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 + aExamples:
1. real numbers
2 + 3 = 3 + 2
2. algebraic expressions
x 2 + x = x + x 2
2. Commutative Property of Multiplication.
a × b = b × aExamples:
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 × cand
(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) = 1Examples:
1. real numbers
reciprocal of 5 is 1/5 and 5 × (1/5) = 1
7. The additive inverse of a is -a.
a + (- a) = 0Examples:
additive inverse of -6 is -(-6) = 6 and - 6 + (6) = 0
8. The additive identity is 0.
and a + 0 = 0 + a = a9. The multiplicative identity is 1.
and a × 1 = 1 × a = a
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