively. Does it mean that positive and negative numbers are on 
opposite sides of zero?
Dada : Correct!
Samir :  Then should we use positive numbers to show height above sea level and negative 
numbers for depth below sea level?
Dada  : You’re right, too! Very good!
The ‘+’ sign is generally not written before positive numbers. However, it is necessary 
to write the ‘- ’ sign of a negative number. Zero does not have any sign.
   
 Take warm water in one beaker, some crushed ice in another and a mixture of salt 
and crushed ice in a third beaker. Ask your teacher for help in measuring the temperature 
of the substance in each of the beakers using a thermometer. Note the temperatures.
Take care!
 
Try this.
14
 
Integers
Positive numbers, zero and negative numbers together form a group 
of numbers called the group of integers.
   
Look at the picture of the kulfi man. In a lift, the ground floor is numbered 0
Why do you think he keeps the kulfi (zero) while the floors below the ground
moulds in a mixture of salt and ice? level are numbered - 1 and - 2.    
  
Showing Integers on the Number Line
 The point on a number line which is marked 0 is called the origin. On the left and right 
of 0, points are marked at equal distances. The numbers shown by points on the right are 
positive numbers and those shown by points on the left are negative numbers. 
    
     Negative numbers  Origin           Positive numbers
Example : Show the numbers - 7 and +8 on the number line.
 - 3 - 2 - 1 0 +1 +6 - 4 - 5 - 6
- 7
Let’s learn.
 
My friend, Maths : At the fair, in the lift.
Let’s learn.
+2 +3 +4 +5 +7
- 6 - 5 - 4 - 3 - 2 - 1 0 +1 +2 +3 +4 +5 +6 +7 +8
Page 4


12
 Count how many boys, flowers and ducks there are in the picture. We have to count 
objects in order to find out the answer to ‘How many?’. Numbers were created because 
of the need to count things in nature. We write the count of things in the form of numbers.   Dada : The numbers 1, 2, 3, 4, ... that we have used up to now for counting are called 
‘counting numbers’. They are also called natural numbers. But is it possible to 
count the stars in the sky or the grains of sand on the beach? They are innumerable 
and so are the natural numbers. Look at this list of natural numbers:
  Natural numbers : 1, 2, 3, 4, ..., 321, 322, ..., 28573, ...
Samir : We have already learnt to add and subtract these natural numbers. But when we 
subtract 5 from 5 nothing remains. The zero that we write to show that, is not 
seen in this list. 
Dada :  Of course, we cannot do without ‘zero’. The set of all natural numbers together 
with zero is the set of whole numbers. 
Whole numbers :  0, 1, 2, 3, 4, ..., 367, 368, ..., 237105, ...
Dada : We need to use some other numbers which are not there in this group.
Salma : Which are those?
Dada : Here’s an example. In Maharashtra, the temperature falls to 10 °C (10 degrees 
Celsius) or even 8 °C in winter, but not down to 0 °C. But in Kashmir, it may 
fall even below 0 °C. To show that, we need numbers that are less than zero.
3
Integers
Let’s recall.
Let’s discuss.
13
Warm water
Crushed ice and salt Crushed ice
Samir : In January, when the papers said that it was snowing in Kashmir, the temperature 
in Srinagar was - 8 °C. How do we read that?
Dada :  It is read as ‘minus eight degrees Celsius’. When we put a minus sign ( - ) before 
any number, the number obtained is less than zero. It is called a negative number. 
On a thermometer, there are increasing numbers like 1, 2, 3, ... above 0. These 
are called positive numbers. The numbers below zero are - 1, - 2, - 3, ... .
Samir :  Can we show negative numbers on the number line?
Dada : Of course! On the right of zero at distances of 1, 2, 3, .... units are the numbers 
1, 2, 3, ... . On the left of zero at 1, 2, 3, ... unit distances are the numbers           
- 1, - 2, - 3, ... . They are called negative numbers. The numbers  1, 2, 3, .... 
on the right are called positive numbers. They can be written as 1, 2, 3, ... and 
also +1, +2, +3, ... .
Salma :  On the thermometer, the positive numbers are above zero and the negative 
numbers are below it. On the number line, they are on the right and left sides 
of zero respectively. Does it mean that positive and negative numbers are on 
opposite sides of zero?
Dada : Correct!
Samir :  Then should we use positive numbers to show height above sea level and negative 
numbers for depth below sea level?
Dada  : You’re right, too! Very good!
The ‘+’ sign is generally not written before positive numbers. However, it is necessary 
to write the ‘- ’ sign of a negative number. Zero does not have any sign.
   
 Take warm water in one beaker, some crushed ice in another and a mixture of salt 
and crushed ice in a third beaker. Ask your teacher for help in measuring the temperature 
of the substance in each of the beakers using a thermometer. Note the temperatures.
Take care!
 
Try this.
14
 
Integers
Positive numbers, zero and negative numbers together form a group 
of numbers called the group of integers.
   
Look at the picture of the kulfi man. In a lift, the ground floor is numbered 0
Why do you think he keeps the kulfi (zero) while the floors below the ground
moulds in a mixture of salt and ice? level are numbered - 1 and - 2.    
  
Showing Integers on the Number Line
 The point on a number line which is marked 0 is called the origin. On the left and right 
of 0, points are marked at equal distances. The numbers shown by points on the right are 
positive numbers and those shown by points on the left are negative numbers. 
    
     Negative numbers  Origin           Positive numbers
Example : Show the numbers - 7 and +8 on the number line.
 - 3 - 2 - 1 0 +1 +6 - 4 - 5 - 6
- 7
Let’s learn.
 
My friend, Maths : At the fair, in the lift.
Let’s learn.
+2 +3 +4 +5 +7
- 6 - 5 - 4 - 3 - 2 - 1 0 +1 +2 +3 +4 +5 +6 +7 +8
15
2016 2021
Practice Set 4
 
My friend, Maths : On the playground.
  
1. Classify the following numbers as positive numbers and negative numbers.  
 - 5, +4, - 2, 7, +26, - 49, - 37, 19, - 25, +8, 5, - 4, - 12, 27
2. Given below are the temperatures in some cities. Write them using the proper signs.
Place Shimla Leh Delhi Nagpur
Temperature 7 °C below 0° 12 °C below 0° 22 °C above 0° 31 °C above 0°
3. Write the numbers in the following examples using the proper signs.
 (1) A submarine is at a depth of 512 metres below sea level. 
 (2) The height of Mt Everest, the highest peak in the Himalayas, is 8848 metres.
 (3) A kite is flying at a distance of 120 metres from the ground.
 (4) The tunnel is at a depth of 2 metres under the ground. 
??	 On the playground, mark a timeline showing the years from 2000 to 2024. With one 
child standing at the position of the current year, ask the following questions :
 (1) While playing this game, what is his/her age? 
 (2) Five years ago, which year was it? And what was his/her age then?
 (3) In which year will he/she go to Std X? How old will he/she be then?
1, 2, 3, ... 
 My class, that is Std VI, is a 
part of my school. My school is 
in my town. My town is a part 
of a taluka. In the same way, 
the taluka is a part of a district, 
and the district is a part of 
Maharashtra State.
 In the same way, what can 
you say about these groups of 
numbers?
Can you tell?
Whole
numbers 
Integers
Natural numbers
0 - 1, - 2, ...
 
1, 2, 3, ... 
Page 5


12
 Count how many boys, flowers and ducks there are in the picture. We have to count 
objects in order to find out the answer to ‘How many?’. Numbers were created because 
of the need to count things in nature. We write the count of things in the form of numbers.   Dada : The numbers 1, 2, 3, 4, ... that we have used up to now for counting are called 
‘counting numbers’. They are also called natural numbers. But is it possible to 
count the stars in the sky or the grains of sand on the beach? They are innumerable 
and so are the natural numbers. Look at this list of natural numbers:
  Natural numbers : 1, 2, 3, 4, ..., 321, 322, ..., 28573, ...
Samir : We have already learnt to add and subtract these natural numbers. But when we 
subtract 5 from 5 nothing remains. The zero that we write to show that, is not 
seen in this list. 
Dada :  Of course, we cannot do without ‘zero’. The set of all natural numbers together 
with zero is the set of whole numbers. 
Whole numbers :  0, 1, 2, 3, 4, ..., 367, 368, ..., 237105, ...
Dada : We need to use some other numbers which are not there in this group.
Salma : Which are those?
Dada : Here’s an example. In Maharashtra, the temperature falls to 10 °C (10 degrees 
Celsius) or even 8 °C in winter, but not down to 0 °C. But in Kashmir, it may 
fall even below 0 °C. To show that, we need numbers that are less than zero.
3
Integers
Let’s recall.
Let’s discuss.
13
Warm water
Crushed ice and salt Crushed ice
Samir : In January, when the papers said that it was snowing in Kashmir, the temperature 
in Srinagar was - 8 °C. How do we read that?
Dada :  It is read as ‘minus eight degrees Celsius’. When we put a minus sign ( - ) before 
any number, the number obtained is less than zero. It is called a negative number. 
On a thermometer, there are increasing numbers like 1, 2, 3, ... above 0. These 
are called positive numbers. The numbers below zero are - 1, - 2, - 3, ... .
Samir :  Can we show negative numbers on the number line?
Dada : Of course! On the right of zero at distances of 1, 2, 3, .... units are the numbers 
1, 2, 3, ... . On the left of zero at 1, 2, 3, ... unit distances are the numbers           
- 1, - 2, - 3, ... . They are called negative numbers. The numbers  1, 2, 3, .... 
on the right are called positive numbers. They can be written as 1, 2, 3, ... and 
also +1, +2, +3, ... .
Salma :  On the thermometer, the positive numbers are above zero and the negative 
numbers are below it. On the number line, they are on the right and left sides 
of zero respectively. Does it mean that positive and negative numbers are on 
opposite sides of zero?
Dada : Correct!
Samir :  Then should we use positive numbers to show height above sea level and negative 
numbers for depth below sea level?
Dada  : You’re right, too! Very good!
The ‘+’ sign is generally not written before positive numbers. However, it is necessary 
to write the ‘- ’ sign of a negative number. Zero does not have any sign.
   
 Take warm water in one beaker, some crushed ice in another and a mixture of salt 
and crushed ice in a third beaker. Ask your teacher for help in measuring the temperature 
of the substance in each of the beakers using a thermometer. Note the temperatures.
Take care!
 
Try this.
14
 
Integers
Positive numbers, zero and negative numbers together form a group 
of numbers called the group of integers.
   
Look at the picture of the kulfi man. In a lift, the ground floor is numbered 0
Why do you think he keeps the kulfi (zero) while the floors below the ground
moulds in a mixture of salt and ice? level are numbered - 1 and - 2.    
  
Showing Integers on the Number Line
 The point on a number line which is marked 0 is called the origin. On the left and right 
of 0, points are marked at equal distances. The numbers shown by points on the right are 
positive numbers and those shown by points on the left are negative numbers. 
    
     Negative numbers  Origin           Positive numbers
Example : Show the numbers - 7 and +8 on the number line.
 - 3 - 2 - 1 0 +1 +6 - 4 - 5 - 6
- 7
Let’s learn.
 
My friend, Maths : At the fair, in the lift.
Let’s learn.
+2 +3 +4 +5 +7
- 6 - 5 - 4 - 3 - 2 - 1 0 +1 +2 +3 +4 +5 +6 +7 +8
15
2016 2021
Practice Set 4
 
My friend, Maths : On the playground.
  
1. Classify the following numbers as positive numbers and negative numbers.  
 - 5, +4, - 2, 7, +26, - 49, - 37, 19, - 25, +8, 5, - 4, - 12, 27
2. Given below are the temperatures in some cities. Write them using the proper signs.
Place Shimla Leh Delhi Nagpur
Temperature 7 °C below 0° 12 °C below 0° 22 °C above 0° 31 °C above 0°
3. Write the numbers in the following examples using the proper signs.
 (1) A submarine is at a depth of 512 metres below sea level. 
 (2) The height of Mt Everest, the highest peak in the Himalayas, is 8848 metres.
 (3) A kite is flying at a distance of 120 metres from the ground.
 (4) The tunnel is at a depth of 2 metres under the ground. 
??	 On the playground, mark a timeline showing the years from 2000 to 2024. With one 
child standing at the position of the current year, ask the following questions :
 (1) While playing this game, what is his/her age? 
 (2) Five years ago, which year was it? And what was his/her age then?
 (3) In which year will he/she go to Std X? How old will he/she be then?
1, 2, 3, ... 
 My class, that is Std VI, is a 
part of my school. My school is 
in my town. My town is a part 
of a taluka. In the same way, 
the taluka is a part of a district, 
and the district is a part of 
Maharashtra State.
 In the same way, what can 
you say about these groups of 
numbers?
Can you tell?
Whole
numbers 
Integers
Natural numbers
0 - 1, - 2, ...
 
1, 2, 3, ... 
16
 The child should find answers to such questions by walking the right number of units 
and in the right direction on the timeline. 
?? 	 Next, the unit on the timeline on the playground should be of 100 years. This will make 
it possible to count the years from 0 to 2100 on it. Important historical events can then 
be shown in the proper centuries. 
Addition of Integers
 On the number line, we shall show the rabbit’s hops to the right with positive signs 
and the ones to the left with negative signs.
Activity :
?? At first the rabbit was at the number 
 
.   ??  It hopped  units to the right.  
?? 	 It is now at the number .
1 + 5 = (+1) + (+5) = +6 
Activity :
?? At first the rabbit was at the number 
 
.  ??  It hopped 
 
 units to the right.
?? 	 It is now at the number .       
(- 2) + (+5) = +3
Activity :
Activity :
?? At first the rabbit was at the number 
 
.   ??  It hopped 
 
 units to the left. 
?? 	 It is now at the number 
 
.   (- 3) + (- 4) = - 7 
- 1 +4 +5 +6 +7 0 +1 +2 +3
+1 0 - 3 - 2 - 1 +2 +3 +4 - 4
To add a positive number to the given number, we move that many units to the 
right on the number line from the given number. 
Now I know -
- 3 - 4 - 7 - 6 - 5 - 2 - 1 0 - 8 +1 +2 +3 - 9 - 10