1. **Problem Statement:** Understand the Real Number System as the first unit of Real Analysis for Delhi University BSc Maths Hons syllabus. 2. **What is the Real Number System?** It is the set of all numbers that can be found on the number line. This includes all rational numbers (fractions, integers) and irrational numbers (numbers that cannot be expressed as fractions). 3. **Key Components:** - **Natural Numbers ($\mathbb{N}$):** Counting numbers starting from 1, 2, 3, ... - **Whole Numbers:** Natural numbers including zero. - **Integers ($\mathbb{Z}$):** All positive and negative whole numbers including zero. - **Rational Numbers ($\mathbb{Q}$):** Numbers that can be expressed as $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$. - **Irrational Numbers:** Numbers that cannot be expressed as a ratio of two integers, e.g., $\sqrt{2}$, $\pi$. 4. **Properties of Real Numbers:** - **Closure:** Real numbers are closed under addition, subtraction, multiplication, and division (except division by zero). - **Commutativity:** $a + b = b + a$ and $ab = ba$ for any real numbers $a, b$. - **Associativity:** $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$. - **Distributivity:** $a(b + c) = ab + ac$. 5. **Important Concepts:** - **Density:** Between any two real numbers, there is always another real number. - **Completeness:** Every non-empty set of real numbers that is bounded above has a least upper bound (supremum). 6. **Why is this important?** Real numbers form the foundation for calculus and analysis. Understanding their properties helps in studying limits, continuity, and other advanced topics. 7. **Summary:** The real number system includes all numbers on the number line, combining rational and irrational numbers, and has important algebraic and order properties essential for higher mathematics. This explanation covers the basics you need to understand the first unit of Real Analysis for your exam.