1 Types of Numbers
Number Sets
| Set | Symbol | Description | Examples |
|---|---|---|---|
| Natural numbers | ℕ | Counting numbers, starting from 1 | 1, 2, 3, 4 … |
| Integers | ℤ | Whole numbers, positive and negative, and zero | … −3, −2, −1, 0, 1, 2, 3 … |
| Rational numbers | ℚ | Any number expressible as a fraction p/q (q ≠ 0) | 1/2, −3, 0.75, 2.333… |
| Irrational numbers | — | Cannot be written as a fraction; infinite non-repeating decimal | √2, π, √5 |
🔑Every integer is a rational number (e.g. 5 = 5/1). But not every rational is an integer.
Absolute Value
The absolute value of a number is its distance from zero — always positive or zero.
Definition
|a| = a if a ≥ 0, and |a| = −a if a < 0
Examples
|7| = 7 |−7| = 7 |0| = 0
Comparing and Ordering Integers
- On a number line, numbers increase from left to right
- −5 < −2 < 0 < 3 < 7 (despite −5 having a larger absolute value than −2)
- A negative number is always less than any positive number
2 Order of Operations (BEDMAS)
🔑BEDMAS: Brackets → Exponents → Division & Multiplication (left to right) → Addition & Subtraction (left to right)
| Step | Operation | Example |
|---|---|---|
| 1st — B | Brackets (innermost first) | (3 + 4) × 2 → 7 × 2 = 14 |
| 2nd — E | Exponents | 2³ + 1 → 8 + 1 = 9 |
| 3rd — DM | Division & Multiplication (left → right) | 12 ÷ 4 × 3 → 3 × 3 = 9 |
| 4th — AS | Addition & Subtraction (left → right) | 10 − 3 + 2 → 7 + 2 = 9 |
Worked Example
✏️
Evaluate: 3 + (2² × 5 − 1) ÷ 3
Step 1 — Brackets: evaluate inside → 2² × 5 − 1
2² = 4
4 × 5 = 20
20 − 1 = 19
Step 2 — Division: 19 ÷ 3 ≈ 6.33
Step 3 — Addition: 3 + 6.33 = 9.33
Step 1 — Brackets: evaluate inside → 2² × 5 − 1
2² = 4
4 × 5 = 20
20 − 1 = 19
Step 2 — Division: 19 ÷ 3 ≈ 6.33
Step 3 — Addition: 3 + 6.33 = 9.33
⚠️Division and multiplication have equal priority — always work left to right. Same for addition and subtraction.
3 Fractions
Fraction Vocabulary
Fraction parts
numerator / denominator (top / bottom)
Equivalent fractions
1/2 = 2/4 = 3/6 (multiply top & bottom by same number)
Simplest form
Divide both by GCF: 6/9 ÷ 3/3 = 2/3
Addition and Subtraction
Fractions must have the same denominator (LCD) before adding or subtracting.
✏️
1/4 + 2/3
LCD = 12
1/4 = 3/12 2/3 = 8/12
3/12 + 8/12 = 11/12
LCD = 12
1/4 = 3/12 2/3 = 8/12
3/12 + 8/12 = 11/12
Multiplication
Multiply numerators together, then denominators. Simplify before or after.
Rule
(a/b) × (c/d) = (a × c) / (b × d)
✏️2/3 × 3/5 = (2 × 3) / (3 × 5) = 6/15 = 2/5
Division
Keep the first fraction, change ÷ to ×, flip the second fraction (find its reciprocal).
💡KCF: Keep · Change · Flip
Rule
(a/b) ÷ (c/d) = (a/b) × (d/c)
✏️3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
Mixed Numbers and Improper Fractions
Mixed → Improper
2 3/4 = (2×4 + 3)/4 = 11/4
Improper → Mixed
11/4 → 11 ÷ 4 = 2 remainder 3 → 2 3/4
4 Decimals & Percentages
Converting Between Forms
| From | To | Method | Example |
|---|---|---|---|
| Fraction | Decimal | Divide numerator by denominator | 3/4 = 3 ÷ 4 = 0.75 |
| Decimal | Fraction | Place over power of 10, simplify | 0.6 = 6/10 = 3/5 |
| Fraction | Percent | Multiply by 100 | 3/4 × 100 = 75% |
| Percent | Decimal | Divide by 100 | 35% ÷ 100 = 0.35 |
| Decimal | Percent | Multiply by 100 | 0.42 × 100 = 42% |
Percentage Calculations
% of a number
part = (percent / 100) × whole
What % is A of B?
percent = (A / B) × 100
Find the whole
whole = part / (percent / 100)
✏️
What is 30% of 80?
part = (30/100) × 80 = 0.30 × 80 = 24
18 is what percent of 60?
percent = (18/60) × 100 = 0.3 × 100 = 30%
part = (30/100) × 80 = 0.30 × 80 = 24
18 is what percent of 60?
percent = (18/60) × 100 = 0.3 × 100 = 30%
5 Square Roots
Perfect Squares
| n | n² | n | n² | n | n² |
|---|---|---|---|---|---|
| 1 | 1 | 5 | 25 | 9 | 81 |
| 2 | 4 | 6 | 36 | 10 | 100 |
| 3 | 9 | 7 | 49 | 11 | 121 |
| 4 | 16 | 8 | 64 | 12 | 144 |
Approximating Non-Perfect Square Roots
🔑Find the two perfect squares the number sits between, then estimate.
✏️
Estimate √50
7² = 49 and 8² = 64, so √50 is between 7 and 8
50 is very close to 49, so √50 ≈ 7.1 (calculator: 7.071…)
7² = 49 and 8² = 64, so √50 is between 7 and 8
50 is very close to 49, so √50 ≈ 7.1 (calculator: 7.071…)
Definition
√a = b means b² = a (a ≥ 0)
Rule
√(a × b) = √a × √b
Cannot simplify
√(a + b) ≠ √a + √b (common mistake!)
6 Common Mistakes to Avoid
| Mistake | What to do instead |
|---|---|
| Wrong BEDMAS order | Always Brackets → Exponents → × and ÷ → + and −. Never add before multiplying. |
| Adding fractions by adding tops and bottoms | 1/2 + 1/3 ≠ 2/5. Find the LCD first: 3/6 + 2/6 = 5/6. |
| Forgetting to flip when dividing fractions | 3/4 ÷ 2/5 ≠ 3/4 × 2/5. Keep · Change · Flip: 3/4 × 5/2. |
| Confusing −a² with (−a)² | −3² = −9 but (−3)² = +9. The bracket changes everything. |
| √(a + b) = √a + √b | This is WRONG. √(9 + 16) = √25 = 5, not 3 + 4 = 7. |
| Forgetting the negative when converting % | 35% = 0.35, not 35. Divide by 100 to convert percent to decimal. |