- Recognise natural numbers and the four arithmetic operations
- Apply the order-of-operations rule
- Evaluate expressions with brackets and powers
- Use the laws of arithmetic to make mental maths easier
Aysel and Murad solved the same problem: 2 + 3 · 4. Aysel added 2 and 3 first and got 20, while Murad multiplied first and got 14. Who is right? The answer lies in a rule that is the same for everyone in mathematics: the order of operations.
Natural numbers and arithmetic operations
The numbers we use to count objects: 1, 2, 3, 4, … The smallest natural number is 1, and there is no largest one.
We perform four arithmetic operations on natural numbers: addition, subtraction, multiplication and division. Each result has its own name: addition gives a sum, subtraction a difference, multiplication a product and division a quotient. To write a number multiplied by itself several times in a short way, we use a power.
- athe base
- nthe exponent — how many times the base is multiplied
For example, 2³ = 2 · 2 · 2 = 8 and 5² = 5 · 5 = 25.
The order-of-operations rule
When an expression contains several operations, we do not carry them out in any order we like, but according to a fixed rule. Otherwise different people would get different answers from the same expression. The rule is:
- Operations inside brackets.
- Powers.
- Multiplication and division — from left to right, in the order they appear.
- Addition and subtraction — again from left to right.
Find the value of 2 + 3 · 4.
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Then add: 2 + 12 = 14.
Murad is right. Aysel actually calculated a different expression: (2 + 3) · 4 = 20.
Evaluate 48 ÷ (2 + 6) · 3 − 2³.
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2) Power: 2³ = 8.
3) Division and multiplication from left to right: 48 ÷ 8 = 6, then 6 · 3 = 18.
4) Subtraction: 18 − 8 = 10.
Leyla bought 3 notebooks at 2 manat each and 2 pens at 1 manat each. She gave the seller 20 manat. How much change should she get?
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Inside the brackets: 3 · 2 + 2 · 1 = 6 + 2 = 8 manat.
Change: 20 − 8 = 12 manat.
The brackets matter here: without them we would get 20 − 6 + 2 = 16, which is wrong.
Laws that make calculation easier
| Law | In symbols | Example |
|---|---|---|
| Commutative law | a + b = b + a, a · b = b · a | 7 · 5 = 5 · 7 = 35 |
| Associative law | (a · b) · c = a · (b · c) | (17 · 25) · 4 = 17 · (25 · 4) = 1700 |
| Distributive law | a · (b + c) = a · b + a · c | 17 · 6 + 17 · 4 = 17 · 10 = 170 |
These laws do not change the value of an expression, but they make calculating much easier. For example, to find 4 · 13 · 25, it is convenient to swap the factors and first compute 4 · 25 = 100: then 100 · 13 = 1300.
Key points
- Natural numbers are the counting numbers 1, 2, 3, …; there is no largest one.
- Order of operations: brackets → powers → multiplication and division → addition and subtraction.
- Multiplication and division, like addition and subtraction, have equal priority and go from left to right.
- The commutative, associative and distributive laws make calculations easier.
Check yourself
10 questions. Every correct answer earns XP.