Multiplication and Division of Integers - Rules

While studying mathematics we notice some arithmetic operations include the processes of addition, subtraction, division, and multiplication. These operations perform on different types of numbers. Numbers could be integers, decimal, rational or real. 

What are Integers?

Integers are the whole numbers that are non-fractional.

In integers, we can find,

The numbers which we count (1,2,3….).

The number 0 

The negative numbers ( -1, -2, -3,...)

Combining the above, we find the integers are (......,-3, -2, -1, 0, 1, 2, 3, …..). 

Multiplication of Integers

Multiplication is the addition of numbers but the rules for the multiplication of integers are different from the addition of integers.

The only thing that makes the multiplication and division different is the sign. You need to be very careful regarding the signs while doing multiplication and division.

Rules for the Multiplication of Integers for Class 7

• Rule 1: When Multiplication occurs between two positive integers the product is always positive. 

For Example 2×2 = 4

• Rule 2: When Multiplication occurs between two negative integers the product is always positive. 

For Example (-2)×(-2) = 4

• Rule 3: The product of a positive integer and a negative integer is negative.

For Example 2×(-2) =  - 4

Proof: When positive integer multiplied with negative integer, it results in negative integer

As we all know that multiplication is nothing but the addition of numbers repetitively. Therefore, it could also rewrite in the form of subtraction. In order to prove the given statement, we will take some examples which follow the pattern of subtraction and multiplication both.

3 × 2 = 6

2 × 2 = 4 = 6 – 2

 1 × 2 = 2 = 4 – 2

 0 × 2 = 0 = 2 – 2 

Continuing the pattern, we get

 –1 × 2 = 0 – 2 = –2, 

–2 × 2 = –2 – 2 = –4

 –3 × 2 = –4 – 2 = –6 

and so on.

Thus we got (–x) × 2 = (–y)

It shows that multiplying a negative integer with a positive integer gives a negative integer. Also, in this proof with the initial three mathematical statements, we can conclude Rule 1 for Multiplication of Integers. 

Problem Sum for Multiplication of Integers for Class 7

Question: Find the product of the following:

1.  (–18) × (–10) × 9

2. (–5) × (–2) × (– 8) × (– 7)

Solved:

1.  (–18) × (–10) × 9 = ((–18)×(–10)) × 9 = 180 × 9 = 1620

2. (–5) × (–2) × (– 8) × (– 7) = ((–5)×(–2)) × ((–8)×(–7)) = 10 × 56 = 560

Properties of Multiplication

In the multiplication of integers, you have the following properties:

• Closure Property: 

The multiplication value of two integers is always an integer. 

According to the closure property of multiplication, if you multiply two integers suppose p x q then the product of p x q is also an integer.

Here, p x q will be an integer, for each integer p and q.

• Commutative Property: 

In the case of any two integers m and n, 

m×n = n×m

Proof: Integers Show Commutative Property

We all know that whole numbers show the commutative property. But in order to prove it for the integers, observe the below given table: 

From the above table, we can conclude that integers also show the commutative property.  

Therefore, a × b = b × a is true for all a, b ∈ Integers.

• Associative Property: 

Multiplication between integers is associative i.e., for any three integers a, b, c, we have: a × ( b × c) = (a × b) × c

Proof: Integers Show Associative Property

To prove associativity for multiplication of integers, i.e., a × ( b × c) = (a × b) × c, let us consider a = 3, b = -4, c = -8.

Put the values of a, b and c in a given equation.

L.H.S. will be 3 × [(-4) × (-8)] and R.H.S. will be [(3) × (-4)] × -8.

In the above two mathematical statements, two different values are combined in pairs. 

Solving the L.H.S. and R.H.S., we get

L.H.S. = 3 × [(-4) × (-8)] = 3 × 32 = 96

R.H.S. = [(3) × (-4)] × -8 = (-12) × -8 = 96

Since, L.H.S. = R.H.S.

Therefore, we came to the conclusion that the integers show the associative property. It means that there will be no effect of grouping the integers. 

• Distributive Property: 

The multiplication of integers is distributive over their addition. For example, if we have three integers, 1, 2, 3.

1 x (2 + 3)  =  (1 x 2) + (1 x 3)

For any three integers, say, a, b, and c, the distributive property could be written as:

 a × ( b + c) =  a × b +  a × c

Therefore, multiplication is distributive over the addition of integers.

The distributive property could be proved as associativity of integers, by considering any random values of a, b and c. 

• Multiplication by Zero: 

When any integer is multiplied by zero(0), we get zero as the product. 

For example: If we have an integer 5

5 × 0 = 0

• Multiplication by Identity: 

When you multiply any integer by 1 you get 1 itself as the result. 

1’ is the multiplicative identity for integers.

In general, for an integer 3, we have

3 x 1  =  1 x 3  =  3

Division of Integers

When you distribute integers you are carrying out the function division of integers. The process of dividing integers is exactly the opposite of multiplying integers. 

In both cases multiplying the integer or dividing the integer, the rules are quite similar. But it’s not necessary to always find integers as your quotient value.

In the division of integers, you have the following properties:

• Suppose a and b are two integers, then a ÷ b does not necessarily have to be an integer.

• If an integer a is not equal to 0, then a ÷ a = 1.

• For every integer a, you have a ÷ 1= a.

•  If an integer a is non-zero, then 0 ÷ a = 0.

•  If a is an integer, then a ÷ 0 is not valid.

• Suppose we have three integers x, y, which are non-zero integers, then (x ÷ y) ÷ z ≠ x÷ (y ÷ z), unless z = 1.

Rules for the Division of Integers for Class 7

• Rule 1: The quotient value of two positive integers will always be a positive integer.

• Rule 2:  For two negative integers the quotient value will always be a positive integer.

• Rule 3: The quotient value of one positive integer and one negative integer will always be a negative integer

One of the things to always remember is when you are dividing you should always divide without the signs but after getting the solution of the integer gives the sign according to the sign given in the problem.

Solved Examples for the Division of Integers for Class 7

Question: Solve –91 ÷ 7, –117 ÷ 13, –98 ÷ –14 

Solution: 

• 91 ÷ 7

= 91/7

=13

• -117 ÷ 13 

= -117/13 

= -9 

• -98 ÷ (-14) 

= -98/-14

= 98/14

= 7 

Summary

In order to summarize the rules for multiplication and division, remember the following points:

1.  Positive × Positive = Positive

2.  Positive × Negative = Negative

3. Negative × Positive = Negative

4. Negative × Negative = Positive

5. Positive ÷ Positive = Positive

6. Positive ÷ Negative = Negative

7. Negative ÷ Positive = Negative

8. Negative ÷ Negative = Positive

Do you Remember?

•  The addition is commutative and associative for integers.

• For every integer a, a + 0 = 0 + a = a, i.e., 0 is the identity under addition. 

The above properties are also shown by the multiplication, and the combined property of addition and multiplication is called the distributive property.