Properties of Natural Numbers

Commutative Property 3 + 5 = 5 + 3 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

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.

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

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?

Associative Property To perform the operation 6 + 8 + 10, we could do it in either of these ways: (6 + 8) + 10 6 + (8 + 10) 14 + 10 6 + 18 24 24

Associative Property Similarly this works for multiplication in the example 3 x 4 x 5: (3 x 4) x 5 3 x (4 x 5) 12 x 5 3 x 20 60 60 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.

Associative Property Next we will try the associative property for division: Example 12 ÷ 6 ÷ 2. (12 ÷ 6) ÷ 2 12 ÷ (6 ÷ 2) 2 ÷ 2 but 12 ÷ 3 1 4 Similarly the associative property fails to work for subtraction: Example 12 – 6 – 3. (12 – 6) – 3 12 – (6 – 3) 6 – 3 12 – 3 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.