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 ↓


Definition of Rational Numbers


Rational numbers are a fundamental concept in mathematics, one we encounter frequently in everyday life and academic studies. Understanding rational numbers provides a solid foundation for further exploration into more complex mathematical ideas. In this explanation, we will discuss what rational numbers are, provide visual representations, and offer plenty of examples to help illustrate this important mathematical concept.

What are rational numbers?

A rational number is defined as any number that can be expressed as a fraction p ⁄ q, where p and q are integers and q is not equal to zero. In other words, a rational number is any number that can be written as a ratio of two integers with non-zero denominators.

Mathematically, the set of rational numbers is represented by the symbol ℚ

The set of rational numbers includes:

  • All positive fractions such 1 ⁄ 2, 3 ⁄ 4, 5 ⁄ 7
  • Negative fractions such -2 ⁄ 3, -4 ⁄ 5
  • Whole numbers such as 0, 1, and -3 (since these can be written as 0 ⁄ 1, 1 ⁄ 1, and -3 ⁄ 1 respectively)
  • Decimals that end in 0.75 (which can be written as 3 ⁄ 4)
  • Recurring or repeating decimals such as 0.333... (which can be expressed as 1 ⁄ 3)

Visual representation

Visual depictions help us understand the position of rational numbers on the number line. Let's explore how rational numbers fit into the broader context of the number line.

-3 -2 -1 0 1 2 3 0 1 2 3 1⁄2​ 3⁄2​

Discovery of rational numbers

To understand rational numbers in more depth, let's consider various categories and examples that illustrate their properties.

Positive and negative fractions

Rational numbers include both positive and negative fractions.

  • Positive Fraction Example: 7 ⁄ 8 is positive because both the numerator and denominator have the same sign.
  • Negative Fraction Example: -5 ⁄ 4 is negative because the numerator is negative while the denominator is positive.
  • Negative Fraction Example: 5 ⁄ -4 is negative because the numerator is positive while the denominator is negative.
Positive: 7⁄8​ Negative: -5⁄4​

Whole numbers as rational numbers

A whole number can also be considered a rational number. This is because any whole number x can be represented as x ⁄ 1. So, a number like 5 can be represented as a rational number 5 ⁄ 1.

3 = 3 ⁄ 1 0 = 0 ⁄ 1 -2 = -2 ⁄ 1

Terminating and recurring decimals

Terminating or repeating decimals can also be considered rational. A terminating decimal has a finite number of digits after the decimal point. For example:

0.25 = 1 ⁄ 4 0.75 = 3 ⁄ 4 0.5 = 1 ⁄ 2

A repeating decimal is a decimal in which some digits are repeated an infinite number of times. For example:

0.3333... = 1 ⁄ 3 0.6666... = 2 ⁄ 3 0.1252525... = 125 ⁄ 990

Rational numbers in real-world context

Rational numbers are important in a variety of real-world contexts. Consider when you're cooking and a recipe calls for 2 ⁄ 3 cups of sugar. You need to carefully measure a fraction, which clearly shows the everyday use of rational numbers.

Another example might be in financial calculations, such as splitting a bill. If a total of $50 is to be divided among four friends, each person contributes 50 ⁄ 4, which simply works out to $12.50. So, although it is presented in decimal form, it is contextually a rational number.

Properties of rational numbers

Rational numbers have several special properties that make them unique:

  • Closure property: The sum or product of any two rational numbers is also a rational number. For example 1 ⁄ 3 + 1 ⁄ 6 = 1 ⁄ 2, a rational number.
  • Commutative property: Rational numbers obey the commutative property under addition and multiplication. Example: a ⁄ b + c ⁄ d = c ⁄ d + a ⁄ b
  • Associative property: The addition and product of rational numbers are associative. Example: (a ⁄ b + c ⁄ d) + e ⁄ f = a ⁄ b + (c ⁄ d + e ⁄ f)
  • Distributive property: Rational numbers obey the distributive law, which is needed to open brackets. Example: a ⁄ b × (c ⁄ d + e ⁄ f) = (a ⁄ b × c ⁄ d) + (a ⁄ b × e ⁄ f)

Conclusion

Rational numbers are versatile and integral to the field of mathematics. Being numbers that can be expressed as fractions of integers, they allow us to represent a wide range of values, from whole numbers to repeating decimals. The notation system, properties (such as closure, commutative, associative, and distributive properties), and their applicability to real-life situations enhance their importance and utility in mathematics. With the examples and visual aids provided, we hope to shed light on the generality and beauty of rational numbers on the number line and beyond.


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Definition of Rational Numbers

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