Introduction to Multiplication Properties of Integers
Multiplication of integers follows specific properties that help in simplifying calculations and understanding mathematical concepts. Here are the key properties:

1. Commutative Property
- The order of multiplication does not affect the product.
- Example: \( a \times b = b \times a \)
- For integers: \( 3 \times 5 = 5 \times 3 = 15 \)

2. Associative Property
- The way in which integers are grouped in multiplication does not change the product.
- Example: \( (a \times b) \times c = a \times (b \times c) \)
- For integers: \( (2 \times 3) \times 4 = 2 \times (3 \times 4) = 24 \)

3. Distributive Property
- Multiplication distributes over addition and subtraction.
- Example: \( a \times (b + c) = a \times b + a \times c \)
- For integers: \( 2 \times (3 + 4) = 2 \times 3 + 2 \times 4 = 14 \)

4. Identity Property
- The identity element for multiplication is 1; multiplying any integer by 1 gives the same integer.
- Example: \( a \times 1 = a \)
- For integers: \( 7 \times 1 = 7 \)

5. Zero Property
- Any integer multiplied by 0 results in 0.
- Example: \( a \times 0 = 0 \)
- For integers: \( 5 \times 0 = 0 \)

Conclusion
Understanding these properties is crucial for performing arithmetic operations efficiently and accurately. These properties not only simplify calculations but also form the foundation for more advanced mathematical concepts.

Properties of Integers
The properties of integers make calculations easier and faster. Integers are a set of numbers that include natural numbers, zero, and negative numbers. They do not have any fractional parts. Let us learn about the properties of integers in this article.

What are the Properties of Integers?
The following points show the list of the various properties of integers:

Closure Property
Associative Property
Commutative Property
Distributive Property
Identity Property

These properties are applicable to whole numbers, natural numbers, rational numbers, and real numbers. Observe the figure given below to have an overview of the applicability of different properties of integers on the four basic arithmetic operations on integers. A detailed explanation of each of the properties is given in the sections below.


Closure Property of Integers
The closure property of integers states that the addition, subtraction, and multiplication of two integers always results in an integer. So, this implies if {a, b} ∈ Z, then c ∈ Z, such that

a + b = c
a - b = c
a × b = c
The closure property of integers does not hold true for the division of integers as the division of two integers may not always result in an integer. For example, we know that 3 and 4 are integers but 3 ÷ 4 = 0.75 which is not an integer. Therefore, the closure property is not applicable to the division of integers. It holds true for the addition, subtraction, and multiplication of integers. Let us look at some examples of closure property of integers given below:

-2 + 7 = 5, where {-2, 5, 7} ∈ Z.
-9 - (-8) = -1, where {-9, -8, -1} ∈ Z.
-7 × 0 = 0, where {-7, 0} ∈ Z.


Associative Property of Integers
The associative property of integers under addition and multiplication states that the result of the addition and multiplication of more than two integers is always the same irrespective of the grouping of integers. This implies that for any three integers a, b, and c, we have,

a + (b + c) = (a + b) + c = (a + c) + b
a × (b × c) = (a × b) × c = (a × c) × b
The associative property of integers does not hold true for subtraction and division of integers, as, in the case of subtraction and division, the order of the numbers is important and cannot be changed. For example, 2 - (8 - 9) = 2 - (-1) = 3. Now, if we change the order as 8 - (2 - 9) = 8 - (-7) = 15. Therefore, 2 - (8 - 9) ≠ 8 - (2 - 9). The examples of associative property of integers are given in the table below:

Operation Example Result
Addition - 8 + (4 + 2) = (-8 + 4) + 2 = -2 Satisfied
Subtraction
- 8 - (4 - 2) = -8 - 2 = -10

(-8 - 4) - 2 = - 12 - 2 = -14

Does not hold true
Multiplication - 8 × (4 × 2) = (-8 × 4) × 2 = - 64 Satisfied
Division
- 8 ÷ (4 ÷ 2) = -4

(-8 ÷ 4) ÷ 2 = -1



Commutative Property of Integers
The commutative property of integers is similar to the associative property; the only difference is that in this property, we take only two integers. The commutative property of integers under addition and multiplication states that the result of the addition and multiplication of two integers is always the same regardless of their order. This implies, if there are two integers a and b, we have,

a + b = b + a
a × b = b × a
This property does not hold true with subtraction and division operations. Let us look at the examples of the commutative property of integers under addition, subtraction, multiplication, and division.

Operation Examples Result
Addition -6 + 3 = 3 + (-6) = -3 Satisfied
Subtraction
-6 - 3 = -9

3 - (-6) = 9

Does not hold true
Multiplication -6 × 3 = 3 × (-6) = -18 Satisfied
Division
-6 ÷ 3 = -2

3 ÷ (-6) = -1/2

Does not hold true


Distributive Property of Integers
The distributive property of integers states that the multiplication operation can be distributed over addition and subtraction to make calculations easier. This implies, for any three integers, a, b, and c, we have,

a × (b + c) = (a × b) + (a × c)
a × (b - c) = (a × b) - (a × c)
This property has many applications in math. We can use it for mental calculations. For example, what is the value of -6 × 98? This can be written as -6 × (100 - 2). Now we can apply the distributive property of integers on this to get (-6 × 100) - (-6 × 2) = -600 - (-12) = - 600 + 12 = -588.

Identity Property of Integers
The identity property of the addition of integers states that any number added to 0 results in the same number. For an instance, if 'a' is any integer, this implies, a + 0 = 0 + a = a. Let us take an example of a negative integer -5. If we add 0 to -5, we will get -5. There is no change in the result. So, we can say that 0 is the identity element of the addition of integers.

In the case of multiplication of integers, can you think of any number which we can multiply with an integer to get the same integer as the product? Yes, the multiplicative identity element for integers is 1. We can multiply 1 to any integer to get the same result. For example, a × 1 = 1 × a = a.

The identity property of integers does not hold true for subtraction and division operations. In the case of subtraction, if we subtract any number from 0, we will get its additive inverse. So, if 'p' is any integer, then p - 0 = p, but 0 - p = -p. In the case of division of integers, if 'm' is any integer, then m ÷ 1 = m, but 1 ÷ m ≠ m. Therefore, there is no identity element for subtraction and division of integers.