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Last updated on July 22nd, 2025

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Integers

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Numbers are arithmetic values used to represent quantities. Integers are a type of number that includes all the positive numbers, negative numbers, and zero. In this topic, we will learn about integers, their types, operations, etc.

Integers for Filipino Students
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What are integers?

Integers are a type of number that includes whole numbers and negative numbers. This means they do not include fractions or decimals. For example,  -6, -5, -3, 0, 1, 2, and 9. Integers are represented using the letter Z. The positive numbers are the numbers greater than zero, that is, 1, 2, 3, 4,… The numbers that are less than zero are the negative numbers, for example, -1, -2, -3, -4,… Zero is a whole number, and it is neither negative nor positive. 
 

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What are the Types of Integers

Based on the type of numbers, integers can be classified into three categories 

  • Zero
  • Positive Integers
  • Negative Integers

 

 

Zero: Zero is a neutral number, as it is neither positive nor negative; it is represented as 0. 

 

 


Positive Integers: Positive integers are numbers that are greater than 0 and represented as Z+. In a number line, it is represented to the right of zero. For example, 1, 2, 3, 4, 5, … 

 

 

Negative Integers: Negative integers are natural numbers with opposite signs, so it is represented as Z-. In a number line, negative integers are listed to the left of zero. For example, -1, -2, -3, -4, -5, …  
 

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Representation of Integers on Number Line

A number line is a straight line that visually represents integers. Integers include all the positive numbers, negative numbers, and zero. In a number line, when representing the integers, zero is marked in the center, representing a neutral point. The numbers on the right are the positive integers, as the values keep increasing. So, the numbers on the left side are negative integers, and the values of the numbers decrease toward the left.   
 

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Operations on Integers

There are four basic arithmetic operations on integers: they are 

  • Addition of Integers
  • Subtraction of Integers
  • Multiplication of Integers
  • Division of Integers

Now, learn them in detail. 

 

 

Addition of Integers


Adding two or more integers is called integer addition; the result we get is the sum. Depending on whether the number is positive or negative, the result may increase or decrease. When adding the integers, make sure to follow these rules. 
When adding the integers of the same sign, add the values and give the same sign to the sum. For example, 5 + 5= 10, -5 + -5 = -10.
When adding the integers of different signs, that is, one positive and the other negative, find the difference of the numbers and give the sign of the larger number to the sum. For example, 5 + (-3) = 2, 9 + (-16) = -7.

 

 

Subtraction of Integers


We subtract two or more numbers to find the difference between two or more numbers. Here, the result depends on the integers being positive or negative. Follow these rules when subtracting two integers,
Change the subtraction to addition by changing the sign of the second number.
Apply the rules of addition of integers.
For example, 5 - 3 = 2
5 - (-3) = 5 + 3 = 8

 

 

Multiplication of Integers


The multiplication of integers is the way of finding the product of two or more numbers. To multiply integers, follow these steps;
(+) × (+) = + – example 5 × 6 = 30
(-) × (-) = + – example (-5) × (-6) = 30
(-) × (+) = - – example (-5) × (6) = -30
(+) × (-) = - – example (5) × (-6) = -30

 

 

Division of Integers


The division of an integer is splitting a number into groups. There are different rules to be followed, such as;
(+) ÷ (+) = + – example 30 ÷ 6 = 5
(-) ÷ (-) = + – example (-30) ÷ (-6) = 5
(-) ÷ (+) = - – example (-30) ÷ (6) = -5
(+) ÷ (-) = - – example (30) ÷ (-6) = -5

Professor Greenline from BrightChamps

Properties of Integers

Integers follow several important properties, such as : 

  • Closure Property
  • Associative Property
  • Commutative Property
  • Distributive Property
  • Additive Inverse Property
  • Multiplicative Inverse Property
  • Identity Property

 

 

Closure Property: The set is closed under an operation if performing that operation on any two elements of the set results in another element within the set. For any two integers, p and q
p + q ∈ Z 
p - q ∈ Z
p × q ∈ Z

 

 

Associative Property: The associative property states that grouping of integers in both addition and multiplication does not affect the result. That is, p + (q + r) = (p + q) + r and p × (q × r) = (p × q) × r. 

 

 

Commutative Property: The Commutative property states that the order of the integers does not alter both addition and multiplication. For example, p + q = q + p and p × q = q × p. 

 

 

Distributive Property: The product of a number with two addends will be the same as the sum of the products of the number with the addends is the distributive property. That is p × (q + r) = (p × q) + (p × r)
 

 

Additive Inverse Property: The sum of any positive integer and its negative integer is zero, which means p + (-p) = 0.

 

 

Multiplicative Inverse Property: Except for 1 and -1, most integers do not have multiplicative inverses within the set of integers. 

 

 

Identity Property: Identity property states that the sum of any number with 0 is equal to the number itself. That is, a + 0 = a. Same as the product of an integer with 1 is equal to the integer itself. That is, a × 1 = a
 

Professor Greenline from BrightChamps

Real-world Applications of Integers

In the real world, we use integers in different fields, from balancing budgets to measuring temperature. Now, let’s learn some applications of integers. 

 

 

  • Integers are used to represent gains and losses in temperature, elevation, navigation, and so on.

 

  • In population studies, we use integers to represent the number of individuals in a region.

 

  • In computer science and programming, we use integers for efficient computation and logical control of algorithms, software applications, and digital systems.

 

  • To record the credits and debits in financial transactions, we use integers. 
     
Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in Integers

When working with integers, students usually tend to confuse them, which leads to errors. Now let’s learn a few common mistakes and ways to avoid them in integers. 
 

Mistake 1

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Confusing Positive and Negative Signs

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Students mostly get confused with positive and negative signs during calculations. So when doing the calculations, always complete the calculations first and double-check the answers. 
 

Mistake 2

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 Errors in the Addition of Negative Numbers
 

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When adding two negative numbers, the common error that they make is adding the absolute values and adding the positive sign to the answer. For example, (-5) + (-8) = 13 instead of -13. Students should always remember that when adding an integer with the same sign, we first add the numbers and give the same sign. 

Mistake 3

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Assuming 0 is Positive or Negative
 

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Students used to think that 0 can be a positive and a negative number, but it is wrong, as 0 is neutral; it is neither a positive nor a negative number. 

Mistake 4

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Ignoring the Rules of Exponents with Negative Bases
 

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Confusion with -42 and (-4)2 is common among students; the placement of the parentheses makes a significant difference. So it is important to understand the parentheses, that is, 42 is interpreted as -(4)2 = -16, and (-4)2 is interpreted as -4 × -4 = 16. 

Mistake 5

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Thinking that Decimals are Integers
 

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Students usually get confused between integers and decimals; they tend to think that decimals and fractions are a part of integers, which is wrong. Integers consist of only positive numbers, negative numbers, and zero. That is -8, -5, -3, 0, 1, 2, 5.

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Solved Examples of Integers

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Problem 1

Multiply -4 by 6.

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 -4 × 6 = -24
 

Explanation

The product of a negative and a positive integer is a negative integer. So, -4 × 6 = -24
 

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Problem 2

Subtract -3 from -8.

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(-8) - (-3) = -5
 

Explanation

(-8) - (-3) = -8 + 3 = -5.
 

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Problem 3

What is the sum of -7 and 4?

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 -7 + 4 = -3
 

Explanation

 The sum of a negative number and a positive number is the difference between the numbers, that is -7 + 4 = -3
 

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Problem 4

Divide -18 by 6.

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 -18 ÷ 6 = -3
 

Explanation

A negative divided by a positive gives a negative result, so -18 ÷ 6 = -3
 

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Problem 5

Find the product of -9 and -3.

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-9 × -3 = 27
 

Explanation

The product of two negative integers is positive, so -9 × -3 = 27

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FAQs on Integers

1.What are integers?

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2.Is zero an integer?

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3.What is the difference between whole numbers and integers?

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4.What are the rules for multiplying integers?

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5.Are fractions and decimals integers?

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6.How can children in Philippines use numbers in everyday life to understand Integers?

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7.What are some fun ways kids in Philippines can practice Integers with numbers?

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8.What role do numbers and Integers play in helping children in Philippines develop problem-solving skills?

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9.How can families in Philippines create number-rich environments to improve 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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