Understanding Surds and the Real Number System

📋 Course Outline

1. Real Number System
2. Surds and Irrational Numbers
3. Simplifying Surds
4. Operations with Surds
5. Rationalising Denominators
6. Classifying Numbers
7. Order of Surds
8. Square and Cube Roots
9. Surd Expressions

📖 1. Real Number System

🔑 Key Concepts & Definitions

• Natural Numbers (N): The set of positive whole numbers used for counting, denoted as $N = \{1, 2, 3, 4, \dots\}$. They are the most basic numbers in the hierarchy of the real number system.

• Integers (Z): The set that includes all positive and negative whole numbers along with zero, expressed as $Z = \{\dots, -2, -1, 0, 1, 2, \dots \}$. It extends natural numbers to include negatives and zero.

• Rational Numbers (Q): Numbers that can be expressed as a ratio of two integers $\frac{a}{b}$, where $b \neq 0$. They include fractions, integers, and finite or recurring decimals, denoted as $Q = \{ \text{all } \frac{a}{b} \mid a, b \in Z, b \neq 0 \}$.

• Irrational Numbers (I): Real numbers that cannot be written as a ratio of two integers. They have non-terminating, non-recurring decimal expansions, such as $\pi$ and $\sqrt{2}$, and are denoted as $I$.

• Set Notation & Subset Relationships: The hierarchy of the real number system is expressed as $N \subset Z \subset Q \subset R$, indicating that natural numbers are a subset of integers, which are a subset of rational numbers, all contained within the set of real numbers $R$.