Delhi Public School Harni: Mathematics
Delhi Public School Harni: Mathematics
Delhi Public School Harni: Mathematics
HARNI
MATHEMATICS
TARGETING MATHEMATICS
Integers
chapter : 1
std : VII
Introduction to integers
We already know that
• Natural numbers include : 1,2,3,………
• These numbers are called positive numbers or counting numbers.
• Whole numbers include : 0,1,2,3,…….
• Whole numbers are also infinite in number.
• Thus ,the smallest natural number is 1 and the smallest whole number is 0.
• 1,2,3….. lie on the right side of zero. There are the numbers lying on the
left side of zero also.
• These numbers are called negative numbers. E.g. -1,-3,-4,-5,-6…….
• So, a set of positive numbers, zero and negative numbers is called integers .
Basic rules for addition and subtraction of integers.
Rule 1: To add any two integers with same signs, we
add their absolute values and assign the same sign.
For example. (-9 ) + (-5 ) = (-14 )
9 + 5 = 14
Rule 2: To add any two integers with different signs,
we subtract their absolute values and assign the sign of
greater number.
For example. (-9 ) + 15 = 6
(- 19) + 12 = - 7
Note- Absolute value is the real value of the number, means we will ignore
the sign of the number.
E.g. absolute value of -7 is 7 and absolute value of +6 is also 6
Properties of integers for addition and
subtraction.
1. Closure property-
If a and b are integers then, (a + b) and (a – b) will also be an
integer.
E.g.1 5 + (-8 ) = -3
Here, we are adding two integers 5 and (-8),
we get (-3) as answer, which is also an integer.
E.g.2 3 – 9 = -6
Here, we are subtracting 9 from 3, we get (-6) as answer, which is
also an integer.
Thus, if we add or subtract any two integers the result will always
be an integer.
2. Commutative property of integers under addition
and subtraction
When we add any two integers in any order, answer
is always same.
E.g.1. 10 + ( -3 ) = 7, (-3 ) + 10 = 7
When we subtract any two integers in any order, answer is different.
E.g.2. 12 - ( -9 ) = 21, (-9 ) - 12 = -21
Thus , commutative property is true for addition of integers i.e. a +
b = b + a.
But same is not always true for subtraction of integers
i.e. a - b ≠ b – a.
Integers are commutative for addition, but integers are not always
commutative for subtraction.
Associative property is also true for addition of integer
i.e. a + ( b + c ) = ( a + b ) + c.
But , it is not true for subtraction of integer.
i.e. a - ( b - c ) ≠ ( a - b ) – c
Case 1. 4 + [( -3) +( -8 )] = [ 4 + ( -3 ) ] + (-8)
LHS RHS
= 4 + (-11 ) = 1 +( -8 )
= (- 7 ) = ( - 7 ) since LHS = RHS so, its true
Case 2. 14 – ( 10 - 8 ) = ( 14 – 10 ) - 8
LHS RHS
= 14 – 2 = 4–8
= 12 = (– 4 ) since LHS ≠ RHS so, its not true
If we add zero to any integer or any integer to
zero the answer remains the same integer.
i.e. a + 0 = a or 0 + a = a
E.g. 3 + 0 = 3 or 0 + 41 = 41
Thus, Zero is the additive identity
Exercise 1.1
Q.1 Write additive inverse of
i) (- 10)
Additive inverse of ( -10 ) is 10
ii) 15
Additive identity of 15 is ( -15)
Pls. note : i) to get additive inverse, we change the sign.
ii) sum of a number and its additive inverse
is always 0.
HOMEWORK
Now, Find additive inverse of
i) (-12 ) ii ) 15 iii) – (-7 )
Exercise : 1.1
Q. 3 Find the value
i) ( -6 ) + ( -3 ) - 4
= ( - 9) – 4
= ( - 9 ) + ( -4 ) ( adding additive inverse of 4 )
= ( - 13 )
ii) 18 – 11 + ( -9)
=7 +(-9)
=(-2)
iii) 11 + ( - 4 ) – ( - 6 )
=7–(-6)
= 7 + 6 ( adding additive inverse of - 6 )
= 13
xii) 6 + 8 + ( -12 ) – (- 4)
= 14 + ( -12 ) – (- 4)
= 14 + ( -12 ) + 4 [ Add additive inverse of -4 ]
= 2+4
= 6
HOMEWORK
1) 6 + ( - 9 ) – 0
2) ( -17 ) + ( - 3) + (-2)
3) 4 + (-11 ) + ( -6) –( - 2)
Now we will observe some pattern and complete it
1 . 6, 11, 16, ____, ___, ___
2. -2, -4 , -6 ___,____,___
Solution
1 . 6, 11, 16 ,21, 26,31 {add 5 to get next number }
2. -2, -4 , -6 -8, -10, -12 {Add -2 to get next number}
HOMEWORK
Now understand the pattern and solve it.
1. 9, 5, 1,____,____,_____
2. -5, -10, -15 ,___ ,_____, _____
3. 3, 2,1, ____ , _____, _____
Put >, < or = sign in the blank.
i (– 9) + (– 7 ) + 1 ____ (– 8 ) + 15)
solution: (– 16 ) + 1 ____ 7
so, ( – 15 ) < 7
ii 14 – 3 – 10 ___ (– 7 ) – 6
solution: 14 – 13 ____ (– 13 )
so, 1 > ( – 13 )
iii ( - 3 ) + 18 ____ 7 + 8
solution: 15 = 15
HOMEWORK
i 19 – 3 + (- 17) ____ ( - 12 ) + 17 + (-5)
ii ( - 14 ) + 12 + ( - 6 ) ____ ( - 5 ) + 19
iii ( - 7 ) + ( - 4 ) _____ ( -16 ) + 5
Now we will apply integers to solve daily problems
Q.1. The sum of 3 integers is 15. If two of them are
( -11 ) and 7 , find the third integer.
(-10) {2 is +ve, (-5) is –ve so, (-10) {(-2) is –ve, 5 is +ve so,
Here
, in bothanswer
the (-10)
cases the
is –ve} product remains same. , So,
answer (-10) is –ve}
in multiplication of integers grouping is not
important.
This is associative property of multiplication of
integers.
Solved problems
Q1. Evaluate : (-4) × 5 × (-11)
This problem can be solved in two different ways
Solution 1. (-4) × 5 × (-11)
{ first multiply (-4) and 5, since (-4) is
, , –ve and 5 is +ve so, product (-20) is –ve}
= (-20) × (-11) { (-20) is –ve and (-11) is –ve so ,
so, product 220 is +ve}
= 220
Solution 2. (-4) × 5 × (-11)
{ first multiply 5 and (-11), since 5 is
, ve and (-11) is –ve so, product (-55) is
–ve}
= (-4) × (-55) { (-4) is –ve and (-55) is –ve so, ,
product 220 is +ve}
= 220
Q2. Evaluate: (-3) × (-4) ×(-5)
Solution: (-3) × (-4) × (-5)
= 12× (-5) { (-3) is –ve and (-4) is also –ve
,
so, their product 12 is +ve}
= -60 { 12 is +ve and (-5) is –ve so, product is –ve}
Exercise – 1.2
Q. Evaluate:
(I) (-2) × (-5) × 27
(ii) 9 × (-11) × 10
(iii) 6 × (-3) × (-5)
(iv) (-7) × (-21) × (-10)
4) Distributive property of multiplication over addition-
If a,b and c are integers then, a×(b + c) = (a × b) + (a × c)
Example: (-4), (-3) and 2 are the integers then,
Exercise – 1.2
Q. Evaluate:
(i) (-2)×(-1)×(-3)×(-4)×(-5)
(ii) (-7)×(-1)×(-1)×(-1)×(-2)×(-2)
(iii) (-2)×(-2)×(-4)×(-8)
(iv) (-1)×(-3)×(-6)×(-5)
Problems based on multiplication of integers.
Q1. Express (-90) as product of two factors.
Solution: Since (-90) is –ve integer so, its one factor ,
will be –ve and other factor will be +ve.
(-90) = 9 × (-10) OR (-90) = 10 × (-9)
OR (-90) = 3 × (-30) OR (-90) = (-3) × 30
OR (-90) = 6 × (-15) OR (-90) = (-6) × 15
5) Division by 1-
For any integer a, a ÷ 1 = a
e.g. (-10) ÷ 1 = (-10) , 8 ÷ 1 = 8
So, if we divide any integer by 1, we always get
the same number.
6) Division of zero –
For any integer a, 0 ÷ a = 0 provided a ≠ 0.
e.g. 0 ÷ (-6) = 0 , 0 ÷ 13 = 0
If we divide zero by any integer except zero we
will always get answer zero.
7) Division by zero –
e.g. a ÷ 0 = not defined
Any number divided by zero is meaningless.
Let’s solve few problems:
Q1. Find the value of: [(-6) + 18 – 8 +14] ÷ (-9)
Solution: [(-6) + 18 – 8 +14] ÷ (-9)
= [-6 +18 -8 +14]÷ (-9)
= [-14 + 32] ÷ (-9)
= 18 ÷ (-9)
= (-2)
Q2. Find the value of: [(-11) × 8 + 8] ÷ (-10)
Solution: [(-11) × 8 + 8] ÷ (-10)
= [-88 + 8] ÷ (-10)
= (-80) ÷ (-10)
=8
Exercise – 1.3
Q2. Divide the sum of (-6), (-8) and (-10) by the product of (-2)
and 3.
Solution: Sum of (-6),(-8) & (-10) = (-6) + (-8) + (-10)
= (-24)
The product of (-2) and 3 = (-2) × 3
= (-6)
Now, as per question = (-24) ÷ (-6)
=4
Exercise- 1.3
Q1. Divide the sum of (-22),(-63) and (-15) by
the product of (-10) and 5.
Q2. Subtract 100 from (-40) and divide the
difference by the product of (-7) and 10.
Q3.Divide (-100) by (-25) and add to the
quotient, the product of (-12) and (-6).
Q4. Divide the product of (-17) and (-10) by the
sum of (-63), (-5) and (-17).
DMAS
We know that when we have to solve any
mathematical expression containing only one
operation, we solve it from left to right.
I. (-6 ) × 2 + 8
II. (-20 ) ÷ (-1) + (-7) ÷ 7
III. 65 × ( -5 ) + 80 ÷ ( -20 )
IV. 12 - 12 ÷ 3 × 4
V. ( -11 ) × ( -4 ) - 16 ÷ 8
More examples based on DMAS
Example.1
19 – 18 ÷ 2 + 14 × (-1 ) ( Division Rule )
Solution : 19 – 9 + 14 × (-1 ) ( Multiplication Rule)
= 19 - 9 - 14 ( Addition Rule)
= 10 - 14 ( Subtraction Rule )
Ans. = ( - 4 )
Example.2
(-16 ) + ( - 45) ÷(- 5 )× (-1 ) ( Division Rule )
Solution: (-16 ) + 9 × (- 1 ) ( Multiplication Rule )
= ( -16 ) + ( - 9 ) ( Addition Rule )
Ans. = ( - 25 )
Homework
1. 65 × (-13 )+ (-81) ÷9 × 9
2. 77 + ( -47) – (-95 )÷ (-19)× 5
3 105÷ ( -15 ) - 5 × 6 + 230
Do your self ( Mental maths )
Fill in the blanks
1. ( -7 ) + ( -2 ) = ___
2. (- 8 ) – ( -4 ) = ____
3. (-11 ) + 11 = _____
4. ( -9 ) ÷ 3 = ____ ( cont. )
5. ( -20 ) × ( -4 )= _____
6. ( - 5 ) + ( -9 ) – 14 = ___
7. 9 ÷ ( -3 ) + 3
8. 20 + _____ = ( - 2 )
9. ( - 18 ) × ___ = 0
10. ____ + ( - 4 ) = ( - 3 )
Solve following puzzles
1. find 5 pairs of integers whose sum is ( - 7 ).
2. find 3 pairs of integers whose product is ( -24)
3. find 4 pairs of integers whose difference is (-9)
THANKS