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 ↓


Rational Numbers


Rational numbers are a fundamental concept in mathematics, especially for Class 8 students. They play an essential role in number theory and arithmetic calculations. But what exactly are these numbers and why are they important? Let's explore the exciting world of rational numbers in detail.

Definition of rational numbers

A rational number is a number that can be expressed as the quotient of two integers or as the fraction a/b, where the numerator a is an integer and the denominator b is a non-zero integer. In simple terms, rational numbers are numbers that can be written as fractions.

Rational numbers include:

  • Positive numbers
  • Negative numbers
  • Zero

Examples of rational numbers

Let us consider some examples to understand rational numbers better:

  • 3/4: This is a simple fraction that represents the rational number three-fourth.
  • -7/1: This fraction represents the rational number -7, which shows that whole numbers are also rational numbers.
  • 0/1: The number zero is a rational number because it can be expressed as a fraction of any non-zero number times zero.

Properties of rational numbers

Rational numbers have specific properties that are important to understand:

Representation of rational numbers

A rational number can be represented on the number line. Consider the fractions 1/2 and -3/4:

-1 0 1 1/2 -3/4

On this number line, the red dot represents the rational number 1/2 and the blue dot represents -3/4.

Closing assets

Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero). This means that if you take two rational numbers and add, subtract, or multiply them, the result will always be a rational number. However, division by zero is undefined.

Addition: (1/2) + (3/4) = (5/4) Subtraction: (2/3) - (1/3) = (1/3) Multiplication: (2/5) * (3/4) = (6/20) = (3/10) Division: (3/5) / (2/3) = (3/5) * (3/2) = (9/10)

Density properties

Rational numbers are dense, that is, between any two rational numbers there is another rational number. This property makes the rational numbers infinite and ensures that the number system is continuous.

Example: A rational number between 1/4 and 1/2 is 3/8.

Operations with rational numbers

Addition and subtraction

To add or subtract rational numbers you need a common denominator. Let's look at the following example:

Example: (1/3) + (2/5) Common Denominator = 15 (1/3) = (5/15) (2/5) = (6/15) (5/15) + (6/15) = (11/15)

Multiplication

Multiplying rational numbers is simple. Multiply their numerators and multiply their denominators.

Example: (3/7) * (2/3) = (3*2)/(7*3) = (6/21) = (2/7)

Division

To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second.

Example: (4/9) ÷ (2/3) = (4/9) * (3/2) = (12/18) = (2/3)

Understanding negative rational numbers

Rational numbers can also be negative when either the numerator or the denominator is negative, but not both. Below is an example of a negative rational number on the number line.

-2 0 2 -1/2

Converting decimals to rational numbers

Any terminating or repeating decimal can be converted to a rational number. For example, 0.75 can be expressed as 3/4, and 0.333... (where the 3 is repeating) can be written as 1/3.

Converting a repeating decimal involves using algebraic methods. Here's an example of how you can convert a repeating decimal into a fraction:

Example: Let x = 0.666... Then 10x = 6.666... By subtracting, we get: 10x - x = 6.666... - 0.666... Which simplifies to: 9x = 6 Therefore, x = 6/9 = 2/3

Applications of rational numbers

It's important to know and understand rational numbers because they appear in everyday life, such as:

  • Measurement: Rational numbers are used to measure quantities such as length, weight, and volume.
  • Finance: Money is managed based on rational numbers, whether through pricing, interest rates, or savings.
  • Data analysis: Statistical data often requires interpretations that involve rational numbers.

Conclusion

Rational numbers are indispensable for understanding fractions, arithmetic operations, and their properties within mathematics. They open our minds to a huge variety of numbers and provide us with important tools for problem-solving in real-world situations. The field of rational numbers is fascinating, providing a bridge from whole numbers to more complex concepts in mathematics.


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Rational Numbers

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