Which operations are commutative and associative, how multiplication distributes, why division is only right-distributive, plus rounding and estimating results.

The properties of operations tell you what may be rearranged and what may not. Addition and multiplication are commutative and associative, subtraction and division are not - and those exceptions are the source of the most common mistakes. The second half of the lesson covers rounding and estimating.

Order of operations

\begin{aligned}5 - [8 + 7(4 - 1) - 9 : 3] &= 5 - [8 + 7 \cdot 3 - 3] \\&= 5 - [8 + 21 - 3] \\&= 5 - 26 = -21\end{aligned}

1
$4 - 1 = 3$ the bracket first
2
$9 : 3 = 3$ and the division
3
$8 + 7 \cdot 3 - 3 = 26$ then inside the square bracket
4
$5 - 26 = -21$ the subtraction last

Commutative property

5 + 8 = 8 + 5 = 13

5 \cdot 8 = 8 \cdot 5 = 40

Addition and multiplication are commutative operations.

4 - 2 \neq 2 - 4

4 : 2 \neq 2 : 4

Subtraction and division are not commutative operations.

Associative property

3 + (4 + 5) = (3 + 4) + 5 = 3 + 4 + 5 = 12

3 \cdot (4 \cdot 5) = (3 \cdot 4) \cdot 5 = 3 \cdot 4 \cdot 5 = 60

Addition and multiplication are associative operations.

3 - (4 - 5) \neq (3 - 4) - 5

4 \neq -6

24 : (4 : 2) \neq (24 : 4) : 2

12 \neq 3

Subtraction and division are not associative operations.

Distributive property

3 \cdot (4 + 5) = 3 \cdot 4 + 3 \cdot 5 = 27

(3 + 4) \cdot 5 = 3 \cdot 5 + 4 \cdot 5 = 35

3 \cdot (4 - 5) = 3 \cdot 4 - 3 \cdot 5 = -3

(3 - 4) \cdot 5 = 3 \cdot 5 - 4 \cdot 5 = -5

Multiplication is distributive over addition and subtraction (both left and right distributive).

(60 + 15) : 5 = 60 : 5 + 15 : 5 = 15

60 : (15 + 5) \neq 60 : 15 + 60 : 5

3 \neq 16

(20 - 10) : 5 = 20 : 5 - 10 : 5 = 2

20 : (10 - 5) \neq 20 : 10 - 20 : 5

4 \neq -2

Division is only right-distributive over addition and subtraction.

Estimation and rounding

Round the following numbers to the nearest 10, 100 and 1,000.

number	to 10	to 100	to 1,000
1846	1850	1800	2000
−638	−640	−600	−1000
445	450	400	0

Estimate the results of the following calculations by rounding to the nearest 10.

18 \cdot 21 - 19 : 11 \approx 20 \cdot 20 - 20 : 10 = 400 - 2 = 398

99 : 101 - 49 \cdot 8 \approx 100 : 100 - 50 \cdot 10 = 1 - 500 = -499

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Frequently asked questions

Which operations are commutative?

Addition and multiplication: the order of the terms does not change the result. Subtraction and division are not commutative, because 4 − 2 is not the same as 2 − 4.

Why is division only right-distributive?

Because you may divide a sum by a number, but not a number by a sum. (60 + 15) : 5 splits into 60 : 5 + 15 : 5, whereas 60 : (15 + 5) equals 3 while 60 : 15 + 60 : 5 equals 16.

How do you round a number to the nearest hundred?

Look at the tens digit. If it is 5 or more, increase the hundreds by one, otherwise leave them. The remaining digits become zeros.

A lesson from the PhiBoard knowledge base: phiboard.com/en/knowledge-base/from-zero-to-engineer/properties-of-operations-and-rounding. On the lesson page you will also find games and practice exercises - free and without an account.

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Natural numbers, integers and the order of operations

Divisors, prime factorization, GCD and LCM