In Algebra it’s important to learn and understand both the associative and commutative property.
Associative Property
The “Associative Property” is a result that applies to both addition and multiplication.Which is that you can add or multiply in any order, regardless of how the numbers are grouped.
Addition:
General Rule: ( a + b ) + c = a + ( b + c )( 1 + 4 ) + 2 = 5 + 2 = 7
1 + ( 4 + 2 ) = 1 + 6 = 7
Multiplication:
General Rule: ( a + b ) + c = a + ( b + c )( 5 × 4 ) × 2 = 20 × 2 = 40
5 × ( 4 × 2 ) = 5 × 8 = 40
Subtraction and Division:
The Property however, does NOT hold for division and subtraction.40 ÷ ( 20 ÷ 2 ) = 40 ÷ 10 = 4
( 40 ÷ 20 ) ÷ 2 = 2 ÷ 2 = 1 , 4 ≠ 1
( 3 − 8 ) − 4 = -5 − 4 = -9
3 − ( 8 − 4 ) = 3 − 4 = -1 , -9 ≠ -1
Summary:
Commutative Property
The commutative Property is also important to learn and understand in Math.
Addition:
General Rule: a + b = b + a2 + 9 = 11 , 9 + 2 = 11 => 2 + 9 = 9 + 2
Multiplication:
General Rule: a × b = b × a2 × 8 = 16 , 8 × 2 = 16 => 2 × 8 = 8 × 2
However, the commutative property does NOT stand for Subtraction and Division.
7 − 4 = 3 , 4 − 7 = -3 => a − b ≠ b − a
6 ÷ 3 = 2 , 3 ÷ 6 = 0.5 => a ÷ b ≠ b ÷ a
Distributive Property
Added to the associative and commutative property, the distributive property is a property that works with multiplication over addition and subtraction.
Addition:
General Rule: a × ( b + c ) = a × b + b × c2 × ( 4 + 3 ) = 2 × 7 = 14
2 × ( 4 + 3 ) = 2 × 4 + 2 × 3 = 8 + 6 = 14
Subtraction:
General Rule: a × ( b − c ) = a × b − b × c2 × ( 5 − 2 ) = 2 × 3 = 6
2 × ( 5 − 2 ) = 2 × 5 − 2 × 2 = 10 − 4 = 6
However, this distributive property doesn’t hold for the operation of division over division.
12 ÷ ( 4 ÷ 2 ) = 12 ÷ 6 = 2
12 ÷ ( 4 ÷ 2 ) = 12 ÷ 4 ÷ 12 ÷ 2 = 3 ÷ 6 ≠ 2
Transitive Property
There is also another property handy to make note of, known as the Transitive Property.
Where if:
If a = b and b = c , then b = c
Though it may seem like an obvious statement, this is a property that can come in useful when dealing with certain proofs.