The Real Number System: Rational & Irrational Numbers

Naming and sorting numbers — natural, whole, integers, rational and irrational — how each set sits inside the next, and how to tell a rational number from an irrational one.

Grade 9 asks you to do something new: not just use numbers, but classify them. Exams ask "is 16 rational or irrational?" and you need a reliable way to answer.

1The five sets, nested inside each other

Each set contains the one inside it. The irrationals sit inside the reals but outside the rationals.

Natural numbers (ℕ) — the counting numbers: 1, 2, 3, 4, …
Whole numbers (ℕ₀) — the naturals plus zero: 0, 1, 2, 3, …
Integers (ℤ) — whole numbers and their negatives: … −2, −1, 0, 1, 2, …
Rational numbers (ℚ) — anything that can be written as a fraction of two integers.
Irrational numbers — those that cannot, such as π and 2.

2The test for a rational number

A number is rational if it can be written as ab where a and b are integers and b ≠ 0.

That definition is wider than most learners expect. Every one of these is rational:

Whole numbers — 7 is 71.
Terminating decimals — 0,25 is 14.
Recurring decimals — 0,333… is 13.
Exact roots — 16 = 4, so it is rational.

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A root is not automatically irrational. 16 = 4 is rational; 15 is irrational. Work the root out first, then decide.

3Spotting an irrational number

An irrational number's decimal goes on forever without repeating. The two you meet in Grade 9 are π and roots that do not come out exactly.

Worked ExampleClassify: 5, −3, ⅔, 25, 7, π, 0

1
5 — natural, whole, integer and rational.
A counting number belongs to every set above it.
2
−3 — integer and rational (not natural or whole).
Negatives are excluded from ℕ and ℕ₀.
3
⅔ — rational only.
Already a fraction of two integers.
4
25 = 5 — natural, whole, integer, rational.
The root is exact.
5
7 and π — irrational.
Their decimals never end or repeat.
6
0 — whole, integer and rational (not natural).
Zero is the one that separates ℕ from ℕ₀.

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Quick check for a square root: is the number under the root a perfect square (1, 4, 9, 16, 25, 36…)? If yes it is rational; if no it is irrational.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
List the natural numbers less than 5.
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1, 2, 3, 4
2
What is the only whole number that is not a natural number?
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0
3
Is −7 an integer? Is it a natural number?
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Yes, it is an integer; no, it is not natural.
4
Define a rational number.
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One that can be written as ab with a, b integers and b ≠ 0.
5
Is 36 rational or irrational? Why?
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Rational — 36 = 6, an exact whole number.
6
Is 20 rational or irrational? Why?
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Irrational — 20 is not a perfect square, so the decimal never ends or repeats.
7
Is 0,45 rational?
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Yes — it terminates, and equals 45100.
8
Is 0,666… rational?
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Yes — it recurs, and equals 23.
9
Name two irrational numbers.
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π and 2 (any non-perfect-square root)