Common Algebra Properties

New properties of mathematical expressions are introduced in Algebra – remembering and being able to distinguish them can be very important.

Commutative property.

Basically, the commutative property is when the operation is the same forwards as backwards (by operation we mean addition, subtraction, multiplication, etc.). So, for example,

4 + 5 = 5 + 4
4 x 5 = 5 x 4

Addition and multiplication both follow the commutative property. However, subtraction and division do not;

10 – 2 ≠ 2 – 10
10 ÷ 2 ≠ 2 ÷ 10

Associative property.

The associative property will always involve three or more numbers and parentheses. It states that the order in which we add or multiply numbers does not matter. So,

4 + (3 + 2) = (4 + 3) + 2
4 x (3 x 2) = (4 x 3) x 2

Note, all the operations are the same in each equation. We can't mix multiplication and division – that is a different property. Once again, the associative property does not hold true for subtraction and division;

5 – (4 – 2) ≠ (5 – 4) – 2
5 ÷ (4 ÷ 2) ≠ (5 ÷ 4) ÷ 2

Distributive property.

The Distributive property looks a lot like the Associative property. It involves distributing (or multiplying) a value by more than on other value contained in parentheses. So,

4 x (5 +2) = (4 x 5) + (4 x 2) = 28

This can be very useful if multiplying large values

8 x (27) = 8 x (20 + 7) = (8 x 20) + (8 x 7) = 160 + 56 = 216

Sometimes it is easier to work with many small products rather than one large one.

The distributive property does work with subtraction, but not division.

7 x (6 – 3) = (7 x 6) – (7 x 3), but
7 ÷ (6 – 3) ≠ (7 ÷ 6) – (7 ÷ 3)

A final note, variables could be substituted for numbers in any one of these expressions. So

a + (b + c) = (a + b) + c = (b + a) + c = b + (a +c), etc.