Let’s start with something simple: counting.

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When you count things like 1 apple, 2 apples, 3 apples — those numbers are called natural numbers. They start at 1 and keep going forever.

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\[
\mathbb{N} = \{1, 2, 3, \dots\}
\]

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Natural numbers are the ones we use every day to count real things. They don’t include zero, negatives, or fractions — just whole, positive counts.

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Now think about what happens when you have nothing to count. If you have 0 apples, that’s still a number — it just means none. To include that idea of “none,” we add zero to our set. These are called whole numbers.

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\[
\mathbb{W} = \{0, 1, 2, 3, \dots\} = \mathbb{N} \cup \{0\}
\]

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Whole numbers are great for situations that can start at zero — like a bank balance, distance, or goals scored in a game.

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But what about going below zero? Imagine the temperature dropping to \(-5^\circ\text{C}\), or owing someone \$3. We need numbers that represent values less than zero too.

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That’s where integers come in. Integers include all the whole numbers and their opposites.

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\[
\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}
\]

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Visualize the difference between ℕ, 𝕎, and ℤ. Drag the slider left and right to see which numbers belong to each set:

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When we put all these sets together, we get a relationship that looks like this:

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\[
\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z}
\]

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That means every natural number is also a whole number, and every whole number is also an integer. But not every integer is whole — because integers include the negatives too.

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You can imagine it on a number line:
- The natural numbers start at 1 and go right \((1, 2, 3, \dots)\).
- The whole numbers start at 0 and go right \((0, 1, 2, 3, \dots)\).
- The integers extend in both directions \((\dots, -3, -2, -1, 0, 1, 2, 3, \dots)\).

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