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.

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)

(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