Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Number System ↓


Rational Numbers


Rational numbers are a fundamental part of mathematics that you will study in Class 7. They play a vital role in understanding how numbers work, and you will use them in various ways to solve math problems. In this lesson, we will learn in detail what rational numbers are, how to identify them, how to perform operations on them, and their properties. The goal is to ensure that you get a solid understanding of rational numbers and their importance in the number system.

Introduction to rational numbers

A rational number is a number that can be expressed as the quotient or fraction of two integers. Simply put, a rational number is written as p/q, where p and q are integers and q is not zero. The integer p is known as the numerator, and q is known as the denominator.

For example, 1/2, -3/4, 5/1, -2/5, and 6/7 are all rational numbers. Even whole numbers like 3 can be considered rational numbers because they can be expressed as 3/1.

Visual examples of rational numbers

number line -1 -1/2 0 1/2 1 3/2 2

Identifying rational numbers

To identify whether a number is rational or not, you should check if it can be expressed in the form p/q where q ≠ 0 Let us look at some examples to make this more clear.

1. The number 8 is rational because it can be written as 8/1.
2. The fraction -5/9 is already in the form p/q, so it is a rational number.
3. The decimal 0.75 can be written as a fraction: 75/100 = 3/4. Therefore, 0.75 is rational.
4. The repeating decimal 0.333... can be expressed as a fraction: 1/3. Therefore, it is rational.

Properties of rational numbers

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, multiply, or divide them (unless you are dividing by zero), the result will also be a rational number.

Example of addition:
Suppose a = 2/3 and b = 1/6.
a + b = 2/3 + 1/6 = (4 + 1)/6 = 5/6, which is a rational number.

Example of subtraction:
Suppose a = 7/4 and b = 3/4.
a - b = 7/4 - 3/4 = (7 - 3)/4 = 4/4 = 1, a rational number.

Example of multiplication:
Suppose a = 5/2 and b = 2/5.
a * b = (5/2) * (2/5) = 10/10 = 1, a rational number.

Partition example:
Suppose a = 3/7 and b = 6/7.
a / b = (3/7) / (6/7) = 3/6 = 1/2, a rational number.
(Note: Division by zero is not allowed.)

Commutative property

Rational numbers are commutative under addition and multiplication. This means that the order of the numbers does not affect the sum or the product.

Example of addition:
a + b = b + a
(1/4) + (2/3) = (2/3) + (1/4) = 11/12

Example of multiplication:
a * b = b * a

(1/4) * (2/3) = (2/3) * (1/4) = 2/12 = 1/6

Associative property

Rational numbers obey the associative property for both addition and multiplication. This means that the way the numbers are grouped does not change the sum or the product.

Example of addition:
(a + b) + c = a + (b + c)
((1/4) + (1/2)) + (3/4) = (1/4) + ((1/2) + (3/4)) = 1.5

Example of multiplication:
(a * b) * c = a * (b * c)

((1/4) * (1/2)) * (2/1) = (1/4) * ((1/2) * (2/1)) = 1/4

Distributive property

The distributive property combines addition and multiplication, allowing you to multiply one number by a group of numbers added together.

a * (b + c) = (a * b) + (a * c)
Example:

(1/3) * ((3/4) + (2/4)) = (1/3) * (5/4) = (1/3) * (3/4) + (1/3) * (2/4) = 5/12

Operations on rational numbers

Addition of rational numbers

To add rational numbers, follow these steps:

  • Find a common denominator.
  • Convert each fraction into an equivalent fraction with the same denominator.
  • Add the numerators, keeping the denominator the same.
  • Simplify the fraction if necessary.
Example:
1/3 + 1/4

1. The common denominator is 12.
2. Conversion: 1/3 = 4/12, 1/4 = 3/12
3. Add: 4/12 + 3/12 = 7/12

Subtraction of rational numbers

To subtract rational numbers, use the same steps as for addition:

  • Find a common denominator.
  • Convert each fraction into an equivalent fraction with the same denominator.
  • Subtract the numerators and keep the denominator the same.
  • Simplify the result.
Example:
5/6 - 1/3

1. The common denominator is 6.
2. Conversion: 1/3 = 2/6
3. Subtract: 5/6 - 2/6 = 3/6 = 1/2

Multiplication of rational numbers

To multiply rational numbers:

  • Multiply the numerators to get a new fraction.
  • Multiply the denominators to get the new denominator.
  • Simplify the fraction if necessary.
Example:
2/5 * 3/4

1. Multiply: 2 * 3 = 6, 5 * 4 = 20
2. Result: 6/20 = 3/10

Division of rational numbers

To divide rational numbers, multiply by the reciprocal of the divisor:

  • Flip the second fraction (the reciprocal).
  • Multiply like multiplying fractions.
  • Simplify the result if necessary.
Example:
7/8 ÷ 3/2

1. The reciprocal of 3/2 is 2/3.
2. Multiply: 7/8 * 2/3 = 14/24 = 7/12

Decimal representation of rational numbers

Rational numbers can also be represented as decimals. The decimal form of a rational number can be either terminating or repeating.

  • Terminate decimal: Decimal that has a finite number of digits. Example: 1/4 = 0.25
  • Repeating decimals: Decimals where one or more digits are repeated an infinite number of times. Example: 1/3 = 0.333...

Comparing rational numbers

When comparing rational numbers, convert them to the same denominator so it's easier to determine which number is larger or smaller. Alternatively, convert them to decimal form for ease of comparison.

Example:
Compare 2/3 and 3/4.

1. The common denominator is 12.
2. Conversion: 2/3 = 8/12, 3/4 = 9/12
3. Compare: 8/12 < 9/12, hence 2/3 < 3/4.

When using decimals, compare the numerical values directly.

Simplification of rational numbers

To simplify a rational number, divide the numerator and denominator by their greatest common divisor (GCD).

Example:
Simplify 8/12.

1. The GCD of 8 and 12 is 4.
2. Divide: 8 ÷ 4 = 2, 12 ÷ 4 = 3
3. Simplified form: 8/12 = 2/3

Summary

Rational numbers are numbers that can be expressed as a fraction of two integers where the denominator is not zero. They are an integral part of the number system and are essential in many mathematical calculations and concepts. Understanding rational numbers involves recognizing their properties, learning how to perform operations, representing them as decimals, and simplifying them. With practice, working with rational numbers can be simple and will greatly benefit your mathematical skills.


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Definition of Rational Numbers

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Rational Numbers

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