Numbers

Real number system

Naturals, integers, rationals, irrationals and the real line. Interval notation and surds as irrationals.

CAPS

Grade 10 · Mathematics · Numbers

This sits on Mathematics Paper 1.

The paper
Paper 1 is algebra, exponents, equations, number patterns, functions, finance and probability. Copy the formula, then substitute.
Cognitive level
CAPS spreads the marks across knowledge, routine procedures, complex procedures and problem solving. A routine question still wants the line of working. A problem-solving question wants you to choose the method, not only to calculate.
The relation
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. Nested number sets
How a marker reads it
NSC wants set notation: √2 ∈ ℝ and √2 ∉ ℚ.

Each Mathematics paper is 150 marks. A correct substitution can earn a method mark even when the last arithmetic slips, if the formula was the right one.

Diagram

Worked example

Classify −3, 0.75, √9, √7 and π as rational or irrational.

Simplified approach

A rational number can be written as ab in lowest terms. Perfect-square surds become integers. π and √7 cannot.

Culture note

Sharing 3 vetkoek among 4 learners in a Soweto tuck shop is 3/4 — rational. The diagonal of a square classroom tile is √2 — irrational.

Formulas

ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ

Nested number sets

√2 ∉ ℚ

Surds that are not perfect squares are irrational

How to hunt the missing letter

Do not start by naming the letters. First hunt the missing one: copy every number you can see (Given). The letter with no number is Missing. The asking word is Find. The formula that still contains that letter is what you Calculate.

How to hunt: (1) Underline every number and its unit — those are Given. (2) Circle the asking word — find, calculate, how far, what is. That word names Find. (3) The letter the asking word names, which has no number yet, is Missing. (4) Write the formula with empty letters. Cross out every given letter. The letter still standing is what you Calculate.

ℕ
natural numbers
How you spot this in the sentence
When the sentence names the natural numbers, that letter is ℕ. Copy the number next to it into Given. If the sentence never names it, ℕ is Missing. Counting numbers: 1, 2, 3, … (CAPS includes 0 in some lists — follow the paper).
Example:5 ∈ ℕ, but −3 ∉ ℕ.
ℤ
integers
How you spot this in the sentence
When the sentence names the integers, that letter is ℤ. Copy the number next to it into Given. If the sentence never names it, ℤ is Missing. … −2, −1, 0, 1, 2 … No fractions.
Example:−3 ∈ ℤ, ½ ∉ ℤ.
ℚ
rationals
How you spot this in the sentence
When the sentence names the rationals, that letter is ℚ. Copy the number next to it into Given. If the sentence never names it, ℚ is Missing. Any number that can be written as p/q with q ≠ 0.
Example:0,75=34 ∈ ℚ. √2 ∉ ℚ.
ℝ
real numbers
How you spot this in the sentence
When the sentence names the real numbers, that letter is ℝ. Copy the number next to it into Given. If the sentence never names it, ℝ is Missing. Every number on the number line, including irrationals like √2 and π.
Example:ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. √2 is real but not rational.

Exam tricks you must know

These are the moves markers look for — not shortcuts that skip the method, but the habits that earn the method mark.

“Factorise fully” means HCF and then the identity — leaving 2(x² − 9) loses a mark
How:Look at the given polynomial. Pull out the biggest number and letter that divides every term (HCF). Then look at what is left: difference of squares, trinomial, or grouping. Both layers go on the page. Stop only when nothing else factorises.
When to use it:Any “factorise fully” on Paper 1, especially KZN.
Memorise exact values for 0°, 30°, 45°, 60°, 90° — compound angles collapse without them
How:Before a calculator, write the triangle: 30-60-90 and 45-45-90. sin 30° = 12, sin 45° = 1√2, sin 60° = √32, and the cosines swap. Compound-angle questions want these as fractions with division lines, not decimals.
When to use it:Reduction, compound angles, and “without a calculator” trig.
Name similar triangles in corresponding order: ΔABC ∼ ΔDEF, not ΔABC ∼ ΔEFD
How:Hunt equal angles first and mark them on the sketch. Then list vertices so that equal angles sit in the same position: A maps to D, B to E, C to F. The ratio statement uses that same order. Wrong order loses the reason mark even if the lengths are right.
When to use it:Similarity, intercept, and 3-D trig that shares a triangle.

Step 1 / 6

Integers and fractions

−3=−31 ∈ ℚ. 0.75=34 ∈ ℚ.

Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

4. Step-by-step working

01
Integers and fractions
−3=−31 ∈ ℚ. 0.75=34 ∈ ℚ.
02
Perfect square
√9=3 ∈ ℤ ⊂ ℚ.
√9=3

More worked examples

Board walkthroughs on this topic. Open one, copy the working, then try the next.

Example 1 / 20

Is √9 rational or irrational? Give the tightest true set.

How we got there

Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.

Explain in another language · 11

1. What is the question asking?

Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Is √9 rational or irrational? Give the tightest true set.

Before any working — hunt first

Given

Numbers and units already in the sentence: 9 rational. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: tightest true set. Circle that word — that is the missing variable.

Find

We are required to find: tightest true set. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write rational (integer 3) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives tightest true set.

In other words — as if you have never seen this before

If you have never seen this before: A rational is pq with integers p, q ≠ 0. Surds that do not simplify to an integer are irrational. We will name a method in words, write the formula with no numbers, then substitute one line at a time.

2. What do we already know?

The number is √9.

3. Which method — and why this one?

A rational is pq with integers p, q ≠ 0. Surds that do not simplify to an integer are irrational.

4. Step-by-step working

01 / 04
Identify the given
√9 is rational (integer 3).
rational (integer 3)
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
02 / 04
Sense-check
Decimals that terminate or repeat are rational. π and unsquared primes under a radical are not.
rational (integer 3)
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
03 / 04
Step F — Box the answer with its unit
rational (integer 3)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
04 / 04
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.

5. The answer

rational (integer 3)

Copy onto your answer sheet — in this order

Formula: rational (integer 3)

Answer: rational (integer 3)

6. Check it like a marker

Integers ⊂ rationals ⊂ reals. Irrationals are reals that are not rational.

7. A trap first-timers fall into

First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”

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