Multiplication of Integers
We know that
multiplication is nothing but repeated addition.
Example: 3 + 3 + 3 + 3 + 3 +
3 + 3 = 7 × 3 = 21
Similarly, (–3) + (–3) + (–3) + (–3) + (–3) + (–3) +
(–3) = 7 × (–3) = –21
Rules for multiplication of two integers are given below:
1. To find the
product of two integers with the same sign, find the product of the absolute
values of the two integers and assign plus sign to the product.
2. To find the
product of two integers having opposite signs, find the product of the absolute
values of the two integers and assign negative sign to the product.
For example,
a. 5 × 8 = 40, (–5) × (–8) = 40
b. 3 × (–8) = –24, (–3) × 8 = –24
Example: Find the product of the
following integers.
a. 4 × 11 b. 3 × (–13) c. (–6) × 15 d. (–8) × (–7)
Solution:
a. 4 × 11 = 44
b. 3 × (–13) = – (3 × 13)
= –39
c. (–6) × 15 = – (6 × 15) = –90
d. (–8) × (–7) = 8 × 7 = 56
Properties of Multiplication of Integers
Closure property
If a and b are any two integers, then a × b is also an integer.
For example, 4 × 2 = 8 and 5 × (–6) = –30, Here,
8 and (–30) are also integers.
Thus, integers are closed for multiplication.
Commutative property
If a and b are any two integers, then a × b = b × a
For example, 5 × (–6) = (–6) × 5 = –30
Thus, integers are commutative for multiplication.
Associative property
If a, b and c are any three integers, then (a × b) × c = a × (b × c)
For example, [2 × 5]
× (–6) = 2 × [5 × (–6)] = –60
Thus, integers are associative for multiplication.
Distributive property
If a, b and c are
any three integers,
then a ×
(b
+
c) = a × b + a × c and a × (b – c) = a × b – a × c
For example, a. 2 × (5
+ 6) = 2 × 5 + 2 × 6 = 10 + 12 = 22
b. 2 × (5
– 6) = 2 × 5 – 2 × 6 = 10 – 12 = –2
Thus, integers are distributive for multiplication
over addition and subtraction
Multiplicative identity
If a is
an integer, then a × 1 = 1 × a = a.
Here, 1
is called the multiplicative identity.
Multiplicative property of zero
If a is
an integer, then a × 0 = 0 × a = 0.
Thus, if
0 is multiplied with any integer, the product is always 0.
Example 1: Take three integers –7, 8 and –5
and check if (a × b) × c
=
a
×
(b
×
c)?
Solution:
L.H.S. = (a
×
b) × c
=
[(–7) × 8] × (–5) = (–56) × (–5) = 280
R.H.S. = a
×
(b
×
c) = (–7)
× [8 × (–5)] = (–7) × (–40) = 280
Hence, (a
×
b) × c
=
a
×
(b
×
c)
Example 2: Evaluate
the following using properties of multiplication.
a. 2 × (–6) × (–3) b. 7 × 97 c. 12 × 104
Solution:
a. 2 × (–6) × (–3) = [2 ×
(–6)] × (–3) (Using associative property)
= (–12) × (–3) = 36
b. 7 × 97 = 7 × (100 – 3)
= 7 × 100 – 7 × 3 (Using distributive property)
= 700 – 21 = 679
c. 12 × 104 = 12 × (100 +
4) = 12 × 100 + 12 × 4 (Using distributive property)
= 1200 + 48 = 1248
Division of Integers
Division is
the inverse of multiplication. The rules for the division of integers are as
follow:
1. Quotient of two positive integers is a positive
integer.
2. Quotient of two negative integers is a positive
integer.
3. Quotient of two integers with
different signs is a negative integer.
Dividing
45 by –9 means finding an integer which when multiplied by –9, gives 45. The
required integer is –5. Therefore, we write 45 ÷ (–9) = –5.
Example: Find the
value of the following:
a. (–18) ÷ (9) b. (–22) ÷ (–2) c. 56 ÷ (–4) d. (–215) ÷ (–5)
Solution:
a. (–18) ÷ (9) = –18/9
= –2
b. (–22) ÷ (–2) = –22/–2 = 11
c. 56 ÷ (–4) = 56/–4
= –14
d. (–215) ÷ (–5) = --215/–5
= 43
Properties of Division of Integers
Closure property
If a and b are any two integers, then a ÷ b is not always an integer.
For example, 16 ÷ 2 = 8 but
15 × (–6) = –2.5, Here, 2.5 is not an integer.
Thus, integers are not closed for division.
Commutative property
If a and b are any two integers, then a ÷ b ≠ b ÷ a
For example, 12 ÷
(–6) ≠ (–6) ÷
12
Thus, integers are not commutative for division.
Associative property
If a, b and c are any three integers, then (a ÷ b) ÷
c ≠ a ÷ (b ÷
c)
For example, [15 ÷ 5]
÷ (–3) = 15 ÷
[5 ÷ (–3)]
Thus, integers are not associative for division.
Division of an integer by itself and by 1
If
a is an integer, then a
÷ a = 1, where a
≠
0
and a ÷ 1 = a
Division property of zero
If
a is an integer
and a
≠
0,
then 0 ÷ a
= 0
Division of an integer by its
additive inverse and –1
If a is an integer, then a ÷ (–a) = –1, (–a) ÷ a = –1, a ÷ (–1) = –a
Word Problems on Integers
Example 1:
A snail climbs 21 cm up a wall and falls back 11 cm
in one hour. How much will it climb in 6 hours?
Solution:
The snail climbs +21 cm. Falls back –11 cm.
In 6 hours, the distance it will cover = 6[21 + (–11)]
cm = 6(21 – 11) = 6 × 10 cm = 60 cm.
Example 2:
The temperature of an AC room was 24 °C at 9 p.m. The
temperature dropped by 3 °C in one hour. What is the temperature at 10 p.m.?
Solution:
Temperature at 9 p.m. = 24 °C and change in
temperature = –3 °C
Thus, temperature at 10 p.m. = 24 °C – 3 °C = 21 °C
Simplification of Integers using BODMAS
To
simplify an expression with two or more basic fundamental operations, grouping
is used. The grouping symbols are called brackets. There
are 4 types of brackets.
1. — is called bar
or
vinculum.
2. ( ) are called parentheses.
3. { } are called braces
or
curly
brackets.
4. [ ] are called square
brackets.
The brackets are removed in the
following order.
a. bar (—)
b. parentheses ( )
c. braces { }
d. square brackets [ ]
To simplify problems containing
brackets, the order of operation is remembered by the word BODMAS
where
the letters have the meaning as shown below:
B → Brackets
O → Of
D → Division
M → Multiplication
A → Addition
S → Subtraction
Example: Simplify
the following.
a. 35 + 20 ÷ {3 – (2
+ 5)} b. 5 – [6 + {19 – (16 – 2)}].
Solution:
a. 35
+ 20 ÷ {3 – (2 + 5)} = 35 + 20 ÷ {3 – 7}
= 35 + 20 ÷ (–4) = 35 – 5 =
30
b.
15 – [6 + {19 –
(16 – 2)}] = 15 – [6 + {19 – 14}] (Using BODMAS)
= 15 – [6 + 5] = 15 – 11 = 4
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