1.3 – AXIOMS FOR THE REAL NUMBERS
Goals
SWBAT apply basic properties of real numbers
SWBAT simplify algebraic expressions
An axiom (or postulate) is a statement that is assumed to be true.
The table on the next slide shows axioms of multiplication and addition in the real number system.
Note: the parentheses are used to indicate order of operations
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.
Example:
and
Use the substitution principle with the statement above.
8 2 10 10 3 7
Identity Elements
In the real number system:
The identity for addition is: 0
The identity for multiplication is: 1
Inverses
For the real number a,
The additive inverse of a is: -a
The multiplicative inverse of a is: 1
a
Axioms of Equality
Let a, b, and c be and elements of .
Reflexive Property: Symmetric Property:
Transitive Property:
a a
If a b, then b a
If a b and b c, then a c
1.4 – THEOREMS AND PROOF: ADDITION
The following are basic theorems of addition. Unlike an axiom, a theorem can be proven.
Theorem
For all real numbers b and c,
b c c b
Theorem
For all real numbers a, b, and c,
If , then a c b c a b
Theorem
For all real numbers a, b, and c, if
or
then
a c b c
c a c b
a b
Property of the Opposite of a Sum
For all real numbers a and b,
That is, the opposite of a sum of real numbers is the sum of the opposites of the numbers.
a b a b
Cancellation Property of Additive Inverses
For all real numbers a,
a a
Simplify
1.
2.
x x 3
y y
1.5 – Properties of Products
Multiplication properties are similar to addition properties.
The following are theorems of multiplication.
Theorem
For all real numbers b and all nonzero real numbers c,
bc 1
cb
Cancellation Property of Multiplication
For all real numbers a and b and all nonzero real numbers c, if
or ,then ac bc ca cb a b
Properties of the Reciprocal of a Product
For all nonzero real numbers a and b,
That is, the reciprocal of a product of nonzero real numbers is the product of the reciprocals of the numbers.
1
ab
1
a1
b
Multiplicative Property of Zero
For all real numbers a,
and a 0 0 0 a 0
Multiplicative Property of -1
For all real numbers a,
and a 1 a 1 a a
Properties of Opposites of Products
For all real numbers a and b,
a b ab
a b ab
a b ab
Explain why the statement is true.
1. A product of several nonzero real numbers of which an even number are negative is a positive number.
Explain why the statement is true.
2. A product of several nonzero real numbers of which an odd number are negative is a negative number.
Simplify
3. 1
6 22 15
Simplify
8. 1
2 8w
1
3
12w 9
Simplify the rest of the questions and then we will go over them together!
1.6 – Properties of Differences
Definition
The difference between a and b, , is defined in terms of addition.
ba
Definition of Subtraction
For all real numbers a and b,
baba
Subtraction is not commutative.
Example:
Subtraction is not associative.
Example:
5775
375375
Simplify the Expression
1. zw 8637
Simplify the expression
2. xyyyx 53743
Your Turn!
Try numbers 3 and 4 and we will check them together!
Evaluate each expression for the value of the
variable.
5. 8;4657 nnn
Evaluate each expression for the value of the
variable.
6. 2;7468 rrrr