1. Properties of Natural Numbers

2. Commutative Property
3 + 5 = shows that we can change the order in which two numbers are added without changing the result. In general we write: a + b = b + a where a, b are Natural Numbers

3. Commutative Property
Similarly 7 x 8 = 8 x 7. In general we write: a x b = b x a where a, b are Natural Numbers. This shows us that both addition and multiplication are commutative.

4. Commutative Property
However Subtraction and division are not commutative:
Subtraction: 12 – 8 = 4 but 8 – 12 = – 4
Division: 6 ÷ 2 = 3 but 2 ÷ 6 = ⅓

5. Real Life Example
John is saving for his holidays. He has saved €1300 and the holiday costs €2175. How much more does he need to save?

6. Associative Property
To perform the operation , we could do it in either of these ways: (6 + 8) (8 + 10)

7. Associative Property
Similarly this works for multiplication in the example 3 x 4 x 5: (3 x 4) x x (4 x 5) x x
Both of the above examples show that addition and multiplication are associative. This means that the way in which the numbers are grouped does not change the result.

8. Associative Property
Next we will try the associative property for division: Example 12 ÷ 6 ÷ (12 ÷ 6) ÷ ÷ (6 ÷ 2) ÷ 2 but ÷
Similarly the associative property fails to work for subtraction: Example 12 – 6 – (12 – 6) – – (6 – 3) – –

9. Remember!
Addition and multiplication are COMMUTATIVE but subtraction and division are not.
Addition and multiplication are ASSOCIATIVE but subtraction and division are not.