Is there a question in all of that?
This is a reference chart for anyone who needs to review algebra 1 real number properties or use in class.
Property (a, b and c are real numbers, variables or algebraic expressions)
Distributive Property
a • (b + c) = a • b + a • c
Sample
3 • (4 + 5) = 3 • 4 + 3 • 5
Commutative Property of Addition
a + b = b + a
Sample
3 + 4 = 4 + 3
Commutative Property of Multiplication
a • b = b • a
Sample
3 • 4 = 4 • 3
Associative Property of Addition
a + (b + c) = (a + b) + c
Sample
3 + (4 + 5) = (3 + 4) + 5
Associative Property of Multiplication
a • (b • c) = (a • b) • c
Sample
3 • (4 • 5) = (3 • 4) • 5
Additive Identity Property
a + 0 = a
Sample
4 + 0 = 4
Multiplicative Identity Property
a • 1 = a
Sample
4 • 1 = 4
Additive Inverse Property
a + (-a) = 0
Sample
4 + (-4) = 0
Multiplicative Inverse Property
a(1/a), where a cannot be zero.
Sample
2(1/2) = 1
Zero Property of Multiplication
a • 0 = 0
Sample
4 • 0 = 0
Closure Property of Addition
a + b is a real number
Sample
10 + 5 = 15 (a real number)
Closure Property of Multiplication
a • b is a real number
Sample
10 • 5 = 50 (a real number)
Addition Property of Equality
If a = b, then a + c = b + c.
Sample
If x = 10,
then x + 3 = 10 + 3
Subtraction Property of Equality
If a = b, then a - c = b - c.
Sample
If x = 10,
then x - 3 = 10 - 3
Multiplication Property of Equality
If a = b, then a • c = b • c.
Sample
If x = 10,
then x • 3 = 10 • 3
Division Property of Equality
If a = b, then a / c = b / c,
assuming c ≠ 0.
Sample
If x = 10,
then x / 3 = 10 / 3
Substitution Property
If a = b, then a may be substituted for b, or conversely.
Sample
If x = 5, and x + y = z,
then 5 + y = z.
Reflexive (or Identity) Property of Equality
a = a
Sample
12 = 12
Symmetric Property of Equality
If a = b, then b = a.
Sample
If 3 = 3*1, then 3*1 = 3
Transitive Property of Equality
If a = b and b = c,
then a = c.
Sample
If 2a = 10 and 10 = 4b,
then 2a = 4b.
Law of Trichotomy
Exactly ONE of the following holds:
a < b, a = b, a > b
Sample
If 8 > 6, then 8 6 and
8 is not < 6.
Those "properties" are all true of the rational numbers as well as the real numbers. To specifically talk about the real numbers you have to add a "completeness property": if a set of real numbers has an upper bound then it has a least upper bound.