Properties of Multiplication of Integers with Solved Examples

Multiplication of Integers is the repeated addition of numbers, that is, a number is multiplied by itself a certain number of times. From the group of positive numbers and negative numbers, including 0, an integer is a number with no decimal or fractional element. Integers include both positive and negative numbers such as -8, 0, 2, 8, 16, 93, and 3,091. Positive Integers include numbers that are bigger than zero like 1, 2, 3, etc. Negative Integers include numbers that are less than zero like -1, -2, -3 etc. Integers can be subjected to the four basic mathematical operations of addition, subtraction, multiplication, and division, as well as the characteristics associated with these operations. In order to multiply any two integers, one must be aware of the properties of multiplication such as commutative property, associative property, etc. 

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Key Words: Multiplication, Integers, Mathematical Operations, Addition, Division, Natural Numbers, Additive Inverse, Multiplicative Identity, Associative Property, Distributive Property

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Multiplication of Integers

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The product of two or more integers is referred to by the characteristics of multiplication of integers. Integers are the set of numbers that consists of natural numbers, the additive inverse of natural numbers, and zero. As a result, integers can be either positive or negative, as seen on a number line. One must always multiply the precise values of integers when doing multiplication, and there are several principles to remember when determining the sign of the final result.

Multiplication is just the process of adding numbers over and over again. For example, 2 multiplied by 3 indicates that 2 has been added to itself three times.

2 x 3 = 2+2+2=6

Like a result, multiplication of integers is simply repeated addition as follows:

a x n= a+a+a………..a (n times)

Read More: Multiplication and Division of Integers

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Properties of Multiplication of Integers

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Integer multiplication has the following properties:

• Closure property
• Commutative property
• Associative property
• Distributive property
• Multiplication by zero
• Multiplicative identity

Some addition qualities, such as commutative and associative properties, are likewise related to multiplication properties. As a result, it is simpler to recall such qualities.

Closure Property of Multiplication

If two integers a and b are multiplied, the resultant ab is also an integer, according to closure property. As a result, integers are closed when multiplied.

a x b is an integer for every integer a and b.

For example: 

• 2 x -1 = -2
• 4 x 5 = 20

Commutative Property of Multiplication

The commutative property of integer multiplication asserts that changing the order of operands or integers has no effect on the multiplication result.

a x b = b x a, where a and b are integers.

For Example:

• 3 x 4 = 4 x 3 (=12)
• 5 x 2 = 2 x 5 (=10)

Associative Property of Multiplication

The outcome of the product of three or more numbers is independent of how these integers are grouped. If a, b, and c are three numbers in general, then

a × (b × c) = (a × b) × c

For example:

• 3 x (4 x 5) = (3 x 4) x 5 (=60)
• -2 x (-1 x -3) = (-2 x -1) x -3 (= -6)

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Distributive Property of Multiplication

If a, b, and c are three integers, then, according to the distributive property of integer multiplication,

a × (b + c) = (a × b) + (a × c)

For example:

2 x (2 + 3) = (2 x 2) + (2 x 3)

2 x 6 = 4 + 6

12 = 12

Multiplication by Zero

The result of multiplying any integer by zero is always zero. If a and b are two integers in general, then

a × 0 = 0 × a = 0

For example:

• 4 x 0 = 0
• -10 x 0 = 0
• 100 x 0 = 0

As can be seen, when any integer, whether the smallest or the largest, is multiplied by zero, the result is always zero.

Multiplicative Identity of Integers

The outcome of multiplying any integer by one is the integer itself. If a and b are two integers in general, then

a × 1 = 1 × a = a

As a result, the Multiplicative Identity of Integers is 1.

For example:

• 23 x 1 = 23
• 44 x 1 = 44
• -79 x 1 = -79
• -105 x 1 = -105

Read More: Additive Identity Vs Multiplicative Identity

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Other Properties of Multiplication

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If a, b, and c are integers, and a > 

b, then a x c > b x c.

For instance, if 5 is more than 4,

5 multiplied by two equals ten.

8 = 4 x 2

Therefore,

4 x 2 >. 5 x 2

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Change of Sign Property

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1. When two positive numbers are multiplied, the result is always positive.
2. When two negative numbers are multiplied, the result is always negative.
3. When a positive integer and a negative integer are multiplied, the result is a negative integer.

For example:

(+2) x (+ 4) = +8

(-2) x (-4) = +8

(-2) x (+4) = -8

Check More: Multiplicative Inverse

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Things to Remember

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• The practise of repeatedly adding positive or negative integers is known as multiplication of integers. Multiplication is defined as the addition of integers over and over again.
• There are three types of integer multiplication, when two positive numbers are multiplied, when two negative numbers are multiplied, the result is a positive integer and when one positive and one negative integer are multiplied.
• Multiplication of Integers has various properties such as Closure property, Commutative property, Associative property, Distributive property, Multiplication by zero and Multiplicative identity. 
• When two positive numbers are multiplied, the result is always positive.
• When two negative numbers are multiplied, the result is always negative.
• When a positive integer and a negative integer are multiplied, the result is a negative integer.

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