Real Numbers

Chapter Overview

Covers the Fundamental Theorem of Arithmetic (unique prime factorisation of any composite integer apart from order); factor tree decomposition (e.g., $32760 = 2^3 \times 3^2 \times 5 \times 7 \times 13$); finding HCF and LCM via lowest and highest prime exponents; the fundamental relationship $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$; proving non-existence of ending digit zero for powers like $4^n$ and $6^n$; cyclical circular path synchronization via LCM (Sonia & Ravi track problem); and formal proof of irrationality by contradiction for $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, and composite linear forms ($a \pm b\sqrt{p}$).