  # Basic Rules and Properties of Algebra

We list the basic rules and properties of algebra and give examples on they may be used.

Let a, b and c be real numbers, variables or algebraic expressions.

## 1. Commutative Property of Addition.

a + b = b + a
Examples:
1. real numbers
2 + 3 = 3 + 2
2. algebraic expressions
x
2 + x = x + x 2

## 2. Commutative Property of Multiplication.

a � b = b � a
Examples:
1. real numbers
5 � 7 = 7 � 5
2. algebraic expressions
(x
3 - 2) � x = x � (x 3 - 2)

## 3. Associative Property of Addition.

(a + b) + c = a + (b + c)
Examples:
1. real numbers
(2 + 3) + 6 = 2 + (3 + 6)
2. algebraic expressions
(x
3 + 2 x) + x = x 3 + (2 x + x)

## 4. Associative Property of Multiplication.

(a � b) � c = a � (b � c)
Examples:
1. real numbers
(7 � 3) � 10 = 7 � (3 � 10)
2. algebraic expressions
(x
2 � 5 x) � x = x 2 � (5 x � x)

## 5. Distributive Properties of Addition Over Multiplication.

a � (b + c) = a � b + a � c
and
(a + b) � c = a � c + b � c
Examples:
1. real numbers
2 � (2 + 8) = 2 � 2 + 2 � 8
(2 + 8) � 10 = 2 � 10 + 8 � 10
2. algebraic expressions
x � (x 4 + x) = x � x 4 + x � x
(x 4 + x) � x 2 = x 4 � x 2 + x � x 2

## 6. The reciprocal of a non zero real number a is 1/a.

and a � (1/a) = 1
Examples:
1. real numbers
reciprocal of 5 is 1/5 and 5 � (1/5) = 1

## 7. The additive inverse of a is -a.

a + (- a) = 0
Examples:
additive inverse of -6 is -(-6) = 6 and - 6 + (6) = 0

## 8. The additive identity is 0.

and a + 0 = 0 + a = a

## 9. The multiplicative identity is 1.

and a � 1 = 1 � a = a