Lesson 1: Classifying Real Numbers

New Concept

Welcome to Algebra! This course is about learning the language of mathematics—using symbols and rules to describe patterns, solve problems, and understand the world.

What’s next

We'll begin by organizing our toolkit—the numbers themselves. Next, you'll dive into classifying different types of real numbers, from natural numbers to irrationals.

Property

Real Numbers are composed of Rational Numbers (can be written as a fraction $\frac{a}{b}$) and Irrational Numbers (cannot). Rational numbers contain Integers, which include Whole Numbers, which in turn include Natural Numbers.

Examples

• The number $9$ is a Natural, Whole, Integer, Rational, and Real number.
• The number $-4.5$ is a Rational and Real number because it can be written as $-\frac{9}{2}$.
• The number $5\pi$ is an Irrational and Real number because its decimal is endless and non-repeating.

Explanation

Think of it like a big family tree! At the top, you have the Real Numbers. This family has two main branches: the Rationals and the Irrationals. The Rational branch has its own smaller families nested inside. The Integers are part of the Rationals, the Wholes are part of the Integers, and the Naturals are the babies of the whole group.

Property

The intersection of sets $A$ and $B$, $A \cap B$, is the set of elements that are in both $A$ and $B$. The union of $A$ and $B$, $A \cup B$, is the set of all elements that are in $A$ or $B$.

Examples

If $A = \{1, 3, 5, 7\}$ and $B = \{5, 7, 9\}$, the intersection is $A \cap B = \{5, 7\}$.
If $A = \{1, 3, 5, 7\}$ and $B = \{5, 7, 9\}$, the union is $A \cup B = \{1, 3, 5, 7, 9\}$.

Explanation

Intersection is like finding out which friends you and your bestie have in common—it’s the overlap! The symbol $\cap$ even looks like a bridge connecting only what's shared. Union is like combining your entire contact list with your friend's list for a huge party. The $\cup$ symbol is like a giant bowl holding every single person from both lists.