**Introduction**

In mathematics, we study about different types of numbers and operations of addition, subtraction, multiplication and division among them. We know about counting numbers are called natural numbers an d Whole Numbers are the set of natural numbers along with the number 0. We also know that is we subtract a whole number from another whole number, the difference may not always be a whole number. In fact when a smaller whole number is subtracted from a larger whole number, we get a whole number. For instance, 14 – 8 = 6, 20 = 13 = 7 are whole numbers. But we have no whole numbers to represent 3 – 7, 14 = 18, 20 – 35 and so on. So, we need another set of numbers, other than natural numbers and whole numbers to represent such differences. This is where we introduce the set of integers. So, what are integers an how do we multiply an divide them? Let us find out.

**What are Integers?**

Corresponding to natural numbers, 1, 2, 3, 4, 5 …… etc, we create new numbers, -1, -2, -3, -4 ,-5 and so on. These numbers are called minus one, minus two, minus three etc. such that –

1 + ( – 1 ) = 0

2 + ( – 2 ) = 0

3 + ( – 3 ) = 0

So, – 1 is called the negative of 1, -2 is called the negative of 2 and each negative number is the opposite of its positive counterpart. If we combine these negative numbers with the positive ones, together we get a set of numbers which we call as integers.

So, an integer is a whole numbers that can be positive, negative, or zero. Hence, we can say that integers are the collection of whole numbers and negative numbers. The set of integers is represented by Z and can be written as –

Z = { …….. – 3, – 2 , – 1 , 0 , 1 , 2 , 3 …….. }

Here the numbers, 1, 2, 3, 4 ….. are natural numbers and are called positive integers while the numbers – 1, – 2 , – 3 etc. ate called negative integers. Let us now learn about division and multiplication of integers.

**What is meant by Multiplication of Integers?**

We know that the process of finding out the product between two or more numbers is called multiplication and the result thus obtained is called the **product**. Multiplication of integers is similar to the multiplication of natural numbers and whole numbers, except for the fact that we have to take care of the multiplication of negative numbers as well. So, what is the process of multiplication of integers? Let us find out.

**How do we multiply integers?**

The following rules are followed for multiplication of integers –

**Case 1 –** When you have two integers of opposite signs – The product of two integers of opposite signs is equal to the additive inverse of the product of their absolute values. This means that in order to find the product of a positive and negative integer we need to find the product of the absolute values and assign minus sign to the product. Let us understand it through an example.

**Example**

Suppose you have two numbers 7 and -4 and wish to find the product. The multiplication of 7 and -4 will be given by

7 x ( – 4 ) = – ( 7 x 4 ) = – 28

Similarly, ( – 6 ) x 9 = – ( 6 x 9 ) = = – 54

Case 2 – The product of two integers with similar signs is equal to the product of their absolute values. This means that in order to find the product of two integers where wither both of the numbers are positive or both are negative, we will have to find the product of their absolute values. Let us understand this through an example.

**Example**

Suppose you have two numbers 7 and -4 and wish to find the product. The multiplication of – 7 and -4 will be given by

( – 7 ) x ( – 4 ) = ( 7 x 4 ) = 28

Similarly, ( 6 ) x 9 = ( 6 x 9 ) = = 54

**What is meant by Division of Integers?**

We know that Division is the process of repetitive subtraction. The same applies for division of integers as well. There are four important terms in the division, namely, divisor, dividend, quotient, and the remainder. The formula for divisor constitutes all of these four terms. In fact, it is the relationship of these four terms among each that defines the formula for division. If we multiply the divisor with the quotient and add the result to the remainder, the result that we get is the dividend. This means,

**Dividend = Divisor x Quotient + Remainder**

**How do we divide integers?**

Recall that a division of whole numbers is an inverse process of multiplication. Let us extend the same idea to the division of integers. The following rules are followed for division of integers –

**Case 1 –** The quotient of two integers both positive or negative is a positive integer equal to the quotient of the corresponding absolute values of the integers. This means that for dividing two integers with like signs we divide the values regardless of the sign and give a positive sign to the quotient. Let us understand it through an example.

**Example**

Suppose you have two numbers – 20 and -4 and wish to divide the first integer by the other. We will have,

-20 ÷ -4 = $\frac{20}{4}$ = 5

**Case 2 –** The quotient of a positive and negative integer is a negative integer and its absolute value is equal to the quotient of the corresponding absolute values of the integers. This means that when dividing integers with unlike signs we divide the value of regardless of the sign and give minus sign to the quotient. Let us understand it through an example.

**Example**

Suppose you have two numbers – 20 and 4 and wish to divide the first integer by the other. We will have,

-20 ÷ 4 = – $\frac{20}{4}$ = – 5

**Properties of Multiplication and Division of Integers**

**Commutative Property **

Commutative Property states that when an operation is performed on two numbers, the order in which the numbers are placed does not matter.

**Commutative Property for Multiplication of Integers** – If one integer is multiplied with the other, it does not matter which integer is marked as the multiplier and which number is marked as the multiplicand. Does this hold true for multiplication of integers? Let us find out.

Let us take two integers 8 and 5.

We first mark 8 as the multiplier and 5 as the multiplicand.

We get, 8 x 5 = 40.

Now let us reverse the order, i.e. 5 becomes the multiplier and 8 becomes the multiplicand.

Now, we get, 5 x 8 = 40.

Both the processes give us the same answer.

Therefore, we can say that multiplication satisfies the communicative property. We get

**a x b = b x a**

**Hence, multiplication of integers is communicative. **

**Commutative Property for Division of Integers – **If one number is divided from the other, it does not matter which number is marked as the divisor and which one is the dividend. Does this hold true for division of integers?

Let us find out.

Suppose we have two numbers 10 and 4.

We want to divide 10 from 4. The quotient would be 2.5 and the remainder would be 0.

Now, if we interchange the divisor and the dividend, will we get the same answer? No.

This means that, if we have two numbers, a and b, then

a ÷ b ≠ b ÷ a

**Hence, division is not commutative for integers. **

**Closure Property**

Closure property states that when an operation is performed on two numbers, the resultant would also be of the same type as the numbers on whom the operation has been performed.

**Closure Property for Multiplication of Integers –** If you multiply one integer with the other, the product would also be an integer. Let us check whether the closure property holds true for multiplication of integers?

Consider two numbers, 15 and 3.

Multiply 15 by 3. We get the product as 45as 15 x 3 = 45. In this case, the product is also an integer.

This means that closure property holds true for multiplication of integers. .

Hence we can say that multiplication **of integers satisfies the closure property.**

**Closure Property for Division of Integers – **If you divide one whole number from the other, the quotient would also be a whole number. Let us check whether the closure property holds true for division of integers?

Consider two numbers, 15 and 3.

Divide 15 by 3. We get the quotient as 5, and the remainder as 0, as 3 x 5 = 15. In this case, the quotient is a whole number.

Now divide 26 by 4. We get the quotient as 6.5 and 0 as the remainder. But, 6.5 is not a whole number.

This means that closure property may or may not hold true for division.

Hence we can say that **division of integers does not satisfy the closure property.**

**Associative Property**

Associative Property states that when an operation is performed on more than two numbers, the order in which the numbers are placed does not matter.

**Associative Property for Multiplication of Integers** – If we want to multiply 3 integers, two of them can be chosen first, one as a multiplier and the second as a multiplicand. The result of the multiplication would serve as a multiplier and the third number as multiplicand to get the final answer. Is multiplication of integers is associative? Let us find out.

Let us take 3 numbers, 8, 5 and 2.

Let us take 8 and 5 first.

We first mark 8 as the multiplier and 5 as the multiplicand.

We get, 8 x 5 = 40.

Now, we multiply 40 by 2, we get 40 x 2 = 80

Now let us reverse the order and choose 5 and 2 first.

We get, 5 x 2 = 10

Now, we multiply this result by 8, we get, 10 x 8 = 80

Both the processes give us the same answer.

Therefore, we can say that multiplication satisfies the associative property. We get

**(a x b) x c = a x (b x c)**

**Hence, multiplication of integers is associative. **

**Associative Property for Division of Integers** – Since division is not commutative it is also onto associative as well.

**Multiplicative Property of “0”**

The product of an integer and 0 is always 0. For example,

45 x 0 = 0 x 45 = 0

**Identity Property**

A number is said to be an identity for multiplication, if, a number when multiplied by this identity number results in the number itself. Here, **1 is the identity element for multiplication of integers**. Let us see why?

4 x 1 = 4

-15 x 1 = -15

-20 x 1 = -20

Therefore, any number when multiplied by 1 number results in the number itself. This is the identity property of multiplication.

**Multiplicative Inverse**

A number is said to be multiplicative inverse if a number is multiplied by the multiplicative inverse, the result obtained is the identity of the operation, in this case, 1 ( 1 is the multiplicative identity of all integers).

This means that for all non-zero integers, the numbers when multiplied by their reciprocals will give the answer as 1, i.e.

4 x 1/4 = 1

**Hence **$\frac{1}{a}$** which is also written as a ^{-1} is the multiplicative inverse of a.**

**Distributive Property of Multiplication over Addition / Subtraction**

When two numbers are added or subtracted and the result is multiplied by another number, they can be multiplied separately.

Therefore, for any three integers, a, b and c, distributive property of multiplication over addition states that

**a x ( b + c) = (a x b) + (a x c)**

Similarly, for any three numbers, a, b and c, the distributive property of multiplication over subtraction states that

**a x ( b – c) = (a x b) – (a x c)**

**For example, let us first consider 10 x (18 + 12)**

There are two ways to solve this.

**First Method**

First, we add 18 and 12, we get 30. Now we multiply, 30 and 10, we get 10 x 30 = 300

**Second Method**

Now, we use the distributive property of multiplication over addition.

We have 10 x (18 + 12)

= (10 x 18) + (10 x 12)

= 180 + 120

= 300

By both methods, we get the same answer. Hence,

10 x (18 + 12) = (10 x 18) + (10 x 12)

**Now, let us consider 10 x (18 – 12)**

Again let us solve this by both methods.

**First method**

We find out the difference between 18 and 12. We have 18 – 12 = 6

Now, we multiply 10 and 6. We get 10 x 6 = 60.

**Second Method**

Now, we use the distributive property of multiplication over subtraction.

We have 10 x (18 – 12)

= (10 x 18) – (10 x 12)

= 180 – 120

= 60

By both methods, we get the same answer. Hence,

10 x (18 – 12) = (10 x 18) – (10 x 12)

**Some other Properties of Division**

- If a and b are integers, then a ÷ b is not necessarily an integer.
- If a is an integer different from 0, then a ÷ a = 1
- For every integer, a we have, a ÷ 1 = a
- If a is a non-zero integer, then a ÷ 0 does not exist.

**Key Facts and Summary**

- An integer is a whole numbers that can be positive, negative, or zero.
- The process of finding out the product between two or more numbers is called multiplication and the result thus obtained is called the product.
- The product of two integers of opposite signs is equal to the additive inverse of the product of their absolute values.
- Multiplication of integers is commutative and associative while the division of integers is neither commutative nor associative.
- Multiplication of integers satisfies the closure property while division of integers does not satisfy the closure property.
- If a and b are integers, then a ÷ b is not necessarily an integer.
- If a is an integer different from 0, then a ÷ a = 1
- For every integer, a we have, a ÷ 1 = a
- If a is a non-zero integer, then a ÷ 0 does not exist.
- All properties of operations of all whole numbers are satisfied by these operations on integers.
- To find the product of two integers, we multiply the absolute values and give the result of plus sign if both the numbers are the same sign or a minus sign otherwise.
- To find the quotient of one integer divided by another non zero integer we divide the absolute values and give the result a plus sign if both the numbers are of the same sign or a minus sign otherwise.