#### Some properties (axioms) of addition

**Closure**is when all answers fall into the original set. If you add two even numbers, the answer is still an even number (2 + 4 = 6); therefore, the set of even numbers*is closed*under addition (has closure). If you add two odd numbers, the answer is not an odd number (3 + 5 = 8); therefore, the set of odd numbers*is not closed*under addition (no closure).

**Commutative**means that the*order*does not make any difference in the operation's result.

Commutative does*Note:**not*hold for subtraction.

**Associative**means that the*grouping*does not make any difference in the operation's result.

The grouping has changed (parentheses moved), but the sides are still equal.

Associative does*Note:**not*hold for subtraction.

- The
**identity element**for addition is 0.Any number added to 0 gives you the original number.

- The
**additive inverse**is the opposite (negative) of the number. Any number plus its additive inverse equals 0 (the identity).

*a*+ (–

*a*) = 0; therefore,

*a*and –

*a*are additive inverses.

**Some properties (axioms) of multiplication**

**Closure**is when all answers fall into the original set. If you multiply two even numbers, the answer is still an even number (2 × 4 = 8); therefore, the set of even numbers is*closed*under multiplication (has closure). If you multiply two odd numbers, the answer is an odd number (3 × 5 = 15); therefore, the set of odd numbers*is closed*under multiplication (has closure).

**Commutative**means that the*order*does not make any difference in the operation's result.

** Note: **Commutative does

*not*hold for division.

**Associative**means that the*grouping*does not make any difference in the operation's result.

The grouping has changed (parentheses moved), but the sides are still equal.

** Note: **Associative does

*not*hold for division.

- The
**identity element**for multiplication is 1. Any number multiplied by 1 gives the original number.

- The
**multiplicative inverse**is the**reciprocal**of the number. Any nonzero number multiplied by its reciprocal equals 1.

; therefore, 2 and are multiplicative inverses, or reciprocals.; therefore,aand are multiplicative inverses, or reciprocals (provideda≠ 0).

#### A property of two operations

The **distributive property** is the process of distributing, using multiplication, the number on the outside of the parentheses to each term on the inside. The terms within the parentheses are separated by either addition or subtraction.

** Note: **You cannot use the distributive property with only one operation.