What is covered in Grade 9 Mathematics Whole Numbers & Integers?

Properties of the real number system, integer calculations, multiples and factors, LCM, HCF, ratio, rate, and proportion.

Is Grade 9 Whole Numbers & Integers tested in Paper 1 or Paper 2?

Grade 9 Mathematics Whole Numbers & Integers is assessed in Paper 1 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Whole Numbers & Integers taught in Grade 9 Mathematics?

Whole Numbers & Integers is taught in Term 1 of Grade 9 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 9 Mathematics Whole Numbers & Integers on the CAPS curriculum?

Yes. Whole Numbers & Integers is part of the official CAPS Mathematics curriculum for Grade 9, Term 1 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 9 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 9 Mathematics

Grade 9 · Term 1Mathematics

Whole Numbers & Integers

We explore the real number system, starting from whole numbers and integers. We learn how to work with all four operations on integers and solve real-world problems involving ratio, rate, and financial contexts.

1.1 The Real Number System 1.2 Integer Calculations 1.3 Multiples, Factors, Ratio & Rate

Week 1

1.1 The Real Number System

Describe the real number system by recognising and defining natural numbers, whole numbers, integers, rational and irrational numbers
Recognise and use commutative, associative and distributive properties

🌍

Real-World Connection

Think of the number system like a set of nesting dolls. The smallest doll is Natural Numbers (1, 2, 3, …). Each bigger doll contains the one before it: Whole Numbers (adds 0), then Integers (adds negatives), then Rational Numbers (adds fractions and decimals), then the biggest doll — Real Numbers (adds square roots of non-perfect squares, like √2). Every number you'll use in life fits inside this system!

Definition

Natural Numbers (ℕ)

The counting numbers: 1, 2, 3, 4, 5, … They do NOT include 0 or negative numbers.

\mathbb{N} = \{1,\, 2,\, 3,\, 4,\, \ldots\}

Definition

Whole Numbers (ℕ₀)

All natural numbers PLUS zero. The only difference from natural numbers is that 0 is included.

\mathbb{N}_0 = \{0,\, 1,\, 2,\, 3,\, \ldots\}

Definition

Integers (ℤ)

All whole numbers AND their negatives: …, −3, −2, −1, 0, 1, 2, 3, … The word 'integer' means 'whole' in Latin.

\mathbb{Z} = \{\ldots,\,-2,\,-1,\,0,\,1,\,2,\,\ldots\}

Definition

Rational Numbers (ℚ)

Any number that can be written as a fraction p/q where p and q are integers and q ≠ 0. This includes all integers, fractions, terminating decimals (like 0.5), and recurring decimals (like 0.333…).

\mathbb{Q} = \left\{\frac{p}{q} \mid p,q \in \mathbb{Z},\; q \neq 0\right\}

Definition

Irrational Numbers

Numbers that CANNOT be written as a fraction. Their decimal expansions go on forever without repeating. Examples: √2, √3, π (pi).

\text{e.g. } \sqrt{2} = 1.41421356\ldots \quad \pi = 3.14159\ldots

The integer number line: negative integers on the left, zero in the middle, positive integers on the right.

Property / Rule

Commutative Property

The order of numbers does NOT change the result for addition and multiplication.

a + b = b + a \qquad a \times b = b \times a

Property / Rule

Associative Property

The way we group numbers does NOT change the result for addition and multiplication.

(a + b) + c = a + (b + c) \qquad (a \times b) \times c = a \times (b \times c)

Property / Rule

Distributive Property

Multiplication distributes over addition. This is one of the most important properties in algebra.

a(b + c) = ab + ac

🚨 Common Mistake

The commutative property does NOT work for subtraction or division. For example, 8 − 3 ≠ 3 − 8, and 12 ÷ 4 ≠ 4 ÷ 12.

Worked Examples

Worked Example

Classifying numbers in the real number system

Problem

Classify each of the following: \quad (a)\; -7 \quad (b)\; \frac{3}{4} \quad (c)\; \sqrt{5} \quad (d)\; 0

Worked Example

Applying number properties

Problem

Use commutative, associative, or distributive properties to simplify calculations mentally.

Worked Example

Determining rational vs irrational

Problem

\sqrt{\frac{16}{25}}

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

State whether the following number is rational or irrational, and give a reason:

\sqrt{9}

L1 · Knowledge2 marks

Write down the set of integers between

-3

and

2

(inclusive).

L2 · Routine Procedures3 marks

Use the distributive property to simplify:

7(3x - 4) + 2(x + 5)

L2 · Routine Procedures3 marks

-15

a natural number, whole number, integer, and/or rational number? List ALL sets it belongs to.

L3 · Complex Procedures4 marks

Between which two consecutive integers does

\sqrt{50}

lie? Without a calculator, explain your reasoning.

L3 · Complex Procedures5 marks

A shopkeeper notices that multiplying the price of an item by

\frac{3}{4}

and then adding R12 gives the same result as multiplying the original price by

\frac{1}{2}

and adding R18. Find the original price.

L4 · Problem Solving4 marks

Prove that the sum of any two rational numbers is always rational. Use

\frac{a}{b}

and

\frac{c}{d}

(where

a,b,c,d \in \mathbb{Z}

b \neq 0

d \neq 0

L4 · Problem Solving5 marks

Sipho saves R150 in Week 1. Each week he saves R20 more than the previous week. How much does he save in total over 6 weeks?

Weeks 2–3

1.2 Integer Calculations

Perform calculations involving all four operations with integers
Recognise and use commutative, associative and distributive properties for integers
Recognise additive and multiplicative inverses for integers
Solve problems in contexts involving multiple operations with integers

🌍

Real-World Connection

Think of integers like a bank account. Positive numbers (+) are deposits, negative numbers (−) are withdrawals. Adding a negative is like making a withdrawal. Subtracting a negative is like reversing a withdrawal — so you get money back! This is why −(−5) = +5.

Property / Rule

Multiplying Integers — Sign Rules

When multiplying or dividing integers, the sign of the answer depends on the signs of the numbers involved.

(+) \times (+) = + \qquad (-) \times (-) = + \qquad (+) \times (-) = - \qquad (-) \times (+) = -

💡 Tip

Memory trick for multiplication signs: 'Same signs → Positive result. Different signs → Negative result.' This works for division too.

Property / Rule

Additive Inverse

The additive inverse of a number is what you add to it to get zero. For any integer a, its additive inverse is −a.

a + (-a) = 0 \qquad \text{e.g. } 7 + (-7) = 0

Property / Rule

Multiplicative Inverse

The multiplicative inverse (reciprocal) of a non-zero number a is 1/a. Multiplying a number by its reciprocal always gives 1.

a \times \frac{1}{a} = 1 \quad (a \neq 0) \qquad \text{e.g. } 5 \times \frac{1}{5} = 1

🚨 Common Mistake

A common mistake: −3² ≠ (−3)². Without brackets, the exponent applies ONLY to 3. So −3² = −(3²) = −9. But (−3)² = (−3)(−3) = +9. The brackets make a huge difference!

Worked Examples

Worked Example

Operations with integers

Problem

Simplify: \quad (a)\; -8 + (-5) \quad (b)\; -6 - (-10) \quad (c)\; (-4) \times (-7) \times 2

Worked Example

Financial context with integers

Problem

Nandi's bank account has a balance of R-240 (overdrawn). She deposits R500 and then withdraws R180. What is her final balance?

Worked Example

Order of operations with integers (BODMAS)

Problem

\quad (a)\; -3 \times (-4)^2 + 18 \div (-3) \quad (b)\; \frac{-20 + (-4)^2}{-6 \div 2}

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge1 mark

Calculate:

(-3) \times (-5)

L1 · Knowledge1 mark

What is the additive inverse of

-11

L2 · Routine Procedures4 marks

Simplify:

\dfrac{(-12) + (-8)}{(-4)} + 3 \times (-2)

L2 · Routine Procedures4 marks

The temperature in Bloemfontein was

-3°C

at midnight. By noon it had risen by

17°C

, then fell by

8°C

in the evening. What was the evening temperature?

L3 · Complex Procedures5 marks

Evaluate

-2(3a - b) - 3(a - 2b)

when

a = -1

and

b = 3

L3 · Complex Procedures5 marks

A submarine is at

-180\text{ m}

(180 m below sea level). It rises at

15\text{ m/min}

. After how many minutes will it be at

-30\text{ m}

? Set up an equation and solve.

L4 · Problem Solving5 marks

The product of two integers is

-36

and their sum is

+5

. Find the two integers.

L4 · Problem Solving6 marks

Submarine A starts at sea level (0 m) and descends at 5 m/min. Submarine B starts 35 m below sea level and ascends at 2 m/min. (a) Write an expression for each submarine's depth below sea level after

t

minutes. (b) Find the time when they are at the same depth. (c) State that depth.

Weeks 2–3

1.3 Multiples, Factors, Ratio & Rate

Find the LCM and HCF of two or more whole numbers using prime factorisation
Solve problems involving ratio and rate, including simplifying and comparing ratios
Solve problems in financial and measurement contexts using ratio, rate, and proportion
Apply ratio and rate to real-world situations including unit rates and scale drawings

🌍

Real-World Connection

A ratio is a comparison — if a school has 3 maths teachers for every 90 learners, the ratio is 1:30. A rate adds units — a car travelling 120 km in 2 hours has a rate of 60 km/h. LCM and HCF are used every day: tiling a floor (HCF gives the largest square tile that fits), scheduling buses (LCM tells you when two buses next depart together), and mixing cement or paint (ratios keep the recipe right).

Definition

Multiple

A multiple of a number n is any number you get by multiplying n by a positive integer. For example, multiples of 6 are: 6, 12, 18, 24, 30, … The Lowest Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of them.

\text{LCM}(a,b) = \text{smallest positive multiple of both } a \text{ and } b

Definition

Factor

A factor of n divides exactly into n with no remainder. The Highest Common Factor (HCF) is the largest factor shared by all the given numbers. Also called the Greatest Common Divisor (GCD).

\text{HCF}(a,b) = \text{largest integer that divides both } a \text{ and } b

Property / Rule

Prime Factorisation Method for LCM and HCF

1. Write each number as a product of prime factors (factor tree or division ladder). 2. HCF: multiply the COMMON prime factors raised to the LOWEST power. 3. LCM: multiply ALL prime factors raised to the HIGHEST power.

\text{HCF}(12,18): 12=2^2\cdot3,\; 18=2\cdot3^2 \Rightarrow \text{HCF}=2^1\cdot3^1=6. \quad \text{LCM}=2^2\cdot3^2=36.

Definition

Ratio

A ratio compares two or more quantities of the SAME kind. It can be written as a:b, a/b, or 'a to b'. A ratio has no units. Always simplify ratios to lowest terms by dividing by the HCF.

a : b = \frac{a}{b} \quad (\text{divide both sides by HCF})

Definition

Rate

A rate compares quantities of DIFFERENT kinds (different units). It always has units. Examples: speed (km/h), unit price (R/kg), fuel consumption (ℓ/100 km). A unit rate has a denominator of 1.

\text{Rate} = \frac{\text{quantity 1 (with unit)}}{\text{quantity 2 (with unit)}} \quad e.g.\; \frac{120 \text{ km}}{2 \text{ h}} = 60 \text{ km/h}

💡 Tip

To divide a quantity in a given ratio a:b:c, first find the total parts (a+b+c), then each share = (part/total) × quantity. For example, dividing R300 in the ratio 2:3:5 — total parts = 10; shares = R60 : R90 : R150.

Worked Examples

Worked Example

Finding HCF and LCM using prime factorisation

Problem

Find the HCF and LCM of 72 and 120.

Worked Example

Dividing in a ratio

Problem

Three friends — Amo, Busi, and Sipho — invest R2 400 in a business in the ratio 1:2:3. How much does each person receive from a profit of R2 400?

Worked Example

Unit rate and comparison

Problem

Store A sells 5 kg of rice for R89.50. Store B sells 3 kg for R55.20. Which store offers better value? How much would 12 kg cost at the better store?

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge4 marks

Find the HCF and LCM of 24 and 36.

L1 · Knowledge2 marks

Simplify the ratio

45 : 75

to its simplest form.

L2 · Routine Procedures4 marks

Divide R1 260 in the ratio 2:5:7.

L2 · Routine Procedures3 marks

A recipe requires flour and sugar in the ratio 5:2. If a baker uses 750 g of flour, how much sugar is needed?

L3 · Complex Procedures4 marks

Two buses leave the same depot. Bus A departs every 12 minutes; Bus B departs every 20 minutes. They depart together at 06:00. At what time will they next depart together?

L3 · Complex Procedures4 marks

A car uses 8 litres of petrol per 100 km. Petrol costs R22.50 per litre. How much does it cost to travel 650 km?

L4 · Problem Solving5 marks

Thabo and Lerato share profits in the ratio 3:5. If Lerato receives R480 more than Thabo, what is the total profit?

L4 · Problem Solving5 marks

A map has a scale of 1:250 000. Two towns are 6.4 cm apart on the map. What is the actual distance in km? If a road between them is built at R1.2 million per km, what is the total cost?

Exponents