Last updated on July 4th, 2025
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.
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:
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:
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. |
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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. |
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Associative |
The associative property states that the way of grouping the integers does not affect the result in addition and multiplication. That is, |
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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 |
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Identity |
The identity property is applicable for addition and multiplication.
That is a + 0 = a, (-a) + 0 = -a, |
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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.
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.
Verify whether the following expression is correct or not (-5) × 8 and 8 × (-5)
The expression (-5) × 8 and 8 × (-5) is correct.
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.
Evaluate the expression using distributive property: (-6 × 5) + (- 6 × 9)
(-6 × 5) + (- 6 × 9) = -84
Distributive property states that,
a × b + a × c = a(b +c)
So, (-6 × 5) + (- 6 × 9) = -6(5 + 9)
= -6 × 14 = -84.
Find the additive inverse of -65.
The additive inverse of -65 is 65
The additive inverse of an integer -a is a. So, the additive inverse of -65 is 65. As -65 + 65 = 0
David has $50, but he doesn’t spend or earn any money. How much money does he have now?
The money David still has is $50
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
Find the value of x in (-12) + x = (-12) + 24
The value of x is 24
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
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.
: She loves to read number jokes and games.