An operation is commutative if a change in the order of the numbers does not change the results. This means the numbers can be swapped.
Numbers can be added in any order.
For example: | 4 + 5 = 5 + 4 |
x + y = y + x |
Numbers can be multiplied in any order.
For example: | 5 × 3 = 3 × 5 |
a × b = b × a |
Numbers that are subtracted are NOT commutative.
For example: | 4 – 5 ≠ 5 – 4 |
x – y ≠ y –x |
Numbers that are divided are NOT commutative.
For example: | 4 ÷ 5 ≠ 5 ÷ 4 |
x ÷ y ≠ y ÷ x |
An operation is associative if a change in grouping does not change the results. This means the parenthesis (or brackets) can be moved.
Numbers that are added can be grouped in any order.
For example: | (4 + 5) + 6 = 5 + (4 + 6) |
(x + y) + z = x + (y + z) |
Numbers that are multiplied can be grouped in any order.
For example: | (4 × 5) × 6 = 5 × (4 × 6) |
(x × y) × z = x × (y × z) |
Numbers that are subtracted are NOT associative.
For example: | (4 – 5) – 6 ≠ 4 – (5– 6) |
(x – y) – z ≠ x – (y – z) |
Numbers that are divided are NOT associative.
For example: | (4 ÷ 5) ÷ 6 ≠ 4 ÷ (5÷ 6) |
(x ÷ y ) ÷ z ≠ x ÷ ( y ÷ z) |
Distributive property allows you to remove the parenthesis (or brackets) in an expression. Multiply the value outside the brackets with each of the terms in the brackets.
For example: | 4(a + b) = 4a + 4b |
7(2c – 3d + 5) = 14c – 21d + 35 |
What happens if you need to multiply (a – 3)(b + 4)?
You do the same thing but with one value at a time.
For example:
Multiply a with each term to get a × b + 4 × a = ab + 4a
Then, multiply 3 with each term to get “ –3b – 12” (take note of the sign operations).
Put the two results together to get “ab + 4a – 3b – 12”
Therefore, (a – 3)(b + 4) = ab + 4a – 3b – 12
The following table summarizes which number properties are applicable to the different operations: multiplication, division, addition and subtraction.
Number Properties | × | ÷ | + | – |
Commutative | Yes | No | Yes | No |
Associative | Yes | No | Yes | No |
Distributive | Yes | No | No | No |
The Associative Property of addition and multiplication.
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