Integers

Introduction to Integers

An integer is any entire number, including positive numbers, negative numbers & zero, without including fractions or decimals. They help represent real-life conditions such as temperature, loss of wealth or profits, and levels. Important rules for the intensification of guide operations such as addition, subtraction, multiplication, and division. Adding integers with the same sign, for example, gives a result with the same sign, while subtracting a larger number from a smaller one gives a negative result. Understanding these integer operations through clear examples such as 4, 0, and 5 strengthens basic mathematics skills.

Table of Contents

• What Are Integers?

• Set of Integers and Notation

• Understanding Integers on a Number Line

• Integer Operations: Addition, Subtraction, Multiplication, Division

  • Addition of Integers

  • Subtraction of Integers

  • Multiplication of Integers

  • Division of Integers

• Properties of Integers with Examples

  • Closure Property

  • Commutative Property

  • Associative Property

  • Distributive Property

  • Identity Property

• Applications of Integers in Daily Life

• Solved Integer Examples and Practice Questions

• Conclusion

• Related Links

• FAQs on Integers
  

What Are Integers?

The most important number is integers in mathematics is one of the sets. They are used to represent all numbers in both directions on the number line, including homogeneous and negative zero. From measuring height to monitoring financial benefits and disadvantages, integers play an important role in daily life. Let's understand the definition of integers, find out what types of integers and learn about the integer note (Z).

Definition of integer

The integer is a set of whole numbers that include:

• All positive numbers (also known as natural numbers),

• All negative numbers, and

• Zero (0).

They do not include parts, decimals, or any part of the whole number.

Types of Integers: Positive, Negative, and Zero

Integers are categorized into 3 main types based on their value :

Set of Integers and Notation

In mathematics, the set of integers is represented by the Capital Letter Z, which is derived from the German word Zahlen, which means "number".

Symbolic representation: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Properties of an integer set:

• Infinite in both directions.

• Closed under addition, subtraction, and multiplication.

• Symmetrical around 0 on the number line.

Key points:

• Z⁺: set of positive integer = {1, 2, 3, ...}

• Z⁻: Set of negative integer = {..., -3, -2, -1}

• Z: Complete set of integers = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Understanding Integers on a Number Line

The integer is best visualized using a number line, which helps students understand the concept of positive and negative values, direction, and distance. The straight line for integers is a straight line where the number is placed in increasing order. This simple representation helps students compare integers, operate, and understand their relative values.

Integer Operations: Addition, Subtraction, Multiplication, Division

Operations on integers follow the specific rules that depend on the signs of the numbers involved. Mastering these integer operations helps to solve real-life problems and create a strong foundation in arithmetic and algebra. Whether you add, subtract, multiply, or divide the integer, it is necessary to use the correct rules for the integer.

Addition of integers

Adding the integer involves a combination of values,​​considering their signals.

Rules for adding integers with the same or different signs

• Same signal: Add absolute values ​​and hold the same signal.

• Different signs: Subtract the smaller absolute value from the larger and place the sign of the numbers with a large absolute value.

Examples:

• (+4) + (+3) = +7

• (-5) + (-2) = -7

• (+6) + (-9) = -3 (since 9 > 6, result is negative)

Subtraction of Integers

Subtracting integers is the process of finding the difference between them. The easiest way is to add the opposite.

Subtracting  integers by adding the opposite:

• Make the subtraction of addition.

• Change the second number symbol.

Examples:

• 5 − 3 = 5 + (−3) = 2

• −4 − (−6) = −4 + 6 = 2

• 7 − (−2) = 7 + 2 = 9

Multiplication of Integers

Multiplying integers depends entirely on the signs of the numbers.

Multiplying Integers with Sign Rules:

Examples:

• (+4) × (+3) = +12

• (−2) × (−5) = +10

• (+6) × (−3) = −18

Division of Integers

Division of integers follows the same sign rules as multiplication.

Division of Integers: Positive and Negative Results:

Examples:

• (+12) ÷ (+4) = +3

• (−15) ÷ (−3) = +5

• (+18) ÷ (−6) = −3

Integer Operations Using a Number Line

The number line helps imagine basic operations with integers.

How to use the number bar:

• Addition: Start at number one, go right to positive, and to the left of negative.

• Subtraction: Convert to the addition of the opposite and follow the same rule.

Examples:

• 3 + (−2): Start at 3, move 2 units left → Result = 1

• −4 − (−3): Change to −4 + 3 → Start at −4, move 3 units right → Result = −1

Quick Reference Table for Integer Operations

Properties of Integers with Examples

The integer follows many basic mathematical properties necessary to simplify and operate mathematical manifestations. These properties of integers use additional arithmetic operations such as addition and multiplication. Understanding these rules allows students to systematically solve problems and have a strong basis for algebra and higher mathematics.

Closure Property

The closure property states that when two integers are added, subtracted, or multiplied, the result is always an integer.

Closure Property Rules:

• Addition: If a and b are integers, then a + b is also an integer.

• Multiplication: If a and b are integers, then a × b is also an integer.

Examples:

• 4 + (−5) = −1 (Integer) 

• (−3) × 2 = −6 (Integer)

• (−8) ÷ 4 = −2 (Integer)

• 5 ÷ 2 = 2.5  (Not an integer)

Commutative Property

The commutative property applies to addition & multiplication, where changing the order of the numbers does not change the result.

Examples:

• 6 + (−2) = 4 = (−2) + 6 

• (−3) × 7 = −21 = 7 × (−3)

Associative Property

The associative property suggests that when you add or multiply integers, the grouping of numbers does not affect the result.

Associative Property Rules:

• Addition: (a + b) + c = a + (b + c)

• Multiplication: (a × b) × c = a × (b × c)

 Examples:

• (−1 + 2) + 3 = 1 + 3 = 4
  = −1 + (2 + 3) = −1 + 5 = 4

• (2 × 3) × 4 = 6 × 4 = 24
  = 2 × (3 × 4) = 2 × 12 = 24 

Distributive Property

The distributive properties combine multiplication and addition or subtraction. This allows us to break down expressions for simple calculations.

Rule: a × (b + c) = a × b + a × c

Examples:

• 3 × (4 + 2) = 3 × 6 = 18
  = 3 × 4 + 3 × 2 = 12 + 6 = 18

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

Identity Property

The identity property identifies the special number that , when used in an operation, keeps the original number unchanged.

Examples:

• 7 + 0 = 7

• (−4) × 1 = −4 

Summary Table: Properties of Integers

Applications of Integers in Daily Life

The integers are not only limited to mathematical concepts; they are deeply woven into real conditions. From recording temperature and measuring the level of management and height of bank accounts, integers in daily life help us describe and interpret both positive and negative values ​​with meaning and clarity.

Using Integers in Real-Life Situations

Integers are used in many practical and everyday scenarios where both direction and quantity are important. They help represent profits and disadvantages, ups and downs of values, or increases and reduce in different systems.

Real-Life Uses of Integers:

• Gains and losses in business

• Rise or drop in stock prices

• Scoring in games

• Levels in video games (positive/negative levels)

• Steps above or below ground level

Integers in Banking, Temperature, and Elevation

1. Banking and Finance

Integers are used in bank transactions to reflect credits and debits.

 

2. Temperature Measurement

Temperature readings often include positive and negative integers depending on the scale and region.

 

3. Elevation and Depth

In geography, elevations above and below sea level are represented using integers.

Solved Integer Examples and Practice Questions

Practicing integer problems helps students understand how to use integer concepts and rules in different mathematical situations. Examples, spreadsheets, and MCQs designed for students from class 5 to class 7 are solved below. These exercises include integer questions, property-based MCQs, and word problems on scenarios in the real world to improve problem-solving.

Worksheet Sample:

Conclusion

Integers are an essential part of mathematics & daily life, which represent both positive and negative values. Understanding their characteristics and mastery in integer operations allows students to solve complex mathematics problems and understand real-world landscapes such as temperature changes, height, and economic transactions. Practice, explore, and apply these concepts to create a strong foundation for learning future mathematics.

Related Links

• Addition Concepts - Understand the concept of addition with clear examples, properties & how it forms the foundation of basic math.

• Division Operations - Learn about the process of division, its rules, and methods, with simple explanations and practice examples.

• Arithmetic Progression - Dive into the concept of arithmetic progression, its formula, and how it applies to sequences in math.
  
  

FAQs on Integers

1. What are integers and examples?
   Integers are whole numbers (no decimals or fractions) that can be positive, negative, or zero. They are a fundamental set of numbers used in mathematics and various real-world applications.

2. Is 1.5 an integer, yes or no?
   An integer, also called a "round number" or “whole number,” is any positive or negative number that does not include decimal parts or fractions. For example, 3, -10, and 1,025 are all integers, but 2.76 (decimal), 1.5 (decimal), and 3 ½ (fraction) are not.

3. Is 7 an integer number?
   Integers are numbers that cannot be decimals or fractions. They are either whole numbers or negative numbers. Some examples are: 2, 7, 0, -9, -12, etc.

4. What are the 7 rules of integers?
   Rules of Integers

• The sum of two positive integers is an integer.

• The sum of two negative integers is an integer.

• The product of two positive integers is an integer.

• The product of two negative integers is an integer.

• The sum of an integer and its inverse is equal to zero.

• The product of an integer and its reciprocal is equal to 1.

5. Is root 2 an integer?
   is not, as 2 is not a perfect square) or irrational. The rational root theorem (or integer root theorem) may be used to show that any square root of any natural number that is not a perfect square is irrational.

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