Pre CalculusSec 1.1 Real Numbers

Objectives:•To review the set of Real Numbers•To review the properties of Algebra•To understand interval and set notation.

Real Numbers

• Natural Numbers: 1,2,3,4,…• Integers: -,…-3,-2,-1,0,1,2,3,…• Rational Numbers: any # that can be written

as a ratio of integers (as a fraction).

• Irrational Numbers: any # that cannot be written as a fraction.

CLASS WORK1. Given the set,

list the elements of the set that are:

a) Natural numbersb) Integersc) Rational numbersd) Irrational numbers

13 151.001,0.333..., , 11,11, , 16,3.14,

15 3

Properties of Real NumbersCommutative Property: a + b = b + a ab = ba order doesn’t matter

Associative Property: (a+b)+c = a+(b+c) (ab)c = a(bc) order doesn’t change

Distributive Property: a(b+c) = ab + ac you can add then multiply

or multiply then add.

CLASS WORKState the property of real numbers being used.

2.

3.

4.

2 3 5 3 5 2

2 2 2A B A B

2 3 2 3p q r p q r

Sets & Elements

• A set is a collection of objects. - the objects are called the elements of the

set.

If S is a set, the notation of means that a is an element of S.

a S

Sets & Elements

means that b is not an element of S.

Ex1. If Z represents the set of integers, then but

b S

3 Z Z

Notation of Sets• Braces { } -

The set A that consists of positive integers less than 7 is

• Set-builder notation –

• Interval notation – These are sets of real numbers and correspond geometrically to line segments.

1,2,3,4,5,6A

is an integer and 0 7A x x x

Union of Sets

• If S and T are sets, then , represents their union. The union of sets consists of all elements in both sets.

Ex 2. Find if

S T

A B

3, 2, 1 , 1,2,3A B

Intersection of Sets

• The intersection of S and T is the set consisting of all elements that are in both sets. It is only what they have in common.

Ex 3. Find if

S T

A B

1,2,3,4,5 , 4,5,6,7,8A B

Ex 4. Find, , ,

A C A B A B

2 , 4 , 1 5A x x B x x C x x

CLASS WORK

If find:

5.

6.

7.

1,2,3,4,5 , 4,5,6,7 , 6,7,8S T V

S T

S T

S V

CLASS WORK

If find

8.

9.

10.

1 , 5 , 3 2P x x Q x x S x x

P Q

P S

Q S

IntervalsNotation Graph Set-builder Notation

(a, b)

[a, b]

[a, b)

(a, b]

(a, )

[a, )

(-, b)

(-, b]

(-, )

b

b

Ex. 5 Express each interval in terms of inequalities then graph the interval.

a) [-1, 2)

b) [1.5, 4]

c) (-3, )

CLASS WORK

Express each interval in terms of inequalities then graph the interval.

11. [2, 8)

12. (-, -5)

CLASS WORK

Express the inequality in interval notation, then graph the interval.

13.

14.

2 5x

3x

HW p. 10 1-9 odd, 33,34,35-59 odd