1.3 – axioms for the real numbers. goals swbat apply basic properties of real numbers swbat...

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