The Real Number System — Grade 9 Maths (CAPS)

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.

Also known as: rational and irrational numbers · number sets · real number classification

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 a/b 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 7/1.
Terminating decimals — 0,25 is 1/4.
Recurring decimals — 0,333… is 1/3.
Exact roots — √16 = 4, so it is rational.

⚠️

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.

💡

Quick check for the square root of a positive whole number: is it a perfect square (1, 4, 9, 16, 25, 36…)? If yes, the root is rational; if no, it is irrational. The check is for whole numbers only — √0,25 = 0,5 and √(1/4) = 1/2 are rational even though 0,25 and 1/4 are not on that list.

4True or false? The statements exams love

Exams test the definitions with short true-or-false statements. Use the nested picture: a set contains every set inside it, but not the other way round.

Worked ExampleDecide whether each statement is true or false

1
Every integer is a rational number — true: an integer such as −4 can be written as (−4)/1.
2
Every rational number is an integer — false: 1/2 is rational but not an integer.
3
0 is a natural number — false: the natural numbers start at 1; 0 is the first whole number.
4
√9 is irrational — false: √9 = 3, a whole number.
Work the root out before you decide.
5
π = 22/7 — false: 22/7 is only an approximation; π is irrational.

5Showing that a number is rational

Worked ExampleShow that 0,75 and −6 are rational numbers

1
0,75 = 75/100 = 3/4.
Two decimal places means hundredths.
2
−6 = (−6)/1.
Any integer can be written over 1.
3
Both can be written as a/b with integers a and b and b ≠ 0, so both are rational.

6Estimating an irrational root

An irrational root can't be written exactly as a decimal, but you can always say which two whole numbers it lies between. Find the perfect squares on either side of the number under the root.

Worked ExampleBetween which two consecutive whole numbers does √30 lie?

1
25 < 30 < 36.
25 and 36 are the perfect squares on either side of 30.
2
So √25 < √30 < √36, that is, 5 < √30 < 6.
3
√30 lies between 5 and 6.
30 is closer to 25 than to 36, so √30 is nearer 5.

⚠️

π is not 3,14. 3,14 and 22 over 7 are rational approximations for calculations. π itself never ends and never repeats, which is why 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.
Show answer ▾Hide answer ▴
1, 2, 3, 4
2
What is the only whole number that is not a natural number?
Show answer ▾Hide answer ▴
0
3
Is −7 an integer? Is it a natural number?
Show answer ▾Hide answer ▴
Yes, it is an integer; no, it is not natural.
4
Define a rational number.
Show answer ▾Hide answer ▴
One that can be written as a/b with a, b integers and b ≠ 0.
5
Is √36 rational or irrational? Why?
Show answer ▾Hide answer ▴
Rational — √36 = 6, an exact whole number.
6
Is √20 rational or irrational? Why?
Show answer ▾Hide answer ▴
Irrational — 20 is not a perfect square, so the decimal never ends or repeats.
7
Is 0,45 rational?
Show answer ▾Hide answer ▴
Yes — it terminates, and equals 45/100.
8
Is 0,666… rational?
Show answer ▾Hide answer ▴
Yes — it recurs, and equals 2/3.
9
Name two irrational numbers.
Show answer ▾Hide answer ▴
π and √2 (or the square root of any positive whole number that is not a perfect square)