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 → Properties of Real Numbers ↓


Associative Property


The associative property is a fundamental feature of numbers that allows us to simplify the way we add or multiply numbers, making complex problems easier to solve. This property is part of a broader category in mathematics known as the properties of real numbers. Here, we will dive into the associative property and understand it in depth with several examples and explanations.

What is associative property?

The associative property refers to the way in which numbers are grouped in operations such as addition and multiplication. It states that the way numbers are grouped has no effect on the sum or product.

Associative property of addition

The associative property of addition states that when we add three or more numbers, the grouping of these numbers does not change their sum. To put it in simple terms:

    (a + b) + c = a + (b + c)

Let's break this down further:

If we have numbers a, b, and c, we can group them as (a + b) + c or a + (b + c), and the result will be the same.

Example 1:

    (2 + 3) + 4 = 2 + (3 + 4)

In this example, first count each group:

  • (2 + 3) + 4 = 5 + 4 = 9
  • 2 + (3 + 4) = 2 + 7 = 9

As you can see, both groups arrive at the same result: 9.

Example 2:

    (1 + 4) + 5 = 1 + (4 + 5)

Computing the grouping, we get:

  • (1 + 4) + 5 = 5 + 5 = 10
  • 1 + (4 + 5) = 1 + 9 = 10

Again, we see the same result of 10 in both cases.

Associative property of multiplication

The associative property of multiplication is the same as that of addition. It ensures that the way the numbers are grouped in the multiplication has no effect on the final product. To put it simply:

    (a × b) × c = a × (b × c)

With this property, given numbers a, b, and c, we can group them as either (a × b) × c or a × (b × c), and we will get the same result.

Example 3:

    (2 × 3) × 4 = 2 × (3 × 4)

Calculate each group:

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

The result in both cases is 24, which confirms the associative property.

Example 4:

    (1 × 2) × 3 = 1 × (2 × 3)

The result of calculating each grouping is:

  • (1 × 2) × 3 = 2 × 3 = 6
  • 1 × (2 × 3) = 1 × 6 = 6

The consistent product of 6 gives further support to associativity in multiplication.

Visual representation of the associative property

Additional scenes

Let us look at the associative property of addition through a simple diagram:

A B C , A B C

Multiplication view

Similarly, let's illustrate the associative property of multiplication:

A B C , A B C

Additional text examples

Examples of totals

Here are some other examples to emphasize the additive property:

    (7 + 2) + 3 = 7 + (2 + 3)

Calculation of both sides:

  • (7 + 2) + 3 = 9 + 3 = 12
  • 7 + (2 + 3) = 7 + 5 = 12

The total is 12 in both cases because grouping does not change the result.

Examples of multiplication

Now, let's reinforce this concept with these multiplication examples:

    (4 × 5) × 6 = 4 × (5 × 6)

Calculate each side:

  • (4 × 5) × 6 = 20 × 6 = 120
  • 4 × (5 × 6) = 4 × 30 = 120

No matter how many groups the numbers are placed in, the result will still be 120.

Why does the associative property matter?

The associative property is incredibly important because it allows us to:

  • Simplify mathematical expressions.
  • Make calculations more manageable by rearranging numbers in a way that makes it easier to calculate.
  • Ensure consistent results in multi-step problems.

Difference between associative and commutative properties

It is worth noting that associative and commutative properties are different. While the associative property emphasizes the grouping of numbers, the commutative property focuses on the order of numbers. For example:

  • Commutative: a + b = b + a
  • Associativity: (a + b) + c = a + (b + c)

These properties together form the basis of many mathematical theories and operations, making mathematical operations more smooth.

Conclusion

Understanding the associative property helps explain how numbers interact in mathematical operations. Whether you are adding or multiplying, remember that grouping does not change the result, which can be clearly demonstrated through various examples. This property lays the foundation for developing more advanced mathematical skills and simplifies the process of solving problems efficiently.


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Commutative Property
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Associative Property

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