Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Number Systems → Operations on Real Numbers ↓


Addition and Subtraction


Addition and subtraction are two fundamental arithmetic operations that are essential in solving a variety of mathematical problems. In everyday life, we use these operations constantly, from simple tasks like calculating the total cost of groceries to advanced scientific calculations. In this lesson, we will explore how addition and subtraction apply to real numbers in the number system, including practical examples and visual aids to simplify the concepts.

Understanding real numbers

Real numbers include a wide range of numbers that we use every day, including whole numbers, fractions, and decimals. The set of real numbers also includes rational numbers (numbers that can be expressed as a ratio of two integers) and irrational numbers (numbers that cannot be expressed as simple fractions, such as π and √2). When performing addition and subtraction, it is important to remember that these operations are valid for all real numbers.

Sum of real numbers

Addition is the process of combining two or more numbers to get a total. For real numbers, addition is straightforward: you just add the values. The result is also a real number. Let's take a deeper look at how addition works with different types of real numbers.

Examples of totals

Example 1: Adding whole numbers

5 + 7 = 12

Here, adding the whole numbers 5 and 7 gives 12.

Example 2: Adding fractions

1/3 + 1/6 = 1/2

To add these fractions:

  • Find a common denominator.
  • Convert each fraction into an equivalent fraction with the same denominator.
  • Add the numerators, keeping the denominator the same.

Example 3: Adding decimals

3.5 + 2.75 = 6.25

Align the decimal points and add them just as you would with whole numbers.

0 12 12 Adding 5 and 7

Subtraction of real numbers

Subtraction is the process of subtracting or removing one number from another number. It can also be seen as adding negative numbers. When you subtract, you determine the difference between numbers. Let's learn about subtraction with different types of real numbers.

Subtraction examples

Example 1: Subtracting whole numbers

15 - 8 = 7

You subtract 8 from 15 and the result is 7.

Example 2: Subtracting fractions

5/6 - 1/3 = 1/2

This involves finding common denominators before subtracting fractions, just like addition.

Example 3: Subtracting decimals

10.75 - 3.25 = 7.5

Align the decimal points and subtract like whole numbers.

0 15 15 Subtract 8

Properties of addition and subtraction

When working with the addition and subtraction of real numbers, it is necessary to keep in mind several key properties that make these operations simple and logical.

Properties of yoga

Commutative property This property states that changing the order of the numbers does not change the sum. In mathematical terms: 
a + b = b + a

Example: 4 + 5 = 5 + 4 both equal 9.

Associative property This property states that no matter which group you place the numbers in while adding them, the sum will be the same. In the language of mathematics: 
(a + b) + c = a + (b + c)

Example: (2 + 3) + 4 = 2 + (3 + 4), both equal 9.

Identity property This property involves adding zero to a number, which does not change the number. 
a + 0 = a

Example: 7 + 0 = 7.

Properties of subtraction

  • Subtraction as the inverse of addition:

    Any subtraction operation can also be expressed as an addition operation using negative numbers.

    a - b = a + (-b)
  • Non-interchangeable:

    Subtraction is not commutative. This means that changing the order changes the result:

    a - b ≠ b - a

Uses of addition and subtraction in practical contexts

The practical application of addition and subtraction is found in a variety of contexts, from simple household budgeting to complex scientific endeavors. Below are some examples illustrating their application:

Budget

When creating a budget, you add up all your projected income and subtract your projected expenses to determine your finances. Let's see this in action:

Income:

  • Job: $3000
  • Investment: $500
Total Income = $3000 + $500 = $3500

Expenditure:

  • Rent: $1000
  • Utilities: $300
  • Groceries: $400
Total Expenses = $1000 + $300 + $400 = $1700
Savings = Income - Expenses = $3500 - $1700 = $1800

Scientific calculations

Scientific calculations, such as determining the balance of chemical equations or the distance traveled by a mass subject to certain forces, rely heavily on addition and subtraction as basic operations.

Summary

Addition and subtraction are fundamental arithmetic operations used in calculating the sum and difference of real numbers. Understanding their practical applications and properties simplifies a wide range of mathematical explorations and real-world calculations. Using examples, properties, and practical illustrations, one can understand the versatility and indispensable nature of these operations.


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Addition and Subtraction

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