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

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