Real Numbers: The Complete Number Line
Big Idea: Real numbers include every number you can place on a number line. They form the foundation of algebra and all higher mathematics.
1. The Number System Hierarchy
Think of the real number system as a series of nested sets, each one containing the previous:
| Set | Symbol | Definition | Examples |
|---|---|---|---|
| Natural Numbers | $ \mathbb{N} $ | Counting numbers | $ 1, 2, 3, 4, 5, \ldots $ |
| Whole Numbers | $ \mathbb{W} $ | Natural numbers + zero | $ 0, 1, 2, 3, 4, \ldots $ |
| Integers | $ \mathbb{Z} $ | Whole numbers + negatives | $ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots $ |
| Rational Numbers | $ \mathbb{Q} $ | Numbers expressible as $ \frac{p}{q} $ where $ p, q \in \mathbb{Z} $ and $ q \neq 0 $ | $ \frac{1}{2}, -\frac{3}{4}, 0.75, 0.\overline{3} $ |
| Irrational Numbers | $ \mathbb{Q'} $ or $ \mathbb{R} \setminus \mathbb{Q} $ | Numbers that cannot be expressed as a fraction of integers | $ \sqrt{2}, \pi, e, \sqrt{5} $ |
| Real Numbers | $ \mathbb{R} $ | All rational and irrational numbers combined | Every point on the number line |