Åk 6–9
 
English/فارسی

8.3 Associative Law
When you calculate with parenthesis, it can be a good idea to check which role they play when applied to different calculations. In chapter 8.1 you can find the order of precedence rules. Sometimes though, parenthesis don’t matter. The calculations look the same anyway. Do you know when?  We will look at a few examples:


Addition

How would you add the following numbers?

5 + 8 + 3 + 2 + 7 = ?

Maybe according to one of the alternatives below?

We will look at one more example:
6 + 4 + 1 = ?

Alternative 1: (6 + 4) + 1 = 10 + 1 = 11
Alternative 2: 6 + (4 + 1) = 6 + 5 = 11

As you can see, it doesn’t matter in which order you add the numbers.  It still becomes the same sum.


Multiplication

How does this apply for multiplication? How would you multiply the following numbers?

5 · 3
· 2 · 2 = ?

We will look at one more example:
7 · 3
· 2 = ?

Alternative 1: (7 · 3)
· 2 = 21 · 2 = 42
Alternative 2: 7 · (3
· 2) = 7 · 6 = 42

As you can see in multiplication, it doesn’t matter in which order you multiply the numbers. It still becomes the same product.


We can summarise the calculation rules for addition and multiplication in something called the associative laws:



and



If you haven’t done calculations with variables earlier, you might think that it looks odd to calculate with letters. You can learn more about this in the section Algebra.



You have learned that the associative laws apply for addition and multiplication. Do they even work concerning subtraction and division?  We will try with an example:

Subtraction

A. 18 - 6 - 4

18 - 6 - 4
18 - 6 = 12
12 - 4 = 8
18 - 6 - 4
6 - 4 = 2
18 - 2 = 16

You do not get the same answer. The associative laws do not apply for subtraction.

Division

A. 45 ÷ 15 ÷ 3

45 ÷ 15 ÷ 3
45 ÷ 15 = 3
3 ÷ 3 = 1
45 ÷ 15 ÷ 3
15 ÷ 3 = 5
45 ÷5 = 9

You do not get the same answer. The associative laws do not apply either for division.