Understanding
Numbers
This guide explains the why behind every rule — not just what to memorise, but how to think about numbers. Work through each section in order, try every checkpoint question, then reveal the answer to check yourself.
You've been counting since you were very young — 1, 2, 3… These are the natural numbers. But over time, mathematicians needed more. They needed zero, then negatives, then fractions, then something to describe √2. Each new type of number was invented to solve a problem that the previous set couldn't.
Imagine you have $5 in your bank account and spend $8. You owe $3. Natural numbers can't express "owing" — you need numbers below zero. Temperature below freezing, floors below ground level, or debt are all real situations that require integers.
You can't evenly share 7 apples among 3 people using whole numbers. You need 7/3. Any measurement that falls between whole numbers — half a litre, three-quarter turns — requires rational numbers.
The number families
| Name | Symbol | Contains | Examples |
|---|---|---|---|
| Natural numbers | ℕ | Counting numbers from 1 upward | 1, 2, 3, 100 |
| Integers | ℤ | All whole numbers (positive, negative, zero) | −5, 0, 3, −100 |
| Rational numbers | ℚ | Any number of the form p/q where q ≠ 0 | 1/2, −3, 0.75, 2.333… |
| Irrational numbers | — | Cannot be written as a fraction — decimals go on forever without repeating | √2, π, √5 |
Absolute value
The absolute value of a number is its distance from zero on the number line. Distance is always positive (or zero).
7/2 = 3.5 → rational (not integer)
√9 = 3 → natural number (also integer, also rational)
π = 3.14159… → irrational
0 → integer (also rational: 0/1)
√3 = 1.732… (non-repeating, non-terminating) → irrational
b) Give one example of a number that is rational but NOT an integer.
c) Which is greater: |−8| or |5|? Why?
a) Yes. Every integer n can be written as n/1, which fits the definition of a rational number (p/q with q ≠ 0).
b) Any example works: 1/2, 0.75, −3/4, etc. — any fraction that doesn't simplify to a whole number.
c) |−8| = 8 and |5| = 5. So |−8| > |5|. Even though −8 < 5, its distance from zero is larger.
If everyone calculated 3 + 4 × 2 in their own order, they'd get different answers. One person gets 14 (adding first), another gets 11 (multiplying first). Mathematics needs one agreed-upon convention — that's BEDMAS.
Think of 3 + 4 × 2 as "3 plus four pairs of 2." The multiplication groups things together — it's a stronger bond. You assemble the groups first, then combine them. This interpretation is why multiplication (and division) take priority over addition and subtraction.
12 ÷ 4 × 3: do 12 ÷ 4 = 3 first, then 3 × 3 = 9. NOT 12 ÷ 12 = 1.
−3² means −(3²) = −9. The exponent only applies to 3, not to the negative sign.
(−3)² means (−3) × (−3) = +9. The bracket makes the negative part of the base.
a) 20 − 4 × 3 + 1
b) (6 + 2)² ÷ 4 − 3
c) −2² + (−2)²
a) 20 − 4 × 3 + 1
= 20 − 12 + 1 (multiply first)
= 8 + 1 = 9
b) (6 + 2)² ÷ 4 − 3
= 8² ÷ 4 − 3 = 64 ÷ 4 − 3 = 16 − 3 = 13
c) −2² + (−2)² = −4 + 4 = 0
(−2² = −(4) = −4; but (−2)² = +4)
Fraction rules feel arbitrary until you understand why they work. Let's build from scratch.
1/4 + 1/3: you're adding "one quarter" and "one third." They're different-sized pieces. Before you can count them together, you need to cut everything into the same size. Converting to twelfths (LCD = 12) gives you 3/12 + 4/12 = 7/12 — now the pieces are the same size and you can just add the numerators.
"How many halves fit in 3?" is the same as 3 ÷ 1/2. Think of it physically: 3 metres ÷ half-metre lengths = 6 lengths. So 3 ÷ 1/2 = 6 = 3 × 2. Dividing by 1/2 is the same as multiplying by 2 (the reciprocal). That's why Keep · Change · Flip works — it converts division into multiplication.
3/4 = 9/12 (× 3/3)
(Hint for c: convert the mixed number to an improper fraction first.)
a) LCD = 20: 12/20 − 5/20 = 7/20
b) 2/3 × 9/10 = 18/30 = 3/5 (cancel: 2×9 / 3×10 → 1×3 / 1×5)
c) 1 1/2 = 3/2 → 3/2 ÷ 3/4 = 3/2 × 4/3 = 12/6 = 2
Every percentage problem is one of three types. Once you can identify which type you're looking at, the rest is mechanical.
After 20% off, you're paying 80% of the original price. So $64 = 80% of the original.
b) A phone's price increased from $500 to $575. What is the percent increase?
c) 12% of a number is 30. What is the number?
a) % = (45 / 180) × 100 = 0.25 × 100 = 25%
b) % change = ((575 − 500) / 500) × 100 = (75/500) × 100 = 15%
c) whole = 30 ÷ (12/100) = 30 ÷ 0.12 = 250
The square root of a number asks: what number, multiplied by itself, gives this? √25 = 5 because 5 × 5 = 25. Geometrically, √A is the side length of a square with area A.
No fraction p/q, when squared, gives exactly 2. Try it: (1/1)² = 1, (3/2)² = 2.25, (7/5)² = 1.96 — you can get close but never exact. The decimal 1.41421356… goes on forever without repeating. This shocked ancient Greek mathematicians who believed all numbers were rational.
Find the two perfect squares n sits between. The square root is between those roots. Then use the position within the gap to refine.
Example: √50. Since 7² = 49 and 8² = 64, we know √50 is between 7 and 8. 50 is very close to 49, so √50 ≈ 7.1.
So √110 is between 10 and 11
Fraction of the way: 10/21 ≈ 0.48
√110 ≈ 10 + 0.48 ≈ 10.5
b) Estimate √30 to one decimal place (no calculator)
c) True or false: √(16 + 9) = √16 + √9. Justify your answer.
a) 48 = 16 × 3 → √48 = √16 × √3 = 4√3
b) 5² = 25, 6² = 36. 30 is 5/11 of the way from 25 to 36. √30 ≈ 5.5. (Calculator: 5.477…)
c) False. √(16 + 9) = √25 = 5. But √16 + √9 = 4 + 3 = 7. These are different. You cannot split a square root over addition.