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Last updated on July 4th, 2025

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Properties of Integers

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Integers are the numbers that include both whole numbers and their negatives, that is …. -8, -5, -3, 0, 3, 5, 8 … To make calculations easier and faster, we use the properties of integers. In this topic, we will learn about the different properties of integers.

Properties of Integers for UAE Students
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What are Integers?

The set of numbers that includes the whole numbers and negative numbers are the integers. For example, -3, -2, -1, 0, 1, 2, 3. Integers are a subset of rational numbers, but they don't include fractions or decimals. The only integer that is neither positive nor negative is 0, and it is denoted by the letter “Z”.


The integers can be classified as: positive integers, negative integers, and zero. Let’s learn about them in detail:

 

  • Positive integers: Whole numbers that are greater than 0, that is 1, 2, 3, 4, 5, …. 

     
  • Negative integers: Numbers that are less than zero, that is -1, -2, -3, -4, … 

     
  • Zero: Zero is the only integer that is neither positive nor negative. It is represented as 0
     
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What are the Properties of Integers?

Now that we know what integers are, let's learn about the properties that integers follow. Integers follow a few mathematical properties that help make calculations easier. These properties include: 

 

  • Closure property
     
  • Associative property
     
  • Commutative property
     
  • Distributive property
     
  • Identity property

     
    Property Definition Example
    Closure

    The closure property of integers states that the sum, difference, and product of integers will always result in another integer. It does not apply to division. 
    That is, a + b ∈ Z,
    a - b ∈ Z,
    a × b ∈ Z.

    • (-3) + 5 = 2, where -3, 2, 5 are integers
    • 7 × (-2) = -14, where -14, -2, 7 are integers 
    • 9 - (-5) = 14, where -5, 9, 14 are integers
    Commutative  The commutative property of integers states that changing the order of numbers does not affect the result in addition and multiplication. It is not applicable for subtraction and division. 
    That is a + b = b + a,
    a × b = b × a,
    a - b ≠ b - a, 
    a ÷ b ≠ b ÷ a. 
     
    • (-6) + 5 = 5 + (-6) = -1
    • (-8) × 6 = 6 × (-8) = -48
    Associative

    The associative property states that the way of grouping the integers does not affect the result in addition and multiplication. That is,
    a + (b + c) = (a + b) + c = (a + c) + b,
    a × (b × c) = (a × b) × c = (a × c) × b.

    • 2 + (3 + 5) = (2 + 3) + 5 = (2 + 5) + 3 = 10
    • 5 × (4 × 6) = (5 × 4) × 6 = (5 × 6) × 4 = 120
    Distributive

    The distributive property of integers states that the product of multiplying a number with the sum or difference of two numbers is equal to the sum or difference of the product of each addend separately. It applies to addition and subtraction. That is 
    a × (b + c) = ab + ac,
    a × (b - c) = ab - ac.

    • 3 × (4 + 5) = (3 × 4) + (3 × 5) = 12 + 15 = 27
    • 2 (6 - 3) = 2 × 6 - 2 × 3
    • = 12 - 6 = 6
    Identity

    The identity property is applicable for addition and multiplication. 

    • The identity property of addition states that the sum of any integer with 0 is the same integer. 
       
    • The identity property of multiplication: In multiplication, the product of any integer with 1 is the integer itself. 

    That is a + 0 = a, (-a) + 0 = -a,
    a × 1 = a, (-a) ×1 = -a.

    • 2 + 0 = 2
    • -5 + 0 = -5
    • 6 × 1 = 6
    • -6 × 1 = -6

     

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Real-world Applications of Properties of Integers

In the real world, integers are used to represent quantities such as temperature, financial transactions, engineering calculations, and more. In this section, let’s learn a few real-world applications of the properties of integers. 

 

  • The commutative property of integers is used when splitting a bill among friends or groups. For example, if three friends need to pay a total of $10, and they contribute $2, $3, and $5, the total remains the same no matter who contributes which amount.  

     
  • To combine the expenses of a group we use associative property For example, when adding the expense, the order of grouping will not affect the result. 

     
  • Integers are used in sports to express the scores, points, players statistics, and ranking. 

     
  • In transportation, integers are used to calculate the fuel consumption and plan the scheduling and logistic controls. 
     
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Common Mistakes and How to Avoid Them in Properties of Integers

Integers and their operations can be tricky for students, as integers have both positive and negative numbers. So let’s learn a few common mistakes and ways to avoid them to master properties of integers. 
 

Mistake 1

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Confusing the sign rule for both addition and subtraction

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Students usually confuse or misapply the rules for adding and subtracting integers, such as (-5) + 3 = -8 instead of -2. To avoid this, students should focus on understanding the basic rules of adding and subtracting positive and negative integers.
 

Mistake 2

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Mixing up the sign rules for multiplication and division

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Mixing up the sign rules for multiplication and division is common among students, for example (-5) × (-7) = -35 instead of 35. That is, they confuse the rules of multiplication and division. To avoid this, students should understand the same sign gives positive and different signs give negative signs. 

Mistake 3

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Errors with the sign when simplifying expressions

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Confusion is common among students when dealing with the negative sign, that is (-2) - 3 + (-4) = -1 instead of -9. To avoid this, students should write the calculation step by step and always try to recheck the answer. 
 

Mistake 4

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Thinking that all operations with integers result in a positive value
 

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Students assume that the operations with integers always result in positive values, such as (-8) - 2 = 6 instead of -10. So try to use the real-world situation and use the number line to solve the integers. 

Mistake 5

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Ignoring the order of operations in integers
 

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It is common among students to ignore the order of operations in integers, that is -3 + 5 × -2, they might do the addition first and multiplication next, which is wrong. They should follow either the BODMAS or PEMDAS and understand the operation with parentheses. 
 

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Solved Examples of Properties of Integers

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Problem 1

Verify whether the following expression is correct or not (-5) × 8 and 8 × (-5)

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The expression (-5) × 8 and 8 × (-5) is correct. 
 

Explanation

As the commutative property is multiplication, the order of the number does not affect the results. 


So, (-5) × 8 = -40


8 × (-5)  = -40


As the product is the same, the expression (-5) × 8 and 8 × (-5) is correct. 
 

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Problem 2

Evaluate the expression using distributive property: (-6 × 5) + (- 6 × 9)

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 (-6 × 5) + (- 6 × 9) = -84
 

Explanation

Distributive property states that, 


a × b + a × c = a(b +c) 


So, (-6 × 5) + (- 6 × 9) = -6(5 + 9) 


= -6 × 14 = -84. 
 

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Problem 3

Find the additive inverse of -65.

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The additive inverse of -65 is 65
 

Explanation

The additive inverse of an integer -a is a. So, the additive inverse of -65 is 65. As -65 + 65 = 0
 

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Problem 4

David has $50, but he doesn’t spend or earn any money. How much money does he have now?

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The money David still has is $50
 

Explanation

The identity property states that the sum of any number with 0 is the number itself. 


As he didn’t earn any money, the money David has is,


$50 + 0 = $50
 

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Problem 5

Find the value of x in (-12) + x = (-12) + 24

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The value of x is 24
 

Explanation

 The commutative property states that the order of the numbers in an addition does not impact the result. 


So, the value of x in (-12) + x = (-12) + 24 is 24
 

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FAQs on Properties of Integers

1.List out the properties of integers.

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2.What are integers?

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3.Can integers be fractions or decimals?

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4.What are the rules for multiplying integers?

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5.What does the symbol Z represent?

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6.How can children in United Arab Emirates use numbers in everyday life to understand Properties of Integers ?

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7.What are some fun ways kids in United Arab Emirates can practice Properties of Integers with numbers?

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8.What role do numbers and Properties of Integers play in helping children in United Arab Emirates develop problem-solving skills?

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9.How can families in United Arab Emirates create number-rich environments to improve Properties of Integers skills?

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

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Fun Fact

: She loves to read number jokes and games.

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