Types of Numbers

Why Different Types?

Not all numbers are the same.

Some numbers count things. Some measure things. Some can’t even be written as fractions.

Understanding the types of numbers helps us know what we’re working with.

The Number Hierarchy

Numbers form a hierarchy. Each type contains the previous one.

Natural Numbers (ℕ)

The counting numbers: 1, 2, 3, 4, 5, …

These are the first numbers humans ever used. Counting sheep, counting days.

• No zero
• No negatives
• No fractions

Integers (ℤ)

Take the natural numbers. Add zero and negatives.

$\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$

The “Z” comes from German “Zahlen” (numbers).

Now we can represent debts, temperatures below zero, floors below ground.

Rational Numbers (ℚ)

Any number that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.

The “Q” stands for “quotient” (the result of division).

Examples:

Key property: A decimal is rational if and only if it terminates or repeats.

Irrational Numbers

Numbers that cannot be written as a fraction.

“Irrational” literally means “not a ratio.”

Famous examples:

• $\sqrt{2} \approx 1.41421356...$
• $\pi \approx 3.14159265...$
• $e \approx 2.71828182...$

Their decimals go on forever without repeating.

Why can’t $\sqrt{2}$ be a fraction?

This was proven by the ancient Greeks. If $\sqrt{2} = \frac{a}{b}$ for some integers, you reach a contradiction.

The proof shocked mathematicians. Numbers exist that cannot be written as ratios.

Real Numbers (ℝ)

All numbers on the number line.

$\mathbb{R} = \text{Rationals} \cup \text{Irrationals}$

Every point on the number line is either rational or irrational. Never both, never neither.

The real numbers “fill in” the number line completely. No gaps.

Summary

Key insight: Every real number is either rational (can be a fraction) or irrational (cannot). There’s no in-between.