Number
Numbers are the foundation of all mathematics. In this topic you will work with integers, fractions, decimals, and percentages, and explore properties like prime numbers and divisibility.
What You'll Learn
- Understand integers and place them on a number line
- Identify prime numbers, factors, and multiples
- Find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM)
- Convert between fractions, decimals, and percentages
- Add, subtract, multiply, and divide fractions
- Simplify and work with ratios
IB Assessment Focus
Criterion A: Select and apply correct methods for calculations with integers, fractions, and percentages.
Criterion B: Discover patterns in factors, multiples, and prime numbers and verify your rules.
Criterion C: Use correct mathematical vocabulary and show clear working.
Criterion D: Apply number skills to real-world problems (e.g., shopping, recipes, sharing).
Key Vocabulary
| Term | Definition | Example |
|---|---|---|
| Integer | A whole number (positive, negative, or zero) | … −2, −1, 0, 1, 2 … |
| Factor | A number that divides exactly into another number | Factors of 12: 1, 2, 3, 4, 6, 12 |
| Multiple | The result of multiplying a number by an integer | Multiples of 3: 3, 6, 9, 12… |
| Prime number | A number greater than 1 with exactly two factors: 1 and itself | 2, 3, 5, 7, 11, 13 |
| Composite number | A number greater than 1 that has more than two factors | 4, 6, 8, 9, 10, 12 |
| LCM | Lowest Common Multiple — the smallest number that is a multiple of two given numbers | LCM of 4 and 6 is 12 |
| HCF | Highest Common Factor — the largest number that is a factor of two given numbers | HCF of 12 and 18 is 6 |
| Ratio | A comparison of two or more quantities | 3 : 5 means 3 parts to 5 parts |
| Percentage | A fraction expressed out of 100 | 45% = 45/100 = 0.45 |
Integers & Prime Numbers
Integers are the building blocks of number work. Understanding prime numbers helps you with factors, multiples, and simplifying fractions.
Understanding Integers
Integers are whole numbers. They can be positive (1, 2, 3…), negative (−1, −2, −3…), or zero. They do NOT include fractions or decimals.
Numbers increase as you move to the right on a number line. For example:
−5 < −3 < −1 < 0 < 2 < 4 < 7
A common mistake is thinking −5 is greater than −3 because 5 is greater than 3. Remember: further left on the number line means smaller.
Rules for Integer Arithmetic
- Adding a positive number moves right on the number line: 3 + 5 = 8
- Adding a negative number moves left: 3 + (−5) = 3 − 5 = −2
- Subtracting a negative number is the same as adding: 3 − (−5) = 3 + 5 = 8
- Positive × Positive = Positive (3 × 4 = 12)
- Negative × Negative = Positive (−3 × −4 = 12)
- Positive × Negative = Negative (3 × −4 = −12)
- The same rules apply for division
Prime Numbers
A prime number is a number greater than 1 that has exactly two factors: 1 and itself. The first ten prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Testing if a Number is Prime
- Check if the number is divisible by 2, 3, 5, 7, etc. (try small primes)
- If none of these divide evenly into the number, it is prime
- You only need to test up to the square root of the number
Example: Is 29 prime?
√29 ≈ 5.4, so test 2, 3, and 5 only.
29 ÷ 2 = 14.5 (not whole) 29 ÷ 3 = 9.67 (not whole) 29 ÷ 5 = 5.8 (not whole)
None divide evenly, so 29 is prime.
Factors, Multiples, HCF & LCM
Factors and multiples help you simplify fractions, find common denominators, and solve real-world problems about grouping and sharing.
Finding All Factors
To find the factors of a number, look for all pairs of numbers that multiply to give that number.
1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Finding the HCF — Factor Method
- List all factors of each number
- Identify the factors that appear in both lists
- The largest common factor is the HCF
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12
HCF = 12
Finding the LCM — List Method
- List the first several multiples of each number
- Find the smallest number that appears in both lists
- That is the LCM
Multiples of 6: 6, 12, 18, 24, 30, 36…
Multiples of 8: 8, 16, 24, 32, 40…
LCM = 24
Prime Factor Decomposition
Every composite number can be written as a product of prime factors. This is called prime factorisation.
60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5
So 60 = 2² × 3 × 5
Fractions & Decimals
Fractions and decimals are two ways of writing parts of a whole. Being able to convert between them and perform operations is essential.
Equivalent Fractions
Equivalent fractions represent the same value. You create them by multiplying (or dividing) both the numerator and denominator by the same number.
- 1/2 = 2/4 = 3/6 = 5/10 (all equal 0.5)
- 2/3 = 4/6 = 6/9 = 8/12 (all equal 0.666…)
Simplifying Fractions
To simplify a fraction, divide both the numerator and denominator by their HCF.
HCF of 18 and 24 is 6.
18 ÷ 6 = 3, 24 ÷ 6 = 4
So 18/24 = 3/4
Operations with Fractions
- Find a common denominator (use the LCM of the denominators)
- Convert each fraction to have that denominator
- Add or subtract the numerators; keep the denominator the same
- Simplify if possible
Example: 2/3 + 1/4
LCM of 3 and 4 = 12
2/3 = 8/12, 1/4 = 3/12
8/12 + 3/12 = 11/12
Multiply the numerators together and the denominators together.
2/3 × 4/5 = (2 × 4) / (3 × 5) = 8/15
Flip the second fraction (find the reciprocal), then multiply.
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Converting Fractions to Decimals
Divide the numerator by the denominator:
- 3/4 = 3 ÷ 4 = 0.75
- 1/3 = 1 ÷ 3 = 0.333… (recurring)
- 7/8 = 7 ÷ 8 = 0.875
Key Fraction-Decimal-Percentage Equivalents
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
| 1/3 | 0.333… | 33.3…% |
Percentages & Ratios
Percentages and ratios help us compare quantities and solve everyday problems like discounts, tips, and sharing.
Converting Between Forms
Percentage → Decimal: divide by 100
Percentage → Fraction: write over 100, then simplify
- 0.35 → 0.35 × 100 = 35%
- 72% → 72 ÷ 100 = 0.72
- 60% → 60/100 = 3/5
Finding a Percentage of an Amount
Method: Convert the percentage to a decimal (or fraction), then multiply.
Example: Find 15% of 80.
15% = 0.15
0.15 × 80 = 12
Expressing One Quantity as a Percentage of Another
(18 ÷ 25) × 100 = 0.72 × 100 = 72%
Ratios
A ratio compares two or more quantities. The ratio 3 : 5 means "for every 3 of one thing, there are 5 of another."
Divide both parts by their HCF.
12 : 18 → HCF = 6 → 12 ÷ 6 : 18 ÷ 6 = 2 : 3
Worked Examples
These examples show the step-by-step reasoning expected in your assessments. Notice how each step is explained clearly.
LCM: Multiples of 12: 12, 24, 36, 48… Multiples of 18: 18, 36, 54… The first common multiple is 36.
Step 2: Convert: 3/5 = 9/15 and 2/3 = 10/15.
Step 3: Add numerators: 9/15 + 10/15 = 19/15 = 1 4/15.
The answer is 19/15 or 1 and 4/15 as a mixed number.
Step 2: Subtract: $45 − $9 = $36.
Alternative method: 100% − 20% = 80%. So the sale price = 0.80 × $45 = $36.
Step 2: Value of one part = $72 ÷ 9 = $8.
Step 3: First share = 4 × $8 = $32. Second share = 5 × $8 = $40.
Check: $32 + $40 = $72 ✓
So 120 = 2³ × 3 × 5
(b) Decimal: 3 ÷ 5 = 0.6.
(c) Percentage: 0.6 × 100 = 60%.
Practice Q&A
Attempt each question before revealing the model answer. Focus on showing clear working.
LCM: Multiples of 15: 15, 30, 45, 60… Multiples of 20: 20, 40, 60… LCM = 60.
First share: 2 × $12 = $24. Second: 3 × $12 = $36. Third: 5 × $12 = $60.
Check: $24 + $36 + $60 = $120 ✓
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.