IGCSE / O Level Mathematics · Topic 1 Number · Chapter 1 · pp. 4–12
Number and language
Before you can calculate with numbers, you need the words for them. This chapter names the different sets of numbers, takes numbers apart into prime factors, and works with powers, roots and negative numbers. Every later chapter uses this vocabulary.
Learning objectives
By the end of this chapter you can:
- Sort numbers into natural numbers, integers, rational, irrational and real numbers.
- Recognise prime, square and cube numbers, and find the reciprocal of a number.
- Write a number as a product of its prime factors, and use them to find the HCF and LCM.
- Find squares, cubes, square roots, cube roots and other powers and roots, with and without a calculator.
- Use directed (positive and negative) numbers in real situations such as temperature and money.
Book exercises: 1.1 to 1.11 and Student assessment 1 (pp. 4–12).
🎯 Core and Extended
Cambridge IGCSE (0580) and O Level Mathematics have a Core and an Extended tier. Everything in this chapter is on both. Extended students should be quick and confident with it, because the harder chapters lean on it all the time.
✏️ How to use this page
Work through the tabs in order. Each has worked examples and a short Try it with hidden answers. The book's full exercises are in this chapter's printable assignment, and there is a quiz at the end.
Vocabulary for sets of numbers
🔑 Key ideas
- Each set of numbers is an extension of the one before it: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ.
- A number is rational if it can be written as a fraction a/b of integers with b ≠ 0.
- An irrational number's decimal never ends and never repeats.
A square is a quadrilateral, a polygon and a 2D shape all at once. Numbers can belong to several sets in the same way.
| Set | Symbol | What is in it | Examples |
|---|---|---|---|
| Natural numbers | ℕ | The counting numbers, with 0 | 0, 1, 2, 3, … |
| Integers | ℤ | Whole numbers, positive, negative and zero | …, −2, −1, 0, 1, 2, … |
| Rational numbers | ℚ | Anything that can be written as a fraction | , 0.2, 7, 0.222… |
| Irrational numbers | — | Numbers that cannot be written as a fraction | √2, √5, π |
| Real numbers | ℝ | All rational and irrational numbers together | every number on the number line |
💡 Some books start the natural numbers at 1. This book, and the Cambridge syllabus, include 0.
Rational or irrational?
Every terminating or recurring decimal is rational, because it can be written as a fraction:
0.2 = 0.3 = 7 = 1.53 = 0.222… =
| Rational numbers include | Irrational numbers include |
|---|---|
| whole numbers and fractions | the square root of any number that is not a square number |
| terminating decimals (0.625) | decimals that never end and never repeat (π = 3.14159…) |
| recurring decimals (0.666…) | sums like 6 − √3, where one part is irrational |
📘 Example 1 · Why is √2 irrational?
- Draw a square with sides of 1 unit. By Pythagoras' rule (Chapter 29), its diagonal is √(1² + 1²) = √2.
- Work out the decimal: √2 = 1.41421356…
- Ask the flowchart's question: does it end or repeat? The digits go on for ever with no repeating pattern.
- So √2 cannot be written as a fraction. √2 is irrational.
📘 Example 2 · Careful with roots
- √25 = 5, so √25 is rational, even though it has a root sign.
- √2 × √8 = √16 = 4, which is rational. Two irrational numbers can multiply to a rational one.
- √2 + √3 = 3.146…, which is irrational.
Primes and reciprocals
- A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11, 13, … are prime. 1 is not prime, because it has only one factor.
- The reciprocal of a number is 1 divided by it. The reciprocal of 5 is . The reciprocal of is : just turn the fraction upside down.
- To find the reciprocal of a mixed number, make it an improper fraction first: = , so its reciprocal is .
🧪 Try it: the sieve of Eratosthenes
Exercise 1.1 asks you to find every prime below 100 this way. Press a prime to cross out all of its multiples, or run the whole sieve.
✏️ Try it
- Is 0.625 rational or irrational?
- Is √7 rational or irrational?
- Write the reciprocal of , of 6, and of .
Answers
1. Rational () · 2. Irrational (7 is not a square number) · 3. , ,
Factors, multiples, HCF and LCM
🔑 Key ideas
- A factor of 12 divides into 12 exactly: 1, 2, 3, 4, 6, 12.
- A multiple of 5 is in the 5 times table: 5, 10, 15, 20, …
- Every whole number above 1 can be written as a product of primes in exactly one way.
Prime factors
The factors of 12 are 1, 2, 3, 4, 6 and 12. Of these, only 2 and 3 are prime, so they are the prime factors of 12. To find them, divide by the primes in order, smallest first, until you reach 1.
📘 Example 3 · Write 24 and 75 as products of primes
Method: divide by the smallest prime that goes in exactly, write the answer underneath, and repeat until you reach 1.
2 | 24 2 | 12 2 | 6 3 | 3 | 1
3 | 75 5 | 25 5 | 5 | 1
- 24 is even, so divide by 2: 24 ÷ 2 = 12. Again: 12 ÷ 2 = 6, and 6 ÷ 2 = 3.
- 3 is not even, so try the next prime, 3: 3 ÷ 3 = 1. Stop.
- Multiply the primes on the left: 24 = 2 × 2 × 2 × 3 = 2³ × 3.
- 75 is odd, so skip 2. Try 3: 75 ÷ 3 = 25.
- 25 does not divide by 3, so try 5: 25 ÷ 5 = 5, then 5 ÷ 5 = 1. Stop.
- 75 = 3 × 5 × 5 = 3 × 5².
Highest common factor (HCF)
The HCF is the biggest number that divides into all of them. Write each number in prime factors, then multiply the primes they share.
📘 Example 4 · HCF of 12 and 18
- Write each number as a product of primes: 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3.
- Pick out the primes they share: one 2 and one 3 (put them in the overlap of the Venn diagram below).
- Multiply the shared primes: HCF = 2 × 3 = 6.
- Check: 12 ÷ 6 = 2 and 18 ÷ 6 = 3, both exact. ✔
Lowest common multiple (LCM)
The LCM is the smallest number that all of them divide into. For small numbers, list multiples of the biggest until the others divide into one. With prime factors, take every prime the highest number of times it appears in any of them.
📘 Example 5 · LCM of 6 and 10
Method A · list multiples
- List multiples of the bigger number, 10: 10, 20, 30, 40, …
- Stop at the first one that 6 divides exactly: 10 ✘, 20 ✘, 30 ✔.
- LCM = 30.
Method B · prime factors
- Write each number in primes: 6 = 2 × 3 and 10 = 2 × 5.
- Take every prime that appears, the most times it appears in either number: 2 once, 3 once, 5 once.
- Multiply them: LCM = 2 × 3 × 5 = 30.
💡 For two numbers, HCF × LCM = the two numbers multiplied. For 6 and 10: 2 × 30 = 60 = 6 × 10.
🧪 Try it: prime factors, HCF and LCM
Type two or three whole numbers (2 to 100,000), separated by commas.
✏️ Try it
- List all the factors of 36.
- Write 56 as a product of prime factors.
- Find the HCF of 36, 63 and 108.
- Find the LCM of 4, 15.
Answers
1. 1, 2, 3, 4, 6, 9, 12, 18, 36 · 2. 2³ × 7 · 3. 9 · 4. 60
Squares, cubes, powers and roots
🔑 Key ideas
- A square number is an integer times itself: 1, 4, 9, 16, 25, …
- A cube number is an integer times itself, times itself again: 1, 8, 27, 64, 125, …
- Roots undo powers: √ undoes squaring, ∛ undoes cubing.
| Written | Read as | Means | Value |
|---|---|---|---|
| 4² | 4 squared | 4 × 4 | 16 |
| 4³ | 4 cubed | 4 × 4 × 4 | 64 |
| 5⁶ | 5 to the power 6 | 5 × 5 × 5 × 5 × 5 × 5 | 15,625 |
| √16 | the square root of 16 | the positive number that squares to 16 | 4 |
| ∛64 | the cube root of 64 | the number that cubes to 64 | 4 |
| ∜625 | the fourth root of 625 | the positive number that, to the power 4, is 625 | 5 |
Square roots
4 × 4 = 16, but (−4) × (−4) = 16 as well, so 16 has two square roots, +4 and −4, written ±4. By convention the symbol √16 means only the positive one, so √16 = 4. A negative number like −16 has no real square root, because any number squared is positive or zero.
📘 Example 6 · Roots of fractions and decimals
(a) √
- Change the mixed number to an improper fraction: = = .
- Root the top and the bottom separately: √25 = 5 and √9 = 3.
- Answer: = .
(b) √0.09
- Write the decimal as a fraction: 0.09 = .
- Root the top and the bottom: √9 = 3 and √100 = 10, so the root is .
- Back to a decimal: 0.3. Check: 0.3 × 0.3 = 0.09. ✔
Cube roots
∛64 = 4, because 4 × 4 × 4 = 64. Unlike square roots, a negative number does have a cube root: ∛−64 = −4, because (−4) × (−4) × (−4) = −64.
⚠️ Book correction
On p. 21 the book says ∜625 “is ±5”. That clashes with its own rule two pages earlier, that a root sign means the positive root. 625 does have two real fourth roots, 5 and −5, but the symbol ∜625 means only 5, in the same way that √16 means only 4. Write ±5 only when you are asked for all the fourth roots.
On the calculator
- Squares and square roots have their own keys, x² and √.
- Any power uses xʸ (on some calculators ^): 5 xʸ 6 = gives 15625.
- Any root uses ʸ√x (often SHIFT then xʸ): 4 ʸ√x 625 = gives 5.
🧪 Try it: powers and roots
✏️ Try it, without a calculator
1. √121 2. √0.25 3. √ 4. ∛0.027 5. ∛−1000 6. (3⁴)²
Answers
1. 11 · 2. 0.5 · 3. · 4. 0.3 · 5. −10 · 6. 3⁸ = 6561
Directed numbers
🔑 Key ideas
- Directed numbers have a sign: positive (+) or negative (−). They measure in two directions from zero.
- Think of a number line or a thermometer: adding moves up, subtracting moves down.
- The difference between two values is the larger minus the smaller, even when one is negative.
Temperatures below freezing, heights below sea level and an overdrawn bank account are all negative numbers. If it is 4 °C at 10 p.m. and the temperature drops by 6 °C, it is 4 − 6 = −2 °C.
📘 Example 7 · Temperature rise
At 04:00 the temperature was −3 °C. By 09:00 it had risen by 8 °C. What was it at 09:00?
- Decide the operation: “risen” means move up, so add.
- Write it down: (−3) + 8.
- Count on the number line: 3 steps up takes you from −3 to 0, and 5 more takes you to 5.
- Answer: 5 °C.
📘 Example 8 · The difference across zero
The highest temperature ever recorded is 58 °C (Libya) and the lowest is −88 °C (Antarctica). What is the difference?
- Difference means larger − smaller: 58 − (−88).
- Subtracting a negative is the same as adding: 58 − (−88) = 58 + 88.
- Add: 58 + 88 = 146 °C.
- Check on the number line: −88 up to 0 is 88 degrees, then 0 up to 58 is 58 more. 88 + 58 = 146. ✔
🧪 Try it: number line
✏️ Try it
- An apartment roof is 130 m above ground. Its car park is 35 m below ground. How high is the roof above the car park floor?
- A submarine is at a depth of 165 m. The ocean floor is 860 m down. How far is the submarine from the ocean floor?
- A bank balance of $105 has rent of $140 taken out. What is the new balance?
Answers
1. 130 + 35 = 165 m · 2. 860 − 165 = 695 m · 3. 105 − 140 = −$35
Chapter summary
- ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. Rational numbers can be written as fractions; that includes terminating and recurring decimals. Irrational numbers, like √2 and π, cannot.
- A prime has exactly two factors, so 1 is not prime. The reciprocal of x is 1 ÷ x: flip a fraction to find it.
- Divide by primes, smallest first, to write a number as a product of prime factors. Multiply the shared primes for the HCF; take every prime at its highest power for the LCM.
- √ means the positive root. Negative numbers have cube roots but no real square roots.
- Directed numbers work like a number line: add to move up, subtract to move down, and find a difference by subtracting the smaller value from the larger.
📄 Practice questions
The book's Exercises 1.1 to 1.11 and Student assessment 1 are in the printable assignment for this chapter.
Mini-quiz
Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 1 (Number and language). Explanations, the Try it answers, the tools and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.