Ch 1 · 1.1–1.2 Types of Number and Sets · IGCSE Mathematics 0580
Topic 1 of 6 in Number — IGCSE Mathematics 0580 — 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!
📖 Definitions
🔍 Worked example — Easy
Given mathscrE = \1, 2, 3, 4, 5, 6, 7, 8\, A = \1, 2, 3, 4\, and B = \3, 4, 5, 6\, find A ∩ B.
🔍 Worked example — Standard
In a class of 20 students (mathscrE), 12 students study Art (A) and 10 students study Music (M). 5 students study both. Find n(A ∪ M)'.
🔍 Worked example — Exam-style
Shade the region representing (A ∪ B)' ∩ C in a 3-set Venn diagram where A, B, C overlap.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
mathscrE = \1, 2, dots, 12\, A = multiples of 3, B = even numbers. A = \3, 6, 9, 12\, B = \2, 4, 6, 8, 10, 12\. A ∩ B = \6, 12\; n(A ∪ B) = 8; A' = \1, 2, 4, 5, 7, 8, 10, 11\.
🔍 Worked example — Easy
Given the universal set mathscrE = \x : 1 ≤ x ≤ 10, x ∈ mathbbZ\, A = \x : x is a prime number\, and B = \x : x is an even number\, list the elements of A ∩ B and A ∪ B.
🔍 Worked example — Exam-style
Two flashing lights flash at intervals of 45 seconds and 60 seconds. If they flash together at 12:00 PM, at what time will they next flash together?
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 2^2 × 3^2 × 5. (2) HCF 12, LCM 180. (3) sqrt5 and π. (4) P = \2, 3, 5, 7\, Q = \1, 3, 5, 7, 9\: P ∩ Q = \3, 5, 7\; P ∪ Q = \1, 2, 3, 5, 7, 9\, so n = 6. (5) 25 + 20 - 9 = 36 like at least one, so 4 like neither.
All chapters in Number
- 1.1–1.2 Types of Number and Sets
- 1.3, 1.7, 1.18 Powers, Roots, Indices and Surds
- 1.4–1.6 Fractions, Decimals, Ordering and the Four Operations
- 1.8–1.10, 1.14 Standard Form, Estimation, Limits of Accuracy and Calculators
- 1.11, 1.12, 1.15, 1.16 Ratio, Rates, Time and Money
- 1.13, 1.17 Percentages and Exponential Growth and Decay