Every number you have ever used — 1, 2, −5, even $\pi$ — belongs to one family: the real numbers. Inside that family the groups are nested, each one sitting inside the next like a set of boxes. This lesson builds those groups one step at a time.

Concept Building the groups, one step at a time

Each group starts from the one before it and adds a new kind of number.

Natural — the counting numbers: 1, 2, 3, 4, … the earliest and most basic group.

Whole — the natural numbers together with 0.

Integers — the whole numbers together with the negatives: … −2, −1, 0, 1, 2, …

Rational — any number that can be written as a fraction, one number divided by another.

Up to the integers there are still no fractions. The rational numbers are the step that adds them.

Concept Each group contains the one before it

The rational numbers include the integers, the integers include the whole numbers, and the whole numbers include the natural numbers. The irrational numbers sit in a separate box. Both boxes together fill the real numbers.

Example Any number from an inner group is rational

Because rational means “a number divided by a number”, every natural number, whole number and integer can be rewritten as a fraction.

8 = \dfrac{16}{2}

-5 = \dfrac{-10}{2}

3 = \dfrac{3}{1}

⟹ every integer is also a rational number.

Concept The irrational numbers

Between the fractions sit numbers that cannot be written as one number divided by another. These are the irrational numbers, such as $\pi$ , $\sqrt{3}$ and $\sqrt{2}$ .

\sqrt{3} \approx 1.73205\ldots

The dots matter: the digits after the decimal point run on forever with no repeating pattern. That is why an irrational number can only be approximated, never written down exactly. The irrational numbers are so plentiful that they far outnumber the rational numbers packed between the fractions.

Concept Putting it together

Start with the natural numbers, then the whole numbers, then the integers, then the rational numbers, which contain all of them. The irrational numbers form a separate group. Combine the two:

\text{rational} \;\cup\; \text{irrational} \;=\; \text{real}

Since these are the real numbers, there must be numbers that are not real. Those are the imaginary numbers, studied later.

Summary

Natural → add 0 → whole → add negatives → integers → add fractions → rational.
Each group contains the one before it: natural ⊂ whole ⊂ integers ⊂ rational.
Rational means it can be written as a number divided by a number, so $8 = \dfrac{16}{2}$ .
Irrational numbers such as $\pi$ and $\sqrt{3}$ have endless non-repeating decimals and can only be approximated.
Rational together with irrational makes the real numbers.

The Set of Real Numbers