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


Real numbers are a huge set of numbers that includes all the numbers you can think of: whole numbers, decimals, fractions, positive numbers, negative numbers, and zero. They form the building blocks for many mathematical operations and have special properties that help simplify arithmetic calculations. Understanding these properties is essential to master mathematical concepts and solve problems efficiently.

Let us look at the properties of real numbers one by one.

Types of real numbers

Before we focus on the properties, it is important to understand what kinds of numbers fall under the real numbers:

  • Natural numbers: These are the numbers we use for counting, like 1, 2, 3, etc. They do not include zero or negative numbers.
  • Whole numbers: These are like natural numbers, but they include zero. So, 0, 1, 2, 3, etc.
  • Integers: These include whole numbers and their negative counterparts. So, ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Rational numbers: Numbers that can be expressed as a fraction a/b where b ≠ 0 and both a and b are integers.
  • Irrational numbers: These are numbers that cannot be expressed as simple fractions. Their decimal expansions are non-recurring and non-terminating. For example, π and √2.

Basic properties of real numbers

1. Exchangeable assets

The commutative property states that the order in which you add or multiply numbers does not change the result.

  • For addition: a + b = b + a
  • For multiplication: a × b = b × a

Example:

If a = 3 and b = 5 :
Addition: 3 + 5 = 5 + 3 = 8
Multiplication: 3 × 5 = 5 × 3 = 15
AB

2. Associative property

The associative property states that when you add or multiply numbers, no matter how you group them, the result doesn't change.

  • For addition: (a + b) + c = a + (b + c)
  • For multiplication: (a × b) × c = a × (b × c)

Example:

If a = 2 , b = 3 , and c = 4 :
Sum: (2 + 3) + 4 = 2 + (3 + 4) = 9
Multiplication: (2 × 3) × 4 = 2 × (3 × 4) = 24

3. Distributive property

The distributive property connects both addition and multiplication. It shows that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

  • Distributive Property: a × (b + c) = (a × b) + (a × c)

Example:

If a = 2 , b = 3 , and c = 4 :
2 × (3 + 4) = (2 × 3) + (2 × 4) = 14
ABC

4. Identity property

The identity property refers to adding or multiplying a number so that the original number remains unchanged.

  • For addition: the identity element is 0 : a + 0 = a
  • For multiplication: the identity element is 1 : a × 1 = a

Example:

If a = 7 :
Addition: 7 + 0 = 7
Multiplication: 7 × 1 = 7

5. Inverse property

The inverse property describes a number that, when added or multiplied with the original number, gives the identity element.

  • For addition: The additive inverse of a is -a so that a + (-a) = 0
  • For multiplication: The multiplicative inverse (or reciprocal) of a is 1/a (for a ≠ 0) such that a × (1/a) = 1

Example:

If a = 6 :
Additive inverse: 6 + (-6) = 0
Multiplicative inverse: 6 × (1/6) = 1

6. Zero product property

The zero product property states that if the product of two numbers is zero, then at least one of the numbers must be zero.

  • Zero Product Property: If a × b = 0, then either a = 0, b = 0, or both.

Example:

If a × 0 = 0, then a can be any number, but one must be the number 0.

7. Closing assets

The closure property states that performing an operation on any two numbers in a set always results in another number from the same set. For real numbers, both addition and multiplication are closed operations.

  • For addition: If a and b are real numbers, then a + b is also a real number.
  • For multiplication: If a and b are real numbers, then a × b is also a real number.

Example:

Adding real numbers: 3.5 + 1.2 = 4.7
Product of real numbers: 4 × 2.5 = 10

Visual examples and illustrations

Commutative property illustrated

AB,BA

Associative property illustrated

(A+B)+C,A+(B+C)

Illustration of the distributive property

BC× A =A × BA × C

In conclusion, the properties of real numbers provide a strong framework for computation and problem-solving in mathematics. By understanding and applying these properties, students can simplify math problems and gain deeper insights into the structure of numbers and operations. These properties are fundamental not only in mathematics but also in various real-world applications where mathematical computations are necessary.


Grade 8 → 1.4


U
username
0%
completed in Grade 8

← Prev (1.3)
Real Numbers
Next (1.4.1) →
Commutative Property

Comments 



Properties of Real Numbers

https://www.buddymath.com/g8/1.4?bmal=en