Introduction to Real Numbers
Lesson 1: Introduction to Real Numbers 📚
Welcome to the first lesson of Class 10 Maths on Real Numbers. This lesson will introduce you to the concept of real numbers and provide a foundation for understanding how rational and irrational numbers fit into the broader real number system.
“Math is not about numbers, equations, computations, or algorithms: it is about understanding.”
Learning Objectives 🎯
- Understand what real numbers are.
- Differentiate between rational and irrational numbers.
- Recognize how rational and irrational numbers form the set of real numbers.
What are Real Numbers? 🔢
A real number is any number that can be found on the number line. This includes both rational numbers (like fractions) and irrational numbers (like √2 or π). The set of real numbers is denoted by the symbol R.
| Type of Number | Definition | Examples |
|---|---|---|
| Rational Numbers | Numbers that can be expressed as a fraction \( p/q \), where both \( p \) and \( q \) are integers, and \( q ≠ 0 \). They have terminating or repeating decimals. | 1, -3, 0.75, \( \frac{1}{2} \) |
| Irrational Numbers | Numbers that cannot be expressed as a fraction. Their decimal expansions are non-terminating and non-repeating. | √2, π |
Types of Real Numbers 🧮
- Rational Numbers:
- ⚖ Definition: A number that can be written as a fraction of two integers.
- Decimal Representation:
- Terminating decimals: 0.5, 0.75
- Non-terminating, repeating decimals: 0.3, 0.142857
- Examples: \( \frac{3}{4} = 0.75 \), 2 = \( \frac{2}{1} \), -5 = \( \frac{-5}{1} \)
- Irrational Numbers:
- ⚠ Definition: Numbers that cannot be written as a fraction of two integers.
- Examples: √2 = 1.414213…, π = 3.141592…
Real Numbers on the Number Line 📈
Both rational and irrational numbers can be represented on the number line. For example:
- ➖ Rational numbers are shown as exact points on the line.
- ➕ Irrational numbers like π or √2 can also be represented but are usually approximated on the number line.
Summary 📝
- Real Numbers = Rational Numbers + Irrational Numbers.
- Rational numbers include fractions, terminating decimals, and repeating decimals.
- Irrational numbers have non-terminating, non-repeating decimal expansions.
📌 Important Note: Please review this material carefully before proceeding to the next lesson. For additional clarity, watch the video below.
Quiz Time! 🎯
- Identify whether the following numbers are rational or irrational:
- 0.333…
- √3
- \( \frac{5}{6} \)
- π
- Plot the following numbers approximately on the number line:
- 0.75
- √2
- -2
Next Lesson 🔜
We will dive deeper into the decimal representation of real numbers and learn how to classify them as rational or irrational.
Recommended Video: For more clarity on real numbers, watch the following video: