Page 1
Sunita’s mother has 8 bananas. Sunita has to
go for a picnic with her friends. She wants to
carry 10 bananas with her. Can 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 ` 12 with him but
the pen costs ` 15. The shopkeeper writes ` 3 as due amount from him. He
writes ` 3 in his diary to remember Ronald’s debit. But how would he remember
whether ` 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
6.1 Introduction
Chapter 6 Chapter 6 Chapter 6 Chapter 6 Chapter 6
Integers Integers
Integers Integers Integers
Rationalised 2023-24
Page 2
Sunita’s mother has 8 bananas. Sunita has to
go for a picnic with her friends. She wants to
carry 10 bananas with her. Can 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 ` 12 with him but
the pen costs ` 15. The shopkeeper writes ` 3 as due amount from him. He
writes ` 3 in his diary to remember Ronald’s debit. But how would he remember
whether ` 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
6.1 Introduction
Chapter 6 Chapter 6 Chapter 6 Chapter 6 Chapter 6
Integers Integers
Integers Integers Integers
Rationalised 2023-24
MATHEMATICS
84
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.
(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
Do This
Rationalised 2023-24
Page 3
Sunita’s mother has 8 bananas. Sunita has to
go for a picnic with her friends. She wants to
carry 10 bananas with her. Can 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 ` 12 with him but
the pen costs ` 15. The shopkeeper writes ` 3 as due amount from him. He
writes ` 3 in his diary to remember Ronald’s debit. But how would he remember
whether ` 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
6.1 Introduction
Chapter 6 Chapter 6 Chapter 6 Chapter 6 Chapter 6
Integers Integers
Integers Integers Integers
Rationalised 2023-24
MATHEMATICS
84
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.
(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
Do This
Rationalised 2023-24
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
INTEGERS
85
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.
(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 :
Number Successor
10
8
– 5
– 3
0
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.
Now write the preceding number of the following :
Number Predecessor
10
8
5
3
0
6.1.1 Tag me with a sign
W e 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.
Do This
Rationalised 2023-24
Page 4
Sunita’s mother has 8 bananas. Sunita has to
go for a picnic with her friends. She wants to
carry 10 bananas with her. Can 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 ` 12 with him but
the pen costs ` 15. The shopkeeper writes ` 3 as due amount from him. He
writes ` 3 in his diary to remember Ronald’s debit. But how would he remember
whether ` 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
6.1 Introduction
Chapter 6 Chapter 6 Chapter 6 Chapter 6 Chapter 6
Integers Integers
Integers Integers Integers
Rationalised 2023-24
MATHEMATICS
84
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.
(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
Do This
Rationalised 2023-24
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
INTEGERS
85
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.
(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 :
Number Successor
10
8
– 5
– 3
0
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.
Now write the preceding number of the following :
Number Predecessor
10
8
5
3
0
6.1.1 Tag me with a sign
W e 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.
Do This
Rationalised 2023-24
MATHEMATICS
86
Following is an account of a shopkeeper which shows profit
and loss from the sale of certain items. Since profit and loss
are opposite situations and if profit is represented by ‘+’ sign,
loss can be represented by ‘–’ sign.
Some of the situations where we may use these signs are :
The height of a place above sea level is denoted by a positive number. Height
becomes lesser and lesser as we go lower and lower. Thus, below the surface
of the sea level we can denote the height by a negative number.
If earnings are represented by ‘+’ sign,
then the spendings may be shown by a
‘–’ sign. Similarly, temperature above
0°C is denoted a ‘+’ sign and temperature
below 0°C is denoted by ‘–’ sign.
For example, the temperature of a place
10° below 0°C is written as –10°C.
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.
Name of items Profit Loss Representation
with proper sign
Mustard oil ` 150 ..............................
Rice ` 250 ..............................
Black pepper ` 225 ..............................
Wheat ` 200 ..............................
Groundnut oil ` 330 ..............................
Write the following numbers with
appropriate signs :
(a) 100 m below sea level.
(b) 25°C above 0°C temperature.
(c) 15°C below 0°C temperature.
(d) any five numbers less than 0.
Rationalised 2023-24
Page 5
Sunita’s mother has 8 bananas. Sunita has to
go for a picnic with her friends. She wants to
carry 10 bananas with her. Can 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 ` 12 with him but
the pen costs ` 15. The shopkeeper writes ` 3 as due amount from him. He
writes ` 3 in his diary to remember Ronald’s debit. But how would he remember
whether ` 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
6.1 Introduction
Chapter 6 Chapter 6 Chapter 6 Chapter 6 Chapter 6
Integers Integers
Integers Integers Integers
Rationalised 2023-24
MATHEMATICS
84
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.
(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
Do This
Rationalised 2023-24
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
INTEGERS
85
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.
(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 :
Number Successor
10
8
– 5
– 3
0
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.
Now write the preceding number of the following :
Number Predecessor
10
8
5
3
0
6.1.1 Tag me with a sign
W e 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.
Do This
Rationalised 2023-24
MATHEMATICS
86
Following is an account of a shopkeeper which shows profit
and loss from the sale of certain items. Since profit and loss
are opposite situations and if profit is represented by ‘+’ sign,
loss can be represented by ‘–’ sign.
Some of the situations where we may use these signs are :
The height of a place above sea level is denoted by a positive number. Height
becomes lesser and lesser as we go lower and lower. Thus, below the surface
of the sea level we can denote the height by a negative number.
If earnings are represented by ‘+’ sign,
then the spendings may be shown by a
‘–’ sign. Similarly, temperature above
0°C is denoted a ‘+’ sign and temperature
below 0°C is denoted by ‘–’ sign.
For example, the temperature of a place
10° below 0°C is written as –10°C.
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.
Name of items Profit Loss Representation
with proper sign
Mustard oil ` 150 ..............................
Rice ` 250 ..............................
Black pepper ` 225 ..............................
Wheat ` 200 ..............................
Groundnut oil ` 330 ..............................
Write the following numbers with
appropriate signs :
(a) 100 m below sea level.
(b) 25°C above 0°C temperature.
(c) 15°C below 0°C temperature.
(d) any five numbers less than 0.
Rationalised 2023-24
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
INTEGERS
87
Let us understand this by the following figures. Let us suppose that the
figures represent the collection of numbers written against them.
Natural numbers Zero
Whole numbers Negative numbers
Integers
Then the collection of integers can be understood by the following diagram
in which all the earlier collections are included :
Integers
6.2.1 Representation of integers on a number line
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)
In order to mark + 2 on the number line, we move 2 points to the right of zero.
(Fig 6.2)
Rationalised 2023-24
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