RS Aggarwal Solutions: Integers | Mathematics for Class 6 PDF Download

 Page 1


Points to Remember :
Integers : The numbers .........., – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, .......... are called integers.
The numbers 1, 2, 3, 4, 5 ............ are called positive integers and the numbers – 1, – 2, – 3, – 4,
– 5 ............. are called negative integers. 0 is an integer which is neither positive nor negative.
Representation of Integers On Number Line :
We draw a line and fix a point almost in the middle of it and call it O. We set off equal distances on
right hand side as well as on left hand side of 0. We name the points of division as 1, 2, 3, 4 etc.on
left hand side as shown below :
Some results :
(i) Zero is less than every positive integer.
(ii) Zero is greater than every negative integer.
(iii) Every positive integer is greater then every negative integer.
(iv) The greater is the number, the lesser is its opposite.
Absolute Value of an Integer. The absolute value of an integer is the numerical value of the
integer regardless of its sign.
(iv) 10 km below sea level
(v) 5°C above the freezing point
(vi) A withdrawl of Rs. 100
(vii) Spending Rs. 500
(viii) Going 6 m to the west
(ix) – 24
(x) 34
Q. 2. Indicate the following using ‘+’ or ‘–’
sign :
(i) A gain of Rs. 600
(ii) A loss of Rs. 800
(iii) 7ºC below the freezing point
(iv) Decrease of  9
(v) 2 km above sea level
(vi) 3 km below sea level
(vii) A deposit of Rs. 200
(viii) A withdrawl of Rs. 300
( ) EXERCISE 4 A
Q. 1. Write the opposite of each of the
following :
(i) An increase of 8
(ii) A loss of Rs. 7
(iii) Gaining a weight of 5 kg
(iv) 10 km above sea level
(v) 5°C below the freezing point
(vi) A deposit of Rs. 100
(vii) Earning Rs. 500
(viii) Going 6 m to the east
(ix) 24
(x) – 34
Sol. (i) A decrease of 8
(ii) A gain of Rs. 7
(iii) Loosing a weight of 5 kg
Page 2


Points to Remember :
Integers : The numbers .........., – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, .......... are called integers.
The numbers 1, 2, 3, 4, 5 ............ are called positive integers and the numbers – 1, – 2, – 3, – 4,
– 5 ............. are called negative integers. 0 is an integer which is neither positive nor negative.
Representation of Integers On Number Line :
We draw a line and fix a point almost in the middle of it and call it O. We set off equal distances on
right hand side as well as on left hand side of 0. We name the points of division as 1, 2, 3, 4 etc.on
left hand side as shown below :
Some results :
(i) Zero is less than every positive integer.
(ii) Zero is greater than every negative integer.
(iii) Every positive integer is greater then every negative integer.
(iv) The greater is the number, the lesser is its opposite.
Absolute Value of an Integer. The absolute value of an integer is the numerical value of the
integer regardless of its sign.
(iv) 10 km below sea level
(v) 5°C above the freezing point
(vi) A withdrawl of Rs. 100
(vii) Spending Rs. 500
(viii) Going 6 m to the west
(ix) – 24
(x) 34
Q. 2. Indicate the following using ‘+’ or ‘–’
sign :
(i) A gain of Rs. 600
(ii) A loss of Rs. 800
(iii) 7ºC below the freezing point
(iv) Decrease of  9
(v) 2 km above sea level
(vi) 3 km below sea level
(vii) A deposit of Rs. 200
(viii) A withdrawl of Rs. 300
( ) EXERCISE 4 A
Q. 1. Write the opposite of each of the
following :
(i) An increase of 8
(ii) A loss of Rs. 7
(iii) Gaining a weight of 5 kg
(iv) 10 km above sea level
(v) 5°C below the freezing point
(vi) A deposit of Rs. 100
(vii) Earning Rs. 500
(viii) Going 6 m to the east
(ix) 24
(x) – 34
Sol. (i) A decrease of 8
(ii) A gain of Rs. 7
(iii) Loosing a weight of 5 kg
Sol. (i) + Rs. 600 (ii) – Rs. 800
(iii) – 7ºC (iv) – 9
(v) + 2 km (vi) – 3 km
(vii) + Rs. 200 (viii) – Rs. 300
Q. 3. Mark the following integers on a number
line :
(i) – 5 (ii) – 2
(iii) 0 (iv) 7
(v) –13
Sol.
Q. 4. Which number is larger in each of the
following pairs.
(i) 0, – 2 (ii) – 3, – 5
(iii) – 5, 2 (iv) – 16, 8
(v) – 365, – 913 (vi) – 888, 8
Sol. (i) 0 (ii) – 3
(iii) 2 (iv) 8
(v) – 365 (vi) 8
Q. 5. Which number is smaller in each of the
following pairs ?
(i) 6, –7 (ii) 0, – 1
(iii) – 13, – 27 (iv) – 26, 17
(v) – 317, – 603 (vi) – 777, 7
Sol. (i) – 7 (ii) – 1
(iii) – 27 (iv) – 26
(v) – 603 (vi) – 777
Q. 6. Write all integers between
(i) 0 and 6 (ii) – 5 and 0
(iii) – 3 and 3   (iv) – 7 and – 5
Sol. (i) The integers between 0 and 6 are
1, 2, 3, 4, 5.
(ii) The integers between – 5 and 0 are
– 4, – 3, – 2, – 1.
(iii) The integers between – 3 and 3 are
– 2, – 1, 0, 1, 2.
(iv) The integer between – 7 and – 5 is – 6.
Q. 7. Fill in the blanks by appropriate symbol
> or < :
(i) 0 .......... 7 (ii) 0 ....... – 3
(iii) – 5 ........ –2 (iv) – 15 ...... 13
(v) – 231 ............ – 132 (vi) – 6 ...... 6
Sol. (i) 0 <  7 (ii) 0  > – 3
(iii)  – 5 < – 2 (iv) – 15 < 13
(v) – 231 < – 132 (vi) – 6 < 6
Q. 8. Write the following integers in the
increasing order :
(i) 5, –7, – 2, 0, 8
(ii) – 23, 12, 0, – 6, – 100, – 1
(iii) – 17, 15, – 363, – 501, 165
(iv) 21, –106, –16, 16, 0, – 2, – 81
Sol. (i) – 7, – 2, 0, 5, 8
(ii) – 100, – 23, – 6, – 1, 0, 12
(iii) – 501, – 363, – 17, 15, 165
(iv) – 106, – 81, – 16, – 2, 0, 16, 21.
Q. 9. Write the following integers in the
decreasing order :
(i) 0, 7, – 3, –9, – 132, 36
(ii) 51, – 53, – 8, 0, – 2
(iii) – 71, – 81, 36, 0, – 5
(iv) – 365, – 515, 102, 413, – 7
Sol. (i) 36, 7, 0, – 3, – 9, – 132
(ii) 51, 0, – 2, – 8, – 53
(iii) 36, 0, – 5, – 71, – 81
(iv) 413, 102, – 7, – 365, – 515.
Q. 10. Using the number line, write the integer
which is
(i) 4 more than 6 (ii) 5 more than – 6
Page 3


Points to Remember :
Integers : The numbers .........., – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, .......... are called integers.
The numbers 1, 2, 3, 4, 5 ............ are called positive integers and the numbers – 1, – 2, – 3, – 4,
– 5 ............. are called negative integers. 0 is an integer which is neither positive nor negative.
Representation of Integers On Number Line :
We draw a line and fix a point almost in the middle of it and call it O. We set off equal distances on
right hand side as well as on left hand side of 0. We name the points of division as 1, 2, 3, 4 etc.on
left hand side as shown below :
Some results :
(i) Zero is less than every positive integer.
(ii) Zero is greater than every negative integer.
(iii) Every positive integer is greater then every negative integer.
(iv) The greater is the number, the lesser is its opposite.
Absolute Value of an Integer. The absolute value of an integer is the numerical value of the
integer regardless of its sign.
(iv) 10 km below sea level
(v) 5°C above the freezing point
(vi) A withdrawl of Rs. 100
(vii) Spending Rs. 500
(viii) Going 6 m to the west
(ix) – 24
(x) 34
Q. 2. Indicate the following using ‘+’ or ‘–’
sign :
(i) A gain of Rs. 600
(ii) A loss of Rs. 800
(iii) 7ºC below the freezing point
(iv) Decrease of  9
(v) 2 km above sea level
(vi) 3 km below sea level
(vii) A deposit of Rs. 200
(viii) A withdrawl of Rs. 300
( ) EXERCISE 4 A
Q. 1. Write the opposite of each of the
following :
(i) An increase of 8
(ii) A loss of Rs. 7
(iii) Gaining a weight of 5 kg
(iv) 10 km above sea level
(v) 5°C below the freezing point
(vi) A deposit of Rs. 100
(vii) Earning Rs. 500
(viii) Going 6 m to the east
(ix) 24
(x) – 34
Sol. (i) A decrease of 8
(ii) A gain of Rs. 7
(iii) Loosing a weight of 5 kg
Sol. (i) + Rs. 600 (ii) – Rs. 800
(iii) – 7ºC (iv) – 9
(v) + 2 km (vi) – 3 km
(vii) + Rs. 200 (viii) – Rs. 300
Q. 3. Mark the following integers on a number
line :
(i) – 5 (ii) – 2
(iii) 0 (iv) 7
(v) –13
Sol.
Q. 4. Which number is larger in each of the
following pairs.
(i) 0, – 2 (ii) – 3, – 5
(iii) – 5, 2 (iv) – 16, 8
(v) – 365, – 913 (vi) – 888, 8
Sol. (i) 0 (ii) – 3
(iii) 2 (iv) 8
(v) – 365 (vi) 8
Q. 5. Which number is smaller in each of the
following pairs ?
(i) 6, –7 (ii) 0, – 1
(iii) – 13, – 27 (iv) – 26, 17
(v) – 317, – 603 (vi) – 777, 7
Sol. (i) – 7 (ii) – 1
(iii) – 27 (iv) – 26
(v) – 603 (vi) – 777
Q. 6. Write all integers between
(i) 0 and 6 (ii) – 5 and 0
(iii) – 3 and 3   (iv) – 7 and – 5
Sol. (i) The integers between 0 and 6 are
1, 2, 3, 4, 5.
(ii) The integers between – 5 and 0 are
– 4, – 3, – 2, – 1.
(iii) The integers between – 3 and 3 are
– 2, – 1, 0, 1, 2.
(iv) The integer between – 7 and – 5 is – 6.
Q. 7. Fill in the blanks by appropriate symbol
> or < :
(i) 0 .......... 7 (ii) 0 ....... – 3
(iii) – 5 ........ –2 (iv) – 15 ...... 13
(v) – 231 ............ – 132 (vi) – 6 ...... 6
Sol. (i) 0 <  7 (ii) 0  > – 3
(iii)  – 5 < – 2 (iv) – 15 < 13
(v) – 231 < – 132 (vi) – 6 < 6
Q. 8. Write the following integers in the
increasing order :
(i) 5, –7, – 2, 0, 8
(ii) – 23, 12, 0, – 6, – 100, – 1
(iii) – 17, 15, – 363, – 501, 165
(iv) 21, –106, –16, 16, 0, – 2, – 81
Sol. (i) – 7, – 2, 0, 5, 8
(ii) – 100, – 23, – 6, – 1, 0, 12
(iii) – 501, – 363, – 17, 15, 165
(iv) – 106, – 81, – 16, – 2, 0, 16, 21.
Q. 9. Write the following integers in the
decreasing order :
(i) 0, 7, – 3, –9, – 132, 36
(ii) 51, – 53, – 8, 0, – 2
(iii) – 71, – 81, 36, 0, – 5
(iv) – 365, – 515, 102, 413, – 7
Sol. (i) 36, 7, 0, – 3, – 9, – 132
(ii) 51, 0, – 2, – 8, – 53
(iii) 36, 0, – 5, – 71, – 81
(iv) 413, 102, – 7, – 365, – 515.
Q. 10. Using the number line, write the integer
which is
(i) 4 more than 6 (ii) 5 more than – 6
(iii) 6 less than 2 (iv) 2 less than – 3
Sol. (i) We want to write an integer 4 more
than 6.
So, we start from 6 and proceed 4 steps
to the right to obtain 10, as shown
below :
 4 more than 6 is 10.
(ii) We want to write an integer 5 more than
– 6.
So, we start from – 6 and proceed 5
steps to the right to obtain – 1, as shown
below :
 5 more than – 6 is – 1.
(iii) We want to write an integer 6 less than
2. So we start from 2 and come back to
the left by 6 steps to obtain – 4, as shown
below :
 6 less than 2 is – 4.
(iv) We want to write an integer 2 less than
– 3. So we start from – 3 and come
back to the left by 2 steps to obtain – 5,
as shown below :
  2 less than – 3 is – 5.
Q. 11. For each of the following statements,
write (T) for true and (F) for false.
(i) The smallest integer is zero.
(ii) Zero is not an integer.
(iii) The opposite of zero is zero.
(iv) – 10 is greater than – 6.
(v) The absolute value of an integer is
always greater than the integer.
(vi) 0 is larger than every negative integer.
(vii) Every negative integer is less than every
natural number.
(viii) The successor of –187 is –188.
(ix) The predecessor of –215 is –214.
Sol. (i) False, as zero is greater than every
negative integer.
(ii) False, as zero is an integer.
(iii) True, as zero is neither positive nor
negative.
(iv) False, as – 10 is to the left of – 6 on a
number line.
(v) False, as absolute value of an integer is
always equal to the integer.
(vi) True, as 0 is to right of every negative
integer, on a number line.
(vii) False, as every natural number is positive.
(viii) False, the successor is –186
(ix) False, the predecessor is –216
Q. 12. Find the value of
(i) | – 9 | (ii) | – 36 |
(iii) | 0 | (iv) | 15 |
(v) – | – 3 | (vi) 7 + | – 3 |
(vii) | 7 – 4 | (viii) 8 – | – 7 |
Sol. (i) | – 9 | = 9 (ii) | – 36 | = 36
(iii) | 0 | = 0 (iv) | 15 | = 15
(v) – | – 3 | = – 3
(vi) 7 + | – 3 | = 7 + 3 = 10
(vii) | 7 – 4 | = | 3 | = 3
(viii) 8 – | – 7 | = 8 – 7 = 1
Q. 13. (i) Write five negative integers greater
than – 7.
(ii) Write five negative integers less than
– 20.
Sol. (i) The required integers are – 6, – 5,
– 4, – 3, – 2.
(ii) The required integers are – 21, – 22,
– 23, – 24, – 25.
Addition And Subtraction of Integers
Rules for Addition of Integers :
1. If two positive integers or two negative
integers are added, we add their values
regardless of their signs and give the sum
their common sign.
Page 4


Points to Remember :
Integers : The numbers .........., – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, .......... are called integers.
The numbers 1, 2, 3, 4, 5 ............ are called positive integers and the numbers – 1, – 2, – 3, – 4,
– 5 ............. are called negative integers. 0 is an integer which is neither positive nor negative.
Representation of Integers On Number Line :
We draw a line and fix a point almost in the middle of it and call it O. We set off equal distances on
right hand side as well as on left hand side of 0. We name the points of division as 1, 2, 3, 4 etc.on
left hand side as shown below :
Some results :
(i) Zero is less than every positive integer.
(ii) Zero is greater than every negative integer.
(iii) Every positive integer is greater then every negative integer.
(iv) The greater is the number, the lesser is its opposite.
Absolute Value of an Integer. The absolute value of an integer is the numerical value of the
integer regardless of its sign.
(iv) 10 km below sea level
(v) 5°C above the freezing point
(vi) A withdrawl of Rs. 100
(vii) Spending Rs. 500
(viii) Going 6 m to the west
(ix) – 24
(x) 34
Q. 2. Indicate the following using ‘+’ or ‘–’
sign :
(i) A gain of Rs. 600
(ii) A loss of Rs. 800
(iii) 7ºC below the freezing point
(iv) Decrease of  9
(v) 2 km above sea level
(vi) 3 km below sea level
(vii) A deposit of Rs. 200
(viii) A withdrawl of Rs. 300
( ) EXERCISE 4 A
Q. 1. Write the opposite of each of the
following :
(i) An increase of 8
(ii) A loss of Rs. 7
(iii) Gaining a weight of 5 kg
(iv) 10 km above sea level
(v) 5°C below the freezing point
(vi) A deposit of Rs. 100
(vii) Earning Rs. 500
(viii) Going 6 m to the east
(ix) 24
(x) – 34
Sol. (i) A decrease of 8
(ii) A gain of Rs. 7
(iii) Loosing a weight of 5 kg
Sol. (i) + Rs. 600 (ii) – Rs. 800
(iii) – 7ºC (iv) – 9
(v) + 2 km (vi) – 3 km
(vii) + Rs. 200 (viii) – Rs. 300
Q. 3. Mark the following integers on a number
line :
(i) – 5 (ii) – 2
(iii) 0 (iv) 7
(v) –13
Sol.
Q. 4. Which number is larger in each of the
following pairs.
(i) 0, – 2 (ii) – 3, – 5
(iii) – 5, 2 (iv) – 16, 8
(v) – 365, – 913 (vi) – 888, 8
Sol. (i) 0 (ii) – 3
(iii) 2 (iv) 8
(v) – 365 (vi) 8
Q. 5. Which number is smaller in each of the
following pairs ?
(i) 6, –7 (ii) 0, – 1
(iii) – 13, – 27 (iv) – 26, 17
(v) – 317, – 603 (vi) – 777, 7
Sol. (i) – 7 (ii) – 1
(iii) – 27 (iv) – 26
(v) – 603 (vi) – 777
Q. 6. Write all integers between
(i) 0 and 6 (ii) – 5 and 0
(iii) – 3 and 3   (iv) – 7 and – 5
Sol. (i) The integers between 0 and 6 are
1, 2, 3, 4, 5.
(ii) The integers between – 5 and 0 are
– 4, – 3, – 2, – 1.
(iii) The integers between – 3 and 3 are
– 2, – 1, 0, 1, 2.
(iv) The integer between – 7 and – 5 is – 6.
Q. 7. Fill in the blanks by appropriate symbol
> or < :
(i) 0 .......... 7 (ii) 0 ....... – 3
(iii) – 5 ........ –2 (iv) – 15 ...... 13
(v) – 231 ............ – 132 (vi) – 6 ...... 6
Sol. (i) 0 <  7 (ii) 0  > – 3
(iii)  – 5 < – 2 (iv) – 15 < 13
(v) – 231 < – 132 (vi) – 6 < 6
Q. 8. Write the following integers in the
increasing order :
(i) 5, –7, – 2, 0, 8
(ii) – 23, 12, 0, – 6, – 100, – 1
(iii) – 17, 15, – 363, – 501, 165
(iv) 21, –106, –16, 16, 0, – 2, – 81
Sol. (i) – 7, – 2, 0, 5, 8
(ii) – 100, – 23, – 6, – 1, 0, 12
(iii) – 501, – 363, – 17, 15, 165
(iv) – 106, – 81, – 16, – 2, 0, 16, 21.
Q. 9. Write the following integers in the
decreasing order :
(i) 0, 7, – 3, –9, – 132, 36
(ii) 51, – 53, – 8, 0, – 2
(iii) – 71, – 81, 36, 0, – 5
(iv) – 365, – 515, 102, 413, – 7
Sol. (i) 36, 7, 0, – 3, – 9, – 132
(ii) 51, 0, – 2, – 8, – 53
(iii) 36, 0, – 5, – 71, – 81
(iv) 413, 102, – 7, – 365, – 515.
Q. 10. Using the number line, write the integer
which is
(i) 4 more than 6 (ii) 5 more than – 6
(iii) 6 less than 2 (iv) 2 less than – 3
Sol. (i) We want to write an integer 4 more
than 6.
So, we start from 6 and proceed 4 steps
to the right to obtain 10, as shown
below :
 4 more than 6 is 10.
(ii) We want to write an integer 5 more than
– 6.
So, we start from – 6 and proceed 5
steps to the right to obtain – 1, as shown
below :
 5 more than – 6 is – 1.
(iii) We want to write an integer 6 less than
2. So we start from 2 and come back to
the left by 6 steps to obtain – 4, as shown
below :
 6 less than 2 is – 4.
(iv) We want to write an integer 2 less than
– 3. So we start from – 3 and come
back to the left by 2 steps to obtain – 5,
as shown below :
  2 less than – 3 is – 5.
Q. 11. For each of the following statements,
write (T) for true and (F) for false.
(i) The smallest integer is zero.
(ii) Zero is not an integer.
(iii) The opposite of zero is zero.
(iv) – 10 is greater than – 6.
(v) The absolute value of an integer is
always greater than the integer.
(vi) 0 is larger than every negative integer.
(vii) Every negative integer is less than every
natural number.
(viii) The successor of –187 is –188.
(ix) The predecessor of –215 is –214.
Sol. (i) False, as zero is greater than every
negative integer.
(ii) False, as zero is an integer.
(iii) True, as zero is neither positive nor
negative.
(iv) False, as – 10 is to the left of – 6 on a
number line.
(v) False, as absolute value of an integer is
always equal to the integer.
(vi) True, as 0 is to right of every negative
integer, on a number line.
(vii) False, as every natural number is positive.
(viii) False, the successor is –186
(ix) False, the predecessor is –216
Q. 12. Find the value of
(i) | – 9 | (ii) | – 36 |
(iii) | 0 | (iv) | 15 |
(v) – | – 3 | (vi) 7 + | – 3 |
(vii) | 7 – 4 | (viii) 8 – | – 7 |
Sol. (i) | – 9 | = 9 (ii) | – 36 | = 36
(iii) | 0 | = 0 (iv) | 15 | = 15
(v) – | – 3 | = – 3
(vi) 7 + | – 3 | = 7 + 3 = 10
(vii) | 7 – 4 | = | 3 | = 3
(viii) 8 – | – 7 | = 8 – 7 = 1
Q. 13. (i) Write five negative integers greater
than – 7.
(ii) Write five negative integers less than
– 20.
Sol. (i) The required integers are – 6, – 5,
– 4, – 3, – 2.
(ii) The required integers are – 21, – 22,
– 23, – 24, – 25.
Addition And Subtraction of Integers
Rules for Addition of Integers :
1. If two positive integers or two negative
integers are added, we add their values
regardless of their signs and give the sum
their common sign.
2. If a positive integer and a negative integer
are added, we find the difference
between their values regardless of their
signs and give the sign of the integer
with more numerical value.
Properties of Addition of Integers :
Property 1 (Closure Property). The
sum of two Integers is always an integer.
Property 2 (Commutative law of
Addition). If a and b are any two
integers then a + b = b + a.
Property 3 (Associative law of
Addition). If a, b, c are any three
integers then
(a + b) + c = a + (b + c)
Property 4. If a is any integer then
a + 0 = 0 + a = a
Property 5. The sum of an integer and
its opposite is 0. Thus, if a is an integer
then a + (– a) = 0.
a and – a are called opposites or
negatives or additive inverse of each
other.
Property 6. If a is any integer than (a +
1) is also an integer, called the successor
of a.
( ) EXERCISE 4 B
Q. 1. Use the number line and add the
following integers :
(i) 9 + (– 6) (ii) (–3) + 7
(iii) 8 + (– 8) (iv) (–1) + (–3)
(v) (– 4) + (– 7) (vi) (– 2) + (– 8)
(vii) 3 + (– 2) + (– 4)
(viii) (– 1) + (– 2) + (– 3)
(ix) 5 + (– 2) + (– 6)
Sol. (i) On the number line we start from 0
and move 9 steps to the right to reach a
point A. Now, starting from A, we move
6 steps to the left to reach a point B, as
shown below :
Now, B represents the integer 3
 9 + (– 6) = 3
(ii) On the number line, we start from 0 and
move 3 steps to the left to reach a point
A. Now, starting from A, we move 7
steps to the right to reach a point B, as
shown below :
And B represents the integer 4
 (– 3) + 7 = 4
(iii) On the number line, we start from 0 and
move 8 steps to the right to reach a point
A. Now, starting from A, we move 8
steps to the left to reach a point B, as
shown below :
And, B represents the integer 0.
 8 + (– 8) = 0
(iv) On the number line, we start from 0 and
move 1 step the left to reach a point A.
Now, starting from point A, we move 3
steps to the left to reach a point B, as
shown below :
And, B represents the integer – 4
 (– 1) + (– 3) = – 4.
(v) On the number line, we start from 0 and
move 4 steps to the left to reach a point
A. Now, starting from point A, we move
Page 5


Points to Remember :
Integers : The numbers .........., – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, .......... are called integers.
The numbers 1, 2, 3, 4, 5 ............ are called positive integers and the numbers – 1, – 2, – 3, – 4,
– 5 ............. are called negative integers. 0 is an integer which is neither positive nor negative.
Representation of Integers On Number Line :
We draw a line and fix a point almost in the middle of it and call it O. We set off equal distances on
right hand side as well as on left hand side of 0. We name the points of division as 1, 2, 3, 4 etc.on
left hand side as shown below :
Some results :
(i) Zero is less than every positive integer.
(ii) Zero is greater than every negative integer.
(iii) Every positive integer is greater then every negative integer.
(iv) The greater is the number, the lesser is its opposite.
Absolute Value of an Integer. The absolute value of an integer is the numerical value of the
integer regardless of its sign.
(iv) 10 km below sea level
(v) 5°C above the freezing point
(vi) A withdrawl of Rs. 100
(vii) Spending Rs. 500
(viii) Going 6 m to the west
(ix) – 24
(x) 34
Q. 2. Indicate the following using ‘+’ or ‘–’
sign :
(i) A gain of Rs. 600
(ii) A loss of Rs. 800
(iii) 7ºC below the freezing point
(iv) Decrease of  9
(v) 2 km above sea level
(vi) 3 km below sea level
(vii) A deposit of Rs. 200
(viii) A withdrawl of Rs. 300
( ) EXERCISE 4 A
Q. 1. Write the opposite of each of the
following :
(i) An increase of 8
(ii) A loss of Rs. 7
(iii) Gaining a weight of 5 kg
(iv) 10 km above sea level
(v) 5°C below the freezing point
(vi) A deposit of Rs. 100
(vii) Earning Rs. 500
(viii) Going 6 m to the east
(ix) 24
(x) – 34
Sol. (i) A decrease of 8
(ii) A gain of Rs. 7
(iii) Loosing a weight of 5 kg
Sol. (i) + Rs. 600 (ii) – Rs. 800
(iii) – 7ºC (iv) – 9
(v) + 2 km (vi) – 3 km
(vii) + Rs. 200 (viii) – Rs. 300
Q. 3. Mark the following integers on a number
line :
(i) – 5 (ii) – 2
(iii) 0 (iv) 7
(v) –13
Sol.
Q. 4. Which number is larger in each of the
following pairs.
(i) 0, – 2 (ii) – 3, – 5
(iii) – 5, 2 (iv) – 16, 8
(v) – 365, – 913 (vi) – 888, 8
Sol. (i) 0 (ii) – 3
(iii) 2 (iv) 8
(v) – 365 (vi) 8
Q. 5. Which number is smaller in each of the
following pairs ?
(i) 6, –7 (ii) 0, – 1
(iii) – 13, – 27 (iv) – 26, 17
(v) – 317, – 603 (vi) – 777, 7
Sol. (i) – 7 (ii) – 1
(iii) – 27 (iv) – 26
(v) – 603 (vi) – 777
Q. 6. Write all integers between
(i) 0 and 6 (ii) – 5 and 0
(iii) – 3 and 3   (iv) – 7 and – 5
Sol. (i) The integers between 0 and 6 are
1, 2, 3, 4, 5.
(ii) The integers between – 5 and 0 are
– 4, – 3, – 2, – 1.
(iii) The integers between – 3 and 3 are
– 2, – 1, 0, 1, 2.
(iv) The integer between – 7 and – 5 is – 6.
Q. 7. Fill in the blanks by appropriate symbol
> or < :
(i) 0 .......... 7 (ii) 0 ....... – 3
(iii) – 5 ........ –2 (iv) – 15 ...... 13
(v) – 231 ............ – 132 (vi) – 6 ...... 6
Sol. (i) 0 <  7 (ii) 0  > – 3
(iii)  – 5 < – 2 (iv) – 15 < 13
(v) – 231 < – 132 (vi) – 6 < 6
Q. 8. Write the following integers in the
increasing order :
(i) 5, –7, – 2, 0, 8
(ii) – 23, 12, 0, – 6, – 100, – 1
(iii) – 17, 15, – 363, – 501, 165
(iv) 21, –106, –16, 16, 0, – 2, – 81
Sol. (i) – 7, – 2, 0, 5, 8
(ii) – 100, – 23, – 6, – 1, 0, 12
(iii) – 501, – 363, – 17, 15, 165
(iv) – 106, – 81, – 16, – 2, 0, 16, 21.
Q. 9. Write the following integers in the
decreasing order :
(i) 0, 7, – 3, –9, – 132, 36
(ii) 51, – 53, – 8, 0, – 2
(iii) – 71, – 81, 36, 0, – 5
(iv) – 365, – 515, 102, 413, – 7
Sol. (i) 36, 7, 0, – 3, – 9, – 132
(ii) 51, 0, – 2, – 8, – 53
(iii) 36, 0, – 5, – 71, – 81
(iv) 413, 102, – 7, – 365, – 515.
Q. 10. Using the number line, write the integer
which is
(i) 4 more than 6 (ii) 5 more than – 6
(iii) 6 less than 2 (iv) 2 less than – 3
Sol. (i) We want to write an integer 4 more
than 6.
So, we start from 6 and proceed 4 steps
to the right to obtain 10, as shown
below :
 4 more than 6 is 10.
(ii) We want to write an integer 5 more than
– 6.
So, we start from – 6 and proceed 5
steps to the right to obtain – 1, as shown
below :
 5 more than – 6 is – 1.
(iii) We want to write an integer 6 less than
2. So we start from 2 and come back to
the left by 6 steps to obtain – 4, as shown
below :
 6 less than 2 is – 4.
(iv) We want to write an integer 2 less than
– 3. So we start from – 3 and come
back to the left by 2 steps to obtain – 5,
as shown below :
  2 less than – 3 is – 5.
Q. 11. For each of the following statements,
write (T) for true and (F) for false.
(i) The smallest integer is zero.
(ii) Zero is not an integer.
(iii) The opposite of zero is zero.
(iv) – 10 is greater than – 6.
(v) The absolute value of an integer is
always greater than the integer.
(vi) 0 is larger than every negative integer.
(vii) Every negative integer is less than every
natural number.
(viii) The successor of –187 is –188.
(ix) The predecessor of –215 is –214.
Sol. (i) False, as zero is greater than every
negative integer.
(ii) False, as zero is an integer.
(iii) True, as zero is neither positive nor
negative.
(iv) False, as – 10 is to the left of – 6 on a
number line.
(v) False, as absolute value of an integer is
always equal to the integer.
(vi) True, as 0 is to right of every negative
integer, on a number line.
(vii) False, as every natural number is positive.
(viii) False, the successor is –186
(ix) False, the predecessor is –216
Q. 12. Find the value of
(i) | – 9 | (ii) | – 36 |
(iii) | 0 | (iv) | 15 |
(v) – | – 3 | (vi) 7 + | – 3 |
(vii) | 7 – 4 | (viii) 8 – | – 7 |
Sol. (i) | – 9 | = 9 (ii) | – 36 | = 36
(iii) | 0 | = 0 (iv) | 15 | = 15
(v) – | – 3 | = – 3
(vi) 7 + | – 3 | = 7 + 3 = 10
(vii) | 7 – 4 | = | 3 | = 3
(viii) 8 – | – 7 | = 8 – 7 = 1
Q. 13. (i) Write five negative integers greater
than – 7.
(ii) Write five negative integers less than
– 20.
Sol. (i) The required integers are – 6, – 5,
– 4, – 3, – 2.
(ii) The required integers are – 21, – 22,
– 23, – 24, – 25.
Addition And Subtraction of Integers
Rules for Addition of Integers :
1. If two positive integers or two negative
integers are added, we add their values
regardless of their signs and give the sum
their common sign.
2. If a positive integer and a negative integer
are added, we find the difference
between their values regardless of their
signs and give the sign of the integer
with more numerical value.
Properties of Addition of Integers :
Property 1 (Closure Property). The
sum of two Integers is always an integer.
Property 2 (Commutative law of
Addition). If a and b are any two
integers then a + b = b + a.
Property 3 (Associative law of
Addition). If a, b, c are any three
integers then
(a + b) + c = a + (b + c)
Property 4. If a is any integer then
a + 0 = 0 + a = a
Property 5. The sum of an integer and
its opposite is 0. Thus, if a is an integer
then a + (– a) = 0.
a and – a are called opposites or
negatives or additive inverse of each
other.
Property 6. If a is any integer than (a +
1) is also an integer, called the successor
of a.
( ) EXERCISE 4 B
Q. 1. Use the number line and add the
following integers :
(i) 9 + (– 6) (ii) (–3) + 7
(iii) 8 + (– 8) (iv) (–1) + (–3)
(v) (– 4) + (– 7) (vi) (– 2) + (– 8)
(vii) 3 + (– 2) + (– 4)
(viii) (– 1) + (– 2) + (– 3)
(ix) 5 + (– 2) + (– 6)
Sol. (i) On the number line we start from 0
and move 9 steps to the right to reach a
point A. Now, starting from A, we move
6 steps to the left to reach a point B, as
shown below :
Now, B represents the integer 3
 9 + (– 6) = 3
(ii) On the number line, we start from 0 and
move 3 steps to the left to reach a point
A. Now, starting from A, we move 7
steps to the right to reach a point B, as
shown below :
And B represents the integer 4
 (– 3) + 7 = 4
(iii) On the number line, we start from 0 and
move 8 steps to the right to reach a point
A. Now, starting from A, we move 8
steps to the left to reach a point B, as
shown below :
And, B represents the integer 0.
 8 + (– 8) = 0
(iv) On the number line, we start from 0 and
move 1 step the left to reach a point A.
Now, starting from point A, we move 3
steps to the left to reach a point B, as
shown below :
And, B represents the integer – 4
 (– 1) + (– 3) = – 4.
(v) On the number line, we start from 0 and
move 4 steps to the left to reach a point
A. Now, starting from point A, we move
7 steps to the left to reach a point B, as
shown below :
And, B represents the integer – 11.
 (– 4) + (– 7) = – 11
(vi) On the number line we start from 0 and
move 2 steps to the left to reach a point
A. Now, starting from A, we move  8
steps to the left to reach a point B, as
shown below :
And, B represents the integer – 10
 (– 2) + (– 8) = – 10
(vii) On the number line we start from 0 and
move 3 steps to the right to reach a point
A. Now, starting from A, we move 2
steps to the left to reach a point B and
again starting from left to reach a point
B and again starting from B, we move 4
steps to the left to reach a point C, as
shown below :
And, C represents the integer – 3
 3 + (– 2) + (– 4) = – 3
(viii) On the number line we start from 0 and
move 1 step to the left to reach a point
A. Now, starting from A, we move 2
steps to the left to reach a point B and
again starting from B, we move 3 steps
to the left to reach point C, as shown
below :
And, C represents the integer – 6
 (– 1) + (– 2) + (– 3) = – 6.
(ix) On the number line we start from 0 and
move 5 steps to the right to reach a point
A. Now, starting from A, we move 2
steps to the left to reach a point B and
again starting from point B, we move 6
steps to the left to reach a point C, as
shown below :
And, C represents the integer – 3.
 5 + (– 2) + (– 6) = – 3
Q. 2. Fill in the blanks :
(i) (– 3) + (– 9) = ..........
(ii) (– 7) + (– 8) = .........
(iii) (– 9) + 16 = ............
(iv) (– 13) + 25 =...........
(v) 8 + (– 17) = ............
(vi) 2 + (– 12) = ............
Sol. (i) (– 3) + (– 9) = – 12
(Using the rule for addition of integers
having like signs)
(ii) (– 7) + (– 8) = – 15
(Using the rule for addition of integers
having like signs)
(iii) (– 9) + 16 = 7
(Using the rule for addition of integers
having unlike signs)
(iv) (– 13) + 25 = 12
(Using the rule for addition of integers
having unlike signs)
(v) 8 + (– 17) = – 9
(Using  the rule for addition of integers
having unlike signs)
(vi) 2 + (– 12) = – 10
(Using the rule for addition of integers
having unlike signs)