ALGEBRAIC PROPERTIES

Commutative Property 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: = 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 ≠ x - y ≠ y - x Numbers that are divided are NOT commutative. For example:4 ÷ 5 ≠ 5 ÷ 4 x ÷ y ≠ y ÷ x

Associative Property 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 ≠ 5 - (4 - 6) (x - y) - z ≠ x - (y - z) Numbers that are divided are NOT associative. For example:(4 ÷ 5) ÷ 6 ≠ 5 ÷ (4 ÷ 6) (x ÷ y ) ÷ z ≠ y ÷ ( x ÷ z)

Distributive Property 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: Number Properties ×÷+ - CommutativeYesNoYesNo AssociativeYesNoYesNo DistributiveYesNo

Closure property Closure is when all answers fall into the original set. When two even numbers are added, the answer will be an even number. Thus, the set of even numbers has closure under addition. When two odd numbers are added, the answer is not an odd number. Therefore, the set of odd numbers does not have closure under addition.

When two even numbers are multiplied, they produce an even number. Thus, the set of even numbers has closure under multiplication. When two odd numbers are multiplied, the answer is an odd number. Therefore, the set of odd numbers has closure under multiplication.

Identity element The identity element for addition is 0. When 0 is added to any number, it gives the original number. The identity element for multiplication is 1. any number multiplied by 1 gives the original number.

Inverse The additive inverse is the opposite or negative of the number. The sum of any number and its additive inverse is (- 2) = 0 The multiplicative inverse is the reciprocal of the number, which means one divided by the number. When the number is multiplied by its reciprocal, the answer is 1. 3 x 1/3 = 1 2/3 x 3/2 = 1