TOPICAL MATHEMATICS - Unit 1.1 Numbers and the Four Operations - Key concepts and worked examples

Aug 25, 2026

Uploaded by Miles

Unit 1.1 focuses on foundational number concepts and the four operations through definitions, classifications, and worked calculations. It introduces natural, whole, integer, positive/negative, rational, irrational, and real numbers, then applies operations to find differences and averages from temperature data. The unit develops prime factorisation, highest common factor (HCF), lowest common multiple (LCM), and extends to squares, square roots, cubes, cube roots, reciprocals, and significant figures with rounding examples.

LanguageEnglish

CategoryExam

FormatPDF

Pages178

Reading Time7 h 29 min

AccessFree to download

Unit 1.1 Numbers and the Four Operations

Numbers and number sets

The four operations with examples

Prime factorisation

Factors and multiples (HCF, LCM)

Squares, square roots, cubes, cube roots

Reciprocal and significant figures

CONTENTS Unit 1.1 Numbers and the Four Operations 1 Unit 1.2 Ratio, Rate and Proportion 15 Unit 1.3 Percentage 21 Unit 1.4 Speed 24 Unit 1.5 Algebraic Representation and Formulae 28 Unit 1.6 Algebraic Manipulation 33 Unit 1.7 Functions and Graphs 41 Unit 1.8 Solutions of Equations and Inequalities 51 Unit 1.9 Applications of Mathematics in Practical Situations 64 Unit 1.10 Set Language and Notation 77 Unit 1.11 Matrices 85 Unit 2.1 Angles, Triangles and Polygons 91 Unit 2.2 Congruence and Similarity 103 Unit 2.3 Properties of Circles 113 Unit 2.4 Pythagoras’ Theorem and Trigonometry 119 Unit 2.5 Mensuration 129 Unit 2.6 Coordinate Geometry 138 Unit 2.7 Vectors in Two Dimensions 143 Unit 3.1 Data Handling 151 Unit 3.2 Data Analysis 156 Unit 3.3 Probability 167 Mathematical Formulae 176 | UNIT
1.1 | Numbers and
the Four Operations | | --- | --- | Numbers 1. The set of natural numbers, 􀀟 = {1, 2, 3, } 2. The set of whole numbers, W = {0, 1, 2, 3, } 3. The set of integers, Z = {…,–2,–1, 0, 1, 2,…} 4. The set of positive integers, Z+ = {1, 2, 3, } 5. The set of negative integers, Z– = {–1,–2,–3,…} 6. The set of rational numbers, Q = { ab , a , b 􀀟 Z , b ≠ 0} 7. An irrational number is a number which cannot be expressed in the form ab , where a , b are integers and b ≠ 0. 8. The set of real numbers R is the set of rational and irrational numbers. 9. Real Numbers Whole Numbers Numbers and the Four Operations 1 Example 1 The temperature at the bottom of a mountain was 22 °C and the temperature at the top was –7 °C. Find (a) the difference between the two temperatures, (b) the average of the two temperatures. Solution (a) Difference between the temperatures = 22 –(–7) = 22 + 7 = 29 °C (b) Average of the temperatures = 22~~ ~~+2(−7) 22 − 7 = ~~ ~~ 2 15 = ~~ ~~ 2 = 7.5 °C Prime Factorisation 10. A prime number is a number that can only be divided exactly by 1 and itself. However, 1 is not considered as a prime number. e.g. 2, 3, 5, 7, 11, 13,… 11. Prime factorisation is the process of expressing a composite number as a product of its prime factors. 2 UNIT 1.1 Example 2 Express 30 as a product of its prime factors. Solution Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Of these, 2, 3, 5 are prime factors. 2 3 5 | 30 | | --- | | 15 | | 5 | 1 ∴ 30 = 2 × 3 × 5 Example 3 Express 220 as a product of its prime factors. Solution 220 2 110 2 55 5 11 􀀛 220 = 22 × 5 × 11 Factors and Multiples 12. The highest common factor (HCF) of two or more numbers is the largest factor that is common to all the numbers . Numbers and the Four Operations 3 Example 4 Find the highest common factor of 18 and 30. Solution 18 = 2 × 32 30 = 2 × 3 × 5 HCF = 2 × 3 = 6 Example 5 Find the highest common factor of 80, 120 and 280. Solution Method 1 2 2 2 5 80, 120, 280 40, 60, 140 20, 30, 70 10, 15, 35 2, 3, 7 HCF = 2 × 2 × 2 × 5 = 40 (Since the three numbers cannot be divided further by a common prime factor, we stop here) Method 2 Express 80, 120 and 280 as products of their prime factors 80 = 23 × 5 120 = 23 × 5 280 = 23 × 5 × 7 HCF = 23 × 5 = 40 4 UNIT 1.1 13. The lowest common multiple (LCM) of two or more numbers is the smallest multiple that is common to all the numbers. Example 6 Find the lowest common multiple of 18 and 30. Solution 18 = 2 × 32 30 = 2 × 3 × 5 LCM = 2 × 32 × 5 = 90 Example 7 Find the lowest common multiple of 5, 15 and 30. Solution Method 1 2 3 5 | 5, 15, 30 | | | | --- | --- | --- | | 5, | 15, | 15 | | 5, | 5, | 5 | (Continue to divide by the prime factors until 1 is reached) 1, 1, 1 LCM = 2 × 3 × 5 = 30 Method 2 Express 5, 15 and 30 as products of their prime factors. 5 = 1 × 5 15 = 3 × 5 30 = 2 × 3 × 5 LCM = 2 × 3 × 5 = 30 Numbers and the Four Operations 5

FAQs

What number sets are introduced in Unit 1.1?

The unit introduces natural numbers, whole numbers, integers, positive/negative integers, rational numbers, irrational numbers, and real numbers, along with their set definitions.

How do prime factorisation, HCF, and LCM connect in the unit?

Prime factorisation breaks composite numbers into prime factors; HCF uses the highest common prime-power factors, while LCM uses the smallest set of prime powers that covers all numbers.

How are significant figures and rounding handled?

The unit states which digits are significant, including zeros between non-zero digits, and shows how to round a number like 2014 to specified numbers of significant figures.

Aug 25, 2026

Uploaded by Miles