Ch 1 · 1.1–1.2 Types of Number and Sets · O Level Mathematics 4024

Topic 1 of 6 in Number — O Level Mathematics 4024 — 47 lessons.

Introduction

Every number belongs to a family: whole numbers for counting, negative numbers for temperatures below zero, fractions for sharing, and numbers like π and sqrt2 that cannot be written as fractions at all. Sets give a precise language for grouping things and Venn diagrams a picture of how groups overlap. (Syllabus sections 1.1 and 1.2.)

Key ideas — Families of numbers

Numbers in words: six billion is 6 000 000 000; 10 007 is ten thousand and seven.

🧑‍🏫 Teacher's explanation

Think of numbers as a massive family tree. At the top, you have 'Natural numbers' (the basic counting numbers like 1, 2, 3). As you expand the family, you get 'Integers', which include the negatives and zero. Some numbers are special because of how they are built: 'Prime numbers' are the 'atoms' that can't be broken down further, while 'Square' and 'Cube' numbers are built by multiplying a number by itself a specific number of times.

📖 Definitions

🔍 Worked example — Easy

Write 120 as a product of its prime factors using index notation.

🔍 Worked example — Standard

Find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of 24 and 36.

🔍 Worked example — Exam-style

In a class of 30 students, 18 study Biology (B), 15 study Chemistry (C), and 7 study both. Find the number of students who study neither Biology nor Chemistry.

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Prime factors, HCF and LCM

Write a number as a product of prime factors with a factor tree or repeated division: 72 = 2 × 2 × 2 × 3 × 3 = 2^3 × 3^2.

🧑‍🏫 Teacher's explanation

Think of prime numbers as the 'atoms' of the number world. Just as every molecule is built from a specific combination of atoms, every integer greater than 1 is built from a unique combination of prime numbers. When we write a number as a product of its prime factors, we are essentially looking at its 'chemical formula'.

📖 Definitions

🔍 Worked example — Easy

Express 120 as a product of its prime factors in index notation.

🔍 Worked example — Standard

Find the HCF and LCM of 48 and 72.

🔍 Worked example — Exam-style

Two lights flash at different intervals. Light A flashes every 45 seconds. Light B flashes every 60 seconds. If they flash together now, after how many seconds will they next flash together?

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Set language

A set is a collection of elements, written in curly brackets.

🧑‍🏫 Teacher's explanation

Imagine you are organizing a collection of items, like a box of colored pens. A 'set' is simply that box. Each individual pen is an 'element' of that set. If you have a box of only blue pens, and you pick up a red pen, that red pen is 'not an element' of your blue pen set.

📖 Definitions

🔍 Worked example — Easy

Let mathscrE = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10\. If A = \2, 4, 6, 8, 10\ and B = \1, 2, 3, 4, 5\, find A ∩ B.

🔍 Worked example — Standard

Given mathscrE = \x : x ∈ mathbbN, 1 ≤ x ≤ 10\, P = \x : x is a prime number\, and Q = \x : x is a multiple of 3\. Find n(P ∪ Q).

🔍 Worked example — Exam-style

In a class of 30 students (mathscrE), 18 study Biology (B), 15 study Chemistry (C), and 7 study both. Find the number of students who study neither Biology nor Chemistry (n(B' ∩ C')).

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Venn diagrams

Draw a rectangle for mathscrE and overlapping circles for the sets (up to three). Fill in the overlap first, then the rest of each circle, then the outside. Shading questions: A ∩ B is the overlap; A ∪ B is everything inside the circles; (A ∪ B)' is outside both.

🧑‍🏫 Teacher's explanation

Think of a Venn diagram as a way to organize different groups of things so you can see exactly what they have in common and what makes them unique. Imagine you are organizing a party guest list. You have a list of 'People who like Pizza' and a list of 'People who like Burgers'. Some people like both—they belong to both lists at once. Some people like only pizza. Some like only burgers. And some people at the party don't like either!

Ch 1 · 1.1–1.2 Types of Number and Sets · O Level Mathematics 4024

All chapters in Number