hawkes learning systems: college algebra

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Copyright © 2010 Hawkes Learning Systems. All rights reserved.

Hawkes Learning Systems: College Algebra

Section 1.1: The Real Number System




Objectives

o Common subsets of real numbers.

o The real number line.

o Order on the real number line.

o Set-builder notation and interval notation.

o Absolute value and distance on the real number line.

o Working with repeating decimals.




Common Subsets of Real Numbers (cont.)

o The Natural (or Counting) Numbers: The set of counting numbers greater than or equal to 1.

o The Whole Numbers: The set of Natural numbers and 0.

o The Integers: The set of natural numbers, their negatives, and 0.

{1,2,3,4,5,...}

{0,1,2,3,4,5,...}W==

{... 4, 3, 2, 1,0,1,2,3,4,...}




Common Subsets of Real Numbers

o The Rational Numbers: The set of ratios of integers, Anyrational number can be written in the form where p and q

are both integers and Rational numbers either terminate or repeat patterns of digits past some point.

Ex:

o The Irrational Numbers: Every real number that is not rational. In decimal form, irrational numbers are non-terminating and non-repeating.

Ex: o The Real Numbers: Every set above is a subset of the set of real

numbers, which is denoted Every real number is either rational or irrational and no real number is both.

,pq

.

.0q

.

3 2912, , 3, 4 8

, 5

, the whole numbers, , , the irrational numbers




Common Subsets of Real Numbers

Real Numbers ( ) Rational Numbers ( )

Decimal term either terminates or repeats

Integers ( )…,-3,-2,-1,0,1,2,3,…

Whole Numbers 0,1,2,3…

Natural Numbers ( ) 1,2,3…

Irrational Numbers

Decimal form is non-terminating and

non-repeating.




Example 1: Common Subsets of Real Numbers

Consider the set 123{ , , , , ,1.95375

3, 4 1013 04.5 , , } S

NaturalNumbers

WholeNumbers

Integers RationalNumbers

IrrationalNumbers

RealNumbers

3,124, 10 12, , 100 4124

,

,

1 0

10

3

12

1.9537

, , ,

, ,

5

10

13

4

03

4.

5

12

,

,

,1.953

5

73

10

0

4

4.5

3

3

,

1

,




The Real Number Line

Ex: Plot the following numbers on the real number line:

| | | | | | | | | | | | |0, , 5, 2, 1

01 2 5

The real number line is a depiction of the set of real numbers as a horizontal line. The real number corresponding to a given point is called the coordinate of that point. The point for the number 0 is called the origin. Points to the right of the origin represent positive numbers and points to the left of the origin represent negative numbers.

Negative Numbers Positive Numbers




Example 2: The Real Number Line

a. Plot the numbers 101, 106, and 107:

b. Plot the numbers , , and

| | | | | | | | | | |101 106 107

34

12

14

| | | | | | | | |0 11 3

4

12

14




Order on the Real Number Line

a b

a b

b a

b a

The two symbols < and > are called strict inequality signs, while the symbols ≤ and ≥ are non-strict inequality signs.

Meaning

a lies to the left of b on the number line.a lies to the left of b or is equal to b.b lies to the right of a on the number line.b lies to the right of a or is equal to a.

Symbol Read

“a is less than b”

“a is less than or equal to b”

“b is greater than a”

“b is greater than or equal to a”




Example 3: Order on the Real Number Line

What can we say about the following relationship? | | | | | | |

a. lies to the left of .

b. , or is strictly less than .

c. , or is less than or equal to .

d. , or is strictly greater than .

e. , or is greater than or equal to .

5 8

5 8

5 58 8

58 5 8

8 5 8 5

8 85 5




Example 4: Order on the Real Number Line

a. The statement “a is less than or equal to b + c” is written: .

b. The negation of the statement a ≤ b is . Why is this true? a ≤ b means that “a is less than or equal to b.” The

negation of a statement is to say that the statement is not true. So, if a is NOT less than or equal to b then a must be greater than b and a cannot be equal to b. Thus, a must be strictly greater than b.

c. If a ≤ b and a ≥ b then it must be the case that .Why is this true? Consider each case on a real number line.a ≤ b | | | | | | |

a ≥ b | | | | | | |

a bWhere can you place a and b in the second case so that the first case holds true? You should notice that this is impossible unless you make a = b.

a b

a b c a b

a b

a b

ab




Set-Builder Notation and Interval Notation

o Set-builder notation is a general method of h describing the elements that belong to a given set. o The notation {x|x has property P} is used to describe a set of real numbers, all of which have the property

P. This can be read “the set of all real numbers x having property P.”o Interval notation is a way of describing certain subsets of the real line.



Copyright © 2010 Hawkes Learning Systems. All rights reserved.Example 5: Set-builder Notation and Interval

NotationWhat sets of real numbers do the following propertiesdescribe?

a. {x|x is an even integer}

= {…,-4,-2,0,2,4…}

b. {x|x is an integer such that -4 ≤ x <1}

= {-4,-3,-2,-1,0}

c. {x|x > 2 and x ≤ -5}

= { } or

This set could also be described as {2n|n is an integer} since every even integer is a multiple of 2.

These symbols denote the empty set. This property describes the empty set because no real numbers satisfy it.




Set-builder Notation and Interval Notation

o Sets that consist of all real numbers bounded by two endpoints are called intervals. Intervals can also extend indefinitely in either direction.

o Intervals of the form (a,b) are called open intervals.o Intervals of the form [a,b] are called closed intervals.o The intervals (a,b] and [a,b) are called half-open or

half-closed. o The symbols and indicate that the interval

extends in the left and right directions, respectively.




Set-builder Notation and Interval Notation

IntervalNotation

(a,b)

[a,b]

(a,b]

( ,b)

[a, ]

Set-Builder Notation

{x|a < x < b}

{x|a ≤ x ≤ b}

{x|a < x ≤ b}

{x|x < b}

{x|x ≥ a}

Meaning

All real numbers strictly between a and b.

All real numbers between a and b, including both a and b.

All real numbers between a and b, including b but not a.

All real numbers less than b.

All real numbers greater than or equal to a.



Copyright © 2010 Hawkes Learning Systems. All rights reserved.Example 5: Set-builder Notation and Interval

NotationDescribe each of the following properties using the chart below:a. All real numbers strictly between -5 and 8.b. All real numbers greater than or equal to 2.c. All real numbers between -10 and 3, including 3 but not -10.d. The entire set of real numbers.

Interval Notation

Set-builder Notation

Interval is ______ at the left endpoint

Interval is ______ at the right endpoint

a.

b.

c.

d.

5,8

2, 10,3

,

| 5 8x x

| 2x x

| 10 3x x

|x x

open

openopen

openopen

open

closedclosed



Copyright © 2010 Hawkes Learning Systems. All rights reserved.Absolute Value and Distance on the Real

Number LineoThe absolute value of a real number a, denoted as |a|, is defined by:

oThe absolute value of a number is also referred to as its magnitude; it is the non-negative number corresponding to its distance from the origin.

oGiven two real numbers, the distance between them is defined to be |a−b|. In particular, the distance between a and 0 is |a−0| or just |a|.

if 0 if 0

a aa

a a



Copyright © 2010 Hawkes Learning Systems. All rights reserved.Absolute Value and Distance on the Real

Number LineProperties of Absolute ValueFor all real numbers a and b:

1.

2.

3.

4.

0a

a a

a a

ab a b

5.

6.

7.

, 0aab

b b

a b a b

a b b a

(This is called the triangle inequality because it is a reflection of the fact that one side of a triangle is never longer than the sum of the other two sides.)



Copyright © 2010 Hawkes Learning Systems. All rights reserved.Example 6: Absolute Value and Distance on

the Real Number LineSimplify the following expressions using yourknowledge of absolute values:

a.

b.

c.

d.

e.

f.

21 7 7 21 14

3 3

11 3 11 3

11 13 13 11

58

5

8

58

Both and – are units from 0.

How does this compare to ? ( 3)

is greater than 3, so this must be a positive number.

11

is less than 13, so this expression must be negative. So, its absolute value is .

11 11 13

Note the properties of absolute value.

and are units apart.14 14 14. 21 7 14




Working with Repeating DecimalsA rational number that appears with a repeating pattern of digits can be written as a ratio of integers by following the procedure outlined below:Suppose we wish to write as a ratio of integers. We know thatNow, let

Substitute

.1 45681.4568

451 0.

1 0.8

0006

0.0068x 100 0.68x 100 0.68 0.0068x

100 0.68x x 99 0.68x

6899

100x

68 179900 2475

x

. 1 456845

1100

172475

144239900

0.0068x

So, together we have

hawkes learning systems: college algebra

Documents

set of real numbers

real number lineex

real number lineplot

following numbers

set of natural numbers

anyrational number

set of ratios

horizontal line