• Integers are numbers that include all positive numbers, negative numbers, and zero. They do not have any decimal or fractional parts. Understanding integers is a basic part of math and helps in learning algebra. Some examples of integers are -5, 0, 3, and 20. Learning the properties of integers helps us understand how they work in different situations like addition, subtraction, and multiplication.

Properties of Integers

Closure Property of Integers

• The closure property means that a set is closed for some mathematical operation. That is, if the operation can always be completed with elements in the set, the set is closed with respect to that operation. As a result, a set is either closed or open with regard to a certain operation.
  • Addition of Integers: The sum of two integers is always an integer. If a and b are any two integers, then a + b = is also an integer. E.g: 19 + 23 = 42.
  • Subtraction of Integers: The difference between two integers is always an integer. If a and b are any two integers, then a – b = is also an integer. E.g: 19 – 23 = -4.
  • Multiplication of Integers: The product between two integers is always an integer. If a and b are any two integers, then a x b = is also an integer. E.g: 3 x 4 = 12.

Division of Integers: The division of two integers is always an integer. If a and b are any two integers, then a/b = is also an integer. Example: 4/2 = 2.

The commutative property is a math rule that says that the order in which we add, multiply, subtract or divide the numbers does not change the product.

• Addition of Integers: Addition of two integers is commutative in nature. If a and b are any two integers, then a + b = b + a E.g: 2 + 4 = 6 & 4 + 2 = 6 So, 2 + 4 = 4 + 2.
• Subtraction of Integers: Subtracting two integers is not commutative in nature. If a and b are any two integers, then a – b ≠ b – a E.g: 2 – 4 = -2 & 4 – 2 = 4 So, 2 – 4 ≠ 4 – 2.
• Multiplication of Integers: Multiplication of two integers is commutative in nature. If a and b are any two integers, then a x b = b x a E.g: 3 x 4 = 12 & 4 x 3 = 12 So, 3 x 4 = 4 x 3
• Division of Integers: Dividing two integers is not commutative in nature. If a and b are any two integers, then a/b≠b/a
  Example: 4/2=2 and 24=0.5, so 4/2≠2/4

Associative Property of Integers

This law simply states that with addition and multiplication of numbers, you can change the grouping of the numbers in the problem and it will not affect the answer.

• Addition of Integers: Addition of integers is associative in nature. If a, b and c are any three integers, then a + (b + c) = (a + b) + c
• Example : 2 + (4 + 1) = 2 + 5 = 7 and (2 + 4) + 1 = 6 + 1 = 7 So, 2 + (4 + 1) = (2 + 4) + 1 using Associative Property of Addition.
• Subtraction of Integers: Subtraction of integers is not associative in nature. If a, b and c are any three integers, then a – (b – c) = (a – b) – c Example : 2 – (4 – 1) = 2 – 3 = -1 and (2 – 4) – 1 = -2 – 1 = -3 So, 2 – (4 – 1) ≠ (2 – 4) – 1
• Multiplication of Integers: Multiplication of integers is associative in nature. If a, b and c are any three integers, then a x (b x c) = (a x b) x c Example : 2 x (4 x 1) = 2 x 4 = 8 and (2 x 4) x 1 = 8 x 1 = 8 So, 2 x (4 x 1) = (2 x 4) x 1
• Division of Integers: Dividing two integers is not commutative in nature. If a and b are any two integers, then a/b≠b/a Example : 8/2=4 and 28=0.25, so 8/2≠2/8

Identity Property of Integers

An identity is a number that when added, subtracted, multiplied or divided with any number (let’s call this number n), allows n to remain the same. This is also a property of whole numbers

• Additive Identity: The sum of any integer and zero is the rational number itself. If a is any integer, then a + 0 = 0 + a = a as zero is the additive identity for rational numbers. E.g : 2 + 0 = 0 + 2 = 2
• Multiplication Identity: The product of any integer and 1 is the integer itself. ‘One’ is the multiplicative identity for integers. If a is any integers, then a x 1 = 1 x a = a E.g: 5 x 1 = 1 x 5 = 5

Distributive Property of Integers

Distributive Property is used to solve expressions easily by distributing a number to the numbers given in brackets.

• Distributive Property of Multiplication over Addition: Multiplication of integers is distributive over addition. If a, b and c are any three rational numbers, then a x (b + c) = a x b + a x c E.g 1 x (2 + 1) = 1 x 3 = 1 then 1 x 2 + 1 x 1 = 2 + 1 = (2 + 1) = 3 Therefore, Multiplication is distributive over addition.
• Distributive Property of Multiplication over Subtraction: Multiplication of integers is distributive over subtraction. If a, b and c are any three rational numbers, then a x (b – c) = a x b – a x c E.g: 1 x (2 – 1) = 1 x 1 = 1 Therefore, 1 x 2 – 1 x 1 = 2 – 1 = (2 – 1) = 1 So 1 x (2 – 1) = 1 x 2 – 1 x 1 Hence, Multiplication is distributive over subtraction.

Types of Integers

Integers are whole numbers that can be positive, negative, or zero. They can be divided into three main types based on how they behave when divided by each other.

1. Positive ÷ Positive = Positive
   When you divide a positive number by another positive number, the result is always positive.
   Example: 6 ÷ 3 = 2

2. Negative ÷ Negative = Positive
   If you divide a negative number by another negative number, the result is positive as well.
   Example: -6 ÷ -3 = 2

3. Positive ÷ Negative = Negative
   When you divide a positive number by a negative number, or a negative number by a positive number, the result is always negative.
   Example:
   6 ÷ -3 = -2
   or
   -6 ÷ 3 = -2

Properties of Integers Solved Examples

Let’s see some examples of Properties of Integers that come in exams.

Solved Example 1: Find two consecutive integers whose sum is equal 129.

Solution: Let x and x + 1 (consecutive integers differ by 1) be the two numbers. Use the fact that their sum is equal to 129 to write the equation.

x + (x + 1) = 129

Solve for x to obtain

x = 64

The two numbers are

x = 64 and x + 1 = 65

We can see that the sum of the two numbers is 129.

Solved Example 2: The sum of three consecutive even integers is equal to 84. Find the numbers.

Solution: The difference between two even integers is equal to 2. Let x, x + 2 and x + 4 be the three numbers. Their sum is equal to 84, hence

x + (x + 2) + (x + 4) = 84

Solve for x and find the three numbers

x = 26 , x + 2 = 28 and x + 4 = 30

The three numbers are even. Check that their sum is equal to 84.

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