Unit 1 · Number
3 Types of Numbers
Numbers live in families, and the families sit inside each other like nesting boxes. Once you can see the boxes, questions like “are all integers rational?” answer themselves.
🎯 By the end of this you can…
- ✓Name the four families: natural, integer, rational, irrational.
- ✓Decide instantly whether a number is rational or irrational.
- ✓Answer always / sometimes / never questions with confidence.
💡 The idea
Four families, three of them nested
Natural numbers — counting numbers: 1, 2, 3, 4, … (0 is often included too).
Integers — the naturals plus zero and the negatives: … −3, −2, −1, 0, 1, 2 …
Rational numbers — anything you can write as a fraction of two integers. That includes all integers (5 = ), all decimals that stop (0.675 = ), and all decimals that repeat forever in a pattern ().
Irrational numbers — everything else. Decimals that go on forever with no pattern, and can never be written as a fraction. Like and .
🎯 The picture that answers everything
RATIONAL NUMBERS
anything that can be a fraction · 0.675 · ·
INTEGERS
whole numbers, including negatives · −13
NATURAL NUMBERS
counting numbers · 1 · 8
sits outside every box. It is irrational.
Read the picture and the rules fall out. Every natural number is inside the integer box, so every natural number is an integer. Every integer is inside the rational box, so every integer is rational. But the boxes are not the same size, so it does not work the other way round.
🗂️ Put each number in its smallest box
0 / 7 correctTap a card, then tap the box you think it belongs in.
⚠️ The classic trap
“It has a decimal point” does not mean irrational. 0.675 is perfectly rational — it stops, so it is a fraction.
The test is not does it have a decimal point. The test is: does the decimal stop, or repeat in a pattern? If yes → rational. If it runs forever with no pattern → irrational.
📋 Is this square root rational or irrational?
- 1Work out the square root, or try to.
- 2Does it come out exact? exactly. That is a whole number, so it is rational.
- 3If it does not come out exact — like — it is irrational.
- 4Shortcut: the square root of a perfect square (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 …) is rational. Everything else is irrational.
🗂️ Rational or irrational?
0 / 5 correctTap a card, then tap the box you think it belongs in.
🧐 One from the worksheet that is a trick
The worksheet also lists . Be careful: it is not rational and it is not irrational.
There is no ordinary number you can square to get a negative, because a positive squared is positive and a negative squared is also positive. So is not a real number at all — it does not live in any of our boxes. Write exactly that if it comes up.
🎮 Quick decisions
⭐ 0 XP🔥 streak 0🤔 Which of these are irrational numbers between 1 and 2?
🎮 True or false?
⭐ 0 XP🔥 streak 0💡 The idea
Always, sometimes, or never?
These look hard and are actually easy — you only need one example to decide.
To prove “sometimes”, find one case that works and one that does not. To prove “never”, you have to argue it can never happen. To claim “always”, one counter-example destroys you — so hunt for one first.
🎮 Always / sometimes / never
⭐ 0 XP🔥 streak 0🏅
You finished this lesson!
Hold the picture: natural inside integer inside rational, with irrational sitting outside all of them. And remember the real test for irrational — the decimal goes on forever with no pattern.
Next: writing gigantic and tiny numbers the short way.
Keep reading — this lesson marks itself done automatically.