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

More pages related to
Maths Problems and Online Self Tests.
Algebra Problems.
Algebra Tutorial.
More Intermediate and College Algebra Questions and Problems with Answers.