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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"
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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
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"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. |
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" |
12. |
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" |
13. |
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." |
14. |
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." |
15. |
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." |
16. |
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 non-zero value will not change truth value of the equation." |
17. |
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." |
18. |
Reflexive (or Identity) Property of Equality
a = a |
12 = 12 |
"a real number is always
equal to itself" |
19. |
Symmetric Property of Equality
If a = b, then b = a. |
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"quantities that are equal can be read forward or backward" |
20. |
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" |
21. |
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) |