Multiplication and Division of Integers

# Multiplication and Division of Integers

## 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
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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