The commutative, associative and distributive properties, the identity roles of 0 and 1, and why dividing by 0 is undefined — the rules that let you rearrange any calculation, with a diagram, worked examples and an exam-style worksheet with memo.
Whole numbers are the counting numbers you have used since Grade 1: 0, 1, 2, 3, 4, … . You already add, subtract, multiply and divide them without thinking. So why give them a whole chapter in Grade 8? Because this is where you stop just getting answers and start understanding the rules that make number work — the very same rules that quietly run every algebra, equation and factorising question for the next five years. Get comfortable here and a surprising amount of later maths suddenly feels obvious.
1The three properties that never break
Mathematicians noticed that whole numbers always obey three rules, no matter which numbers you choose. These are called properties, and they are the reason you are allowed to rearrange and regroup a calculation to make it easier.
a + b = b + a and a × b = b × a
order doesn't matter for + and ×
(a + b) + c = a + (b + c)
grouping doesn't matter for + and ×
The third one — the distributive property — is the most powerful of all. It is the bridge between multiplication and addition, and it is exactly what you will use to expand brackets in algebra. It is worth really seeing why it is true:
a × (b + c) = a × b + a × c
- 1Split 104 into 100 + 4.Numbers close to a round number are easy to break up.
- 26 × 104 = 6 × (100 + 4).Same value, friendlier form.
- 3= 6 × 100 + 6 × 4 = 600 + 24.Distribute the 6 into each part.
- 4= 624.Done in your head — no calculator.
Subtraction and division are not commutative or associative: 8 − 3 is not 3 − 8, and 12 ÷ 4 is not 4 ÷ 12. Only + and × get to rearrange freely.
20 and 1 — the two numbers with superpowers
Two whole numbers behave so specially that they earn their own names.
- 0 is the identity for addition: adding 0 changes nothing — a + 0 = a (so 5 + 0 = 5).
- 1 is the identity for multiplication: multiplying by 1 changes nothing — a × 1 = a (so 5 × 1 = 5).
3Dividing by zero
Division by 0 is undefined. 6 ÷ 0 has no answer — ask "what times 0 gives 6?" and nothing works, because anything times 0 is 0. But 0 ÷ 6 = 0 (that one is fine). Never divide by 0.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Name the property shown: 7 × 5 = 5 × 7
Show answer ▾Hide answer ▴
Commutative property of multiplication (the order was swapped). - 2Name the property shown: (2 + 9) + 1 = 2 + (9 + 1)
Show answer ▾Hide answer ▴
Associative property of addition (the grouping changed). - 3Use the distributive property to work out 7 × 98 without a calculator.
Show answer ▾Hide answer ▴
7 × (100 − 2) = 700 − 14 = 686 - 4Use the distributive property to work out 8 × 53.
Show answer ▾Hide answer ▴
8 × (50 + 3) = 400 + 24 = 424 - 5Fill in the missing numbers: 46 + ___ = 46 and 46 × ___ = 46
Show answer ▾Hide answer ▴
0 (additive identity) and 1 (multiplicative identity). - 6What is 0 ÷ 8, and what is 8 ÷ 0?
Show answer ▾Hide answer ▴
0 ÷ 8 = 0; 8 ÷ 0 is undefined.
Quick Quiz
4 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 8 past papers and worksheets on this topic.