Properties of Integers: Closure, Commutative & Associative Property

Integers have a few characteristics that affect how they are applied and equations can be solved using these principles and properties. Any positive or negative number, including 0, is an integer. The properties of these integers will assist in efficiently simplifying and answering a series of integer operations. All of the integers share the same properties and identities as the addition, subtraction, multiplication, and division of numbers. Positive numbers, zero, and negative numbers are all referred to as integers, and they are represented by the symbol Z.

Read Also: Math Formulas

Key takeaways: Properties of Integers, Addition, integers, subtraction, multiplication

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What are Integers?

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Any number that comes in the set or category of whole numbers, natural numbers, or negative numbers, are known as integers. We can also say that whole numbers, natural numbers, and negative numbers are all integers.

Integers can be represented as:

(…………..-5,-4,-3,-2,-1,0,1,2,3,4,5,……………….)

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Properties of Integers

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Five properties are possessed by the integers. They include:

• Closure Property
• Associative Property
• Commutative Property
• Distributive Property
• Identity Property

Closure Property

The closure property in the integers defines that in performing any operation be it addition, subtraction or multiplication if m and n are two integers then the result that is generated will also be an integer. That is, m+n, m-n, and mn all three would be an integer.

Example of closure property in addition:

(-3)+5=2

Example of closure property in subtraction:

1-2=1+(-2)=-1

Example of closure property in multiplication:

3×4=12

-3×4=-12

However, in the division, the closure property is not applicable as the result generated is not an integer.

Example to show that closure property does not work in case of division:

(-2)÷(-4)=12 which is not an integer.

Associative Property

Associative property of integers in case of addition and multiplication states that if there are any three integers and we form the group of any two integers, the result will be the same, irrespective of the selection of integers that is made for the grouping. For an instance, if p,q, and r are two integers,

• p+(q+r)=(p+q)+r
• p× (q×r)=(p×q) ×r

Example of associative property in addition:

2+(5+8)=(2+5)+8

2+(13)=(7)+8

15=15

Example of associative property in multiplication:

9×(2×4)=(9×2) ×4

9×(8)=(18) ×4

72=72

However, it is not true for the two operations that are division and subtraction as

• p-(q-r) ≠ (p-q)-r
• p÷ (q÷r) ≠ (p÷q) ÷r

To demonstrate why the associative property does not work in the case of subtraction, consider the following example:

2-(4-8) and (2-4)-8

2-(-4) and (-2)-8

6 and -10 which is not the same

Example to show that associative property does not work in case of division:

2÷(4÷8) and (2÷4) ÷8

2÷(1/2) and (1/2) ÷8

4 and 1/16 which is not the same

Commutative Property

The commutative property of the integers in case of addition and multiplication defines that whatever be the order of integers in the operation, the result obtained will be the same, that is it will remain unchanged. Swapping of any integer in the operation will not impact the result. For example, m and n are the two integers, then:

• m+n = n+m
• m×n = n×m

Example of commutative property in addition:

3+(-6)=(-6)+3=-3

Example of commutative property in multiplication:

8×3=3×8=24

However, it is not applicable in the case of division and subtraction as

• m-n ≠ n-m
• m/n ≠ n/m

Example to show that commutative property does not work in case of subtraction:

7-4 and 4-7

3 ≠ -3

Example to show that commutative property does not work in case of division:

10÷2 and 2÷10

5≠1/5

Check Important Differences Between in Maths

Distributive Property

According to the distributive property of integers, the operation can be performed over other operations within a bracket. It can be applied in both the distributive property of multiplication over addition and the distributive property of multiplication over subtraction. The operations can be first multiplied followed by the addition or subtraction or can be first added or subtracted followed by the multiplication. If p,q,r can be represented as integers:

• p×(q+r)=p×q+q+r
• p×(q-r)=p×q-p×r

Example to depict distributive property:

6×(1+9)= 6×1+6×9=6+54=60

7×(3-2)=7×3-7×2=21-14=7

Identity Property

According to the identity property, the additive identity property is when in addition zero is added to an integer, it yields the same integer. Therefore, additive identity also refers to zero. If m is an integer:

• m+0=0+m

Ex: 3+0=3+0=3

In the case of multiplication, if any integer is multiplied by 1, it yields the same integer. It is termed multiplicative identity. Therefore, multiplicative identity also refers to one. If m is an integer:

• m×1=1×m

Ex: 5×1=1×5=5

When zero is multiplied with any integer, it gives the product of 0.

• m×0=0×m

Ex: 8×0=0×8=0

When -1 is multiplied with any integer, it gives the result as an opposite integer.

• m×(-1)=-m=(-1) ×m

Ex: 9×(-1)=-9=(-1) ×9

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Things to Remember

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• Integers are the numbers that come in the set of whole numbers, integers, or natural numbers.
• For closure property, the addition or subtraction of any two integers is always an integer.
• In commutative property, the order of the terms in an operation does not change the result.
• An associative property, the grouping of numbers does not yield a different result irrespective of the choices of the numbers.
• In distributive property, the operations can be performed over the other operations in a particular bracket.

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