Numerical Sets: $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$
A Complete Lesson for Pre-University Algebra
1. Introduction & Motivation
Mathematics is built upon numbers, but not all numbers are alike. Over thousands of years, mathematicians progressively expanded the number system to solve increasingly complex problems. Each expansion was driven by a concrete need: a problem that could not be solved within the existing number system.
Why does this matter? Every equation you solve, every function you graph, and every proof you write requires you to know which numbers you are working with. The set of numbers you choose determines what operations are valid, what solutions exist, and what properties you can rely on.
In this lesson, we study four foundational numerical sets:
$\mathbb{N}$ Naturals
$\mathbb{Z}$ Integers
$\mathbb{Q}$ Rationals
$\mathbb{R}$ Reals
The Hierarchy at a Glance: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$
Every natural number is an integer, every integer is rational, and every rational number is real.
