
Property (a, b and c are real numbers, variables or algebraic expressions) 
Examples 
Verbal hints 
1. 
Distributive Property
a • (b + c) = a • b + a • c 
3 • (4 + 5) = 3 • 4 + 3 • 5 
"multiplication distributes across addition"

2. 
Commutative Property of Addition
a + b = b + a 
3 + 4 = 4 + 3 
"commute = to get up and move to a new location : switch places" 
3. 
Commutative Property of Multiplication
a • b = b • a 
3 • 4 = 4 • 3 
"commute = to get up and move to a new location: switch places" 
4. 
Associative Property of Addition
a + (b + c) = (a + b) + c 
3 + (4 + 5) = (3 + 4) + 5 
"regroup  elements do not physically move, they simply group with a new friend." 
5. 
Associative Property of Multiplication
a • (b • c) = (a • b) • c 
3 • (4 • 5) = (3 • 4) • 5 
"regroup  elements do not physically move, they simply group with a new friend." 
6. 
Additive Identity Property
a + 0 = a 
4 + 0 = 4 
"the value that returns the input unchanged" 
7. 
Multiplicative Identity Property
a • 1 = a 
4 • 1 = 4 
"the value that returns the input unchanged" 
8. 
Additive Inverse Property
a + (a) = 0 
4 + (4) = 0 
"the value that brings you back to the identity element under addition" 
9. 
Multiplicative Inverse Property


"the value that brings you back to the identity element under multiplication" 
10. 
Zero Property of Multiplication
a • 0 = 0 
4 • 0 = 0 
"zero times any value is 0" 
11. 
Zero Property of Division


"zero divided by any value is 0"
BUT, "any value divided by zero is undefined"
"NO DIVISION BY ZERO"

12. 
Closure Property of Addition
a + b is a real number 
10 + 5 = 15 (a real number) 
"the sum of any two real numbers
is another real number" 
13. 
Closure Property of Multiplication
a • b is a real number 
10 • 5 = 50 (a real number) 
"the product of any two real
numbers
is another real number" 
14. 
Addition Property of Equality
If a = b, then a + c = b + c. 
If x = 10,
then x + 3 = 10 + 3 
"adding the same value to both sides of an equation will not change the truth value of the equation." 
15. 
Subtraction Property of Equality
If a = b, then a  c = b  c. 
If x = 10,
then x  3 = 10  3 
"subtracting the same value from both sides of an equation will not change the truth value of the equation." 
16. 
Multiplication Property of Equality
If a = b, then a • c = b • c. 
If x = 10,
then x • 3 = 10 • 3 
"multiplying both sides of an equation by the same value will not change the truth value of the equation." 
17. 
Division Property of Equality
If a = b, then a / c = b / c, assuming c ≠ 0. 
If x = 10,
then x / 3 = 10 / 3 
"dividing both sides of an equation by the same nonzero value will not change truth value of the equation." 
18. 
Substitution Property
If a = b, then a may be substituted for b, or conversely. 
If x = 5, and x + y = z,
then 5 + y = z. 
"a value may be substituted for its equal." 
19. 
Reflexive (or Identity) Property of Equality
a = a 
12 = 12 
"a real number is always
equal to itself" 
20. 
Symmetric Property of Equality
If a = b, then b = a. 

"quantities that are equal can be read forward or backward" 
21. 
Transitive Property of Equality
If a = b and b = c,
then a = c. 
If 2a = 10 and 10 = 4b,
then 2a = 4b. 
"if two numbers are equal to the same number, then the two numbers are equal to each other" 
22. 
Law of Trichotomy
Exactly ONE of the following holds: a < b, a = b, a > b 
If 8 > 6, then 8 6 and
8 is not < 6. 
"for two real numbers a and b, a is either equal to b, greater than b, or less than b." (common sense) 