Introduction to Real Numbers

Lesson: Real Numbers, Rational and Irrational Numbers

Learning Objective

To understand the structure of real numbers, differentiate between rational and irrational numbers, and identify these numbers based on decimal representations.

1. Introduction to Real Numbers 📘

Real Numbers \( \mathbb{R} \): Real numbers include all numbers commonly used in mathematics, such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

2. Classification of Real Numbers

• Natural Numbers \( \mathbb{N} \): Counting numbers starting from 1, such as 1, 2, 3, \( \ldots \)
• Whole Numbers \( \mathbb{W} \): Natural numbers including 0, such as 0, 1, 2, 3, \( \ldots \)
• Integers \( \mathbb{Z} \): Positive and negative whole numbers, such as \( -3, -2, -1, 0, 1, 2, 3, \ldots \)
• Rational Numbers \( \mathbb{Q} \): Numbers that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
• Irrational Numbers: Numbers that cannot be expressed as a fraction \( \frac{p}{q} \) and have non-terminating, non-repeating decimals (e.g., \( \sqrt{2}, \pi \)).

3. Rational and Irrational Numbers

3.1 Rational Numbers \( \mathbb{Q} \)

Definition: A number is rational if it can be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q \neq 0 \).

Decimal Representation:

• Terminating: Decimal representation ends after a finite number of digits.
  Example: \( 0.75 = \frac{3}{4} \)
• Repeating: Decimal representation repeats after a certain point.
  Example: \( 0.666\ldots = \frac{2}{3} \)

3.2 Irrational Numbers

Definition: An irrational number cannot be written as \( \frac{p}{q} \) and has a non-terminating, non-repeating decimal representation.

Examples: \( \sqrt{2}, \pi, \sqrt{3} \)

4. Key Differences Between Rational and Irrational Numbers

5. Properties of Real Numbers

• Closure Property: Rational + Rational = Rational.
  Example: \( \frac{2}{3} + \frac{1}{6} = \frac{5}{6} \)
• Commutative Property: Addition: \( a + b = b + a \).
  Example: 5 + 3 = 3 + 5
• Distributive Property: \( a(b + c) = ab + ac \).
  Example: 4(3 + 2) = 4 × 3 + 4 × 2 = 20

6. Practice Questions 📝

Based on R.D. Sharma, R.S. Aggarwal, and Previous Board Papers:

1. Basic Identification: Classify the following numbers as rational or irrational:
   • 0.25
   • \( \sqrt{3} \)
   • \( \pi \)
   • -7.125
2. Decimal Representations: Determine if the following numbers have a terminating or non-terminating decimal representation:
   • \( \frac{3}{4} \)
   • \( \frac{7}{3} \)
   • 0.333…
3. True or False:
   • \( \sqrt{16} \) is an irrational number.
   • Every integer is a rational number.
   • The sum of a rational and an irrational number is irrational.
4. Classify Numbers (from Sample Papers): Identify if the following numbers are rational or irrational:
   • \( \frac{5}{8} \)
   • \( \sqrt{11} \)
   • 3.1415…
5. Advanced Problems:
   • Express 0.142857142857… as a fraction.
   • If \( a \) is rational and \( b \) is irrational, prove that \( a + b \) is irrational.

7. Real-World Applications of Rational and Irrational Numbers 🌍

• Rational Numbers: Used in measurements, such as cutting a rope into \( \frac{3}{4} \) meter pieces.
• Irrational Numbers: Used to calculate areas involving circles where \( \pi \) is used.

8. Quick Review and Summary 📝

• Rational numbers are numbers that can be expressed as \( \frac{p}{q} \) and have either terminating or repeating decimals.
• Irrational numbers have non-terminating, non-repeating decimals and cannot be expressed as \( \frac{p}{q} \).
• Use key properties such as closure, commutativity, and distributivity to simplify operations involving rational and irrational numbers.