Honors Algebra 2 – PROOF UNIT PROPERTIES (Mr. Duarte)
Properties of Real Numbers
*Let a , b , and c denote any real numbers
1. a  b is a unique real number.
Closure Property for Addition
2. (a  b)  c  a  (b  c)
Associative Property for Addition
3. a  b  b  a
Commutative Property for Addition
4. There exists an element 0   such that for each
a  ,
0 a  a&a 0  a
Identity Property for Addition
5. There exists an element  a  , for each a   ,
such that
a  (a)  0 & (a)  a  0
Inverse Property of Addition
6. ab is a unique real number.
Closure Property for Multiplication
7. ( ab)c  a (bc )
Associative Property for Multiplication
8. ab  ba
Commutative Property for Multiplication
9. There exists an element 1  , such that for each
a ,
a 1  a &1 a  a
Identity Property for Multiplication
10. There exists an element
a   such that
1
 , for each nonzero
a
Inverse Property of Multiplication
1
1
 a  1& a 
1
a
a
11.
a (b  c)  ab  ac & (b  c)a  ba  ca
Distributive Property
SUBSTITUTION PRINCIPLE
Since a  b and ab are unique, changing the numeral by which a number is named in an
expression involving sums or products does not change the value of the expression.
For example, since
7  3  10 and 10  8  2 ,
you know that
(7  3)  8  2
Properties of Equality
For all real numbers a , b , and c , if a  b, then:
1.
2.
a  c  b  c; Addition Property
of Equality
ca cb
ac  bc;
Multiplication
ca  cb Property of Equality
3.
4.
ac bc
Subtraction Property
of Equality
a
b

c
c
(provided c  0 )
Division Property
of Equality
The “R. S. T.” Properties:
Let a , b , and c be any elements of ,
1.
aa
Reflexive Property (of equality)
2.
If a  b, then b  a
Symmetric Property (of equality)
3.
If a  b and b  c, then a  c
Transitive Property (of equality)
Properties for Simplifying Expressions
Subtraction Rule (or Definition of Subtraction)
For all real numbers a and b ,
a  b  a  ( b)
Division Rule (or Definition of Division)
For all real numbers a and all nonzero real numbers b ,
a
1
 a
b
b
Multiplication Property of Zero
For all real numbers a ,
a  0  0 and 0  a  0
.
Multiplication Property of
-1
For all real numbers a ,
a  ( 1)   a and
( 1)  a   a
Properties of Inequalities
Transitive Property of Inequality
If a , b , and c are real numbers, and if
a  b and b  c, then a  c
Addition (Subtraction) Property of Inequality
If a , b , and c are real numbers, and if
a  b then a  c  b  c
Multiplication (Division) Property of Inequality
Let a, b, and c  ,
1.
If a  b and c  0 then ac  bc
2.
If a  b and c  0 then ac  bc
Derived Properties or Axioms
Cancellation Property of Addition (Subtraction)
For all real numbers a , b , and c if
ac bc
or
ca cb
then
a b
Cancellation Property of Multiplication (Division)
For all real numbers a and b , and all nonzero real numbers c, if
ac  bc or
ca  cb
then
ab
Opposite of an Opposite Property
For all real numbers a ,
 (a)  a
Property of the Opposite of a Sum (Difference)
For all real numbers a and b ,
 ( a  b)  (  a )  ( b)
That is, the opposite of a sum of real numbers is the sum of the opposites of the numbers.
Properties of Opposites in Products
For all real numbers a and b ,
(  a )b   ab, a ( b)   ab , (  a )( b)  ab