Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.1 - 1

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.1 - 1


Review of the Real Number System

Chapter 1


1.1

Basic Concepts

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 4

1.1 Basic Concepts

Objectives

1. Write sets using set notation.

2. Use number lines.

3. Know the common sets of numbers.

4. Find additive inverses.

5. Use absolute value.

6. Use inequality symbols.


1.1 Basic Concepts

Write Sets Using Set Notation

A set is a collection of objects called the elements, or members, of the set.

• Set braces, { }, are used to enclose the elements.

• For example, 4 is an element of the set, {3, 4, 11, 19}.

• {3, 4, 11, 19} is an example of a finite set since we can count the number of elements in the set.

• A set containing no elements is called the empty set, or the null set, denoted by Ø.


Examples of Sets

Natural numbers Whole numbers Empty set

N = {1,2,3,4,5,6,...} W = {0,1,2,3,4,5,6,

…} Ø (a set with no

elements)

1.1 Basic Concepts

Certain sets of numbers have names:

Caution:

Ø is the empty set; {Ø} is the set with one element, Ø.

Note: N and W are infinite sets. The three dots, called an ellipsis, mean “continue on in the pattern that has been established.”


Set-Builder Notation

1.1 Basic Concepts

Sometimes instead of listing the elements of a set, we use a notation called set-builder notation.

{x | x has property P }

such th has a given pthe all eleme set nts ropetf yo a rtxx P


Listing the Elements in Sets

1.1 Basic Concepts

a) {x | x is a whole number less than 3} The whole numbers less than 3 are 0, 1, and 2. This is the set {0, 1, 2}.

b) {x | x is one of the first five odd whole numbers} = {1, 3, 5, 7, 9}.

c) {z | z is a whole number greater than 11} This is an infinite set written with three dots as {12, 13, 14, 15, … }.


Using Set-Builder Notation to Describe Sets

a) { 0, 1, 2, 3 } can be described as

{m | m is one of the first four whole numbers}.

b) { 7, 14, 21, 28, … } can be described as

{s | s is a multiple of 7 greater than 0}.

1.1 Basic Concepts


Using Number Lines

A number line is a way to picture a set

of numbers:

1.1 Basic Concepts

0 5–5 –4 –3 –2 –1 1 2 3 4

0 is neither positive nor negative

Negative numbers Positive numbers


Using Number Lines

The set of numbers identified on this

number line is the set of integers:

I = {…,–3, –2, –1, 0, 1, 2, 3, …}

1.1 Basic Concepts

0 5–5 –4 –3 –2 –1 1 2 3 4

Each number on the number line is called a coordinate of the point it labels.

Graph of 3


Rational Numbers 1.1 Basic Concepts

Rational numbers can be expressed as the quotient of two integers, with a denominator that is not 0.

The set of all rational numbers is written:

and are integers, 0 .

pp q q

qRational numbers


Rational Numbers 1.1 Basic Concepts

Rational numbers can be written in decimal form as:

Terminating decimals:

Repeating decimals:

4 14.8 and 2.8

5 5= =

2 40.66666... 0.66 or 0.363636... 0.36

3 11

Bar means repeating digits.


Irrational Numbers 1.1 Basic Concepts

Irrational numbers have decimals that neither terminate nor repeat:

3 1.7320508...

11 3.316624...

3.141592...


Graphs of Rational and Irrational Numbers

11

1.1 Basic Concepts

0 5–5 –4 –3 –2 –1 1 2 3 4

Irrational Numbers

Rational Numbers

p3

0.363

2


Real Numbers

1.1 Basic Concepts

Rational numbers

Integers

Whole numbers

Natural numbers

Irrational numbers

8, 15, , 4

4 5, - , 0.6, 1.75

9 8

11, 6, 4

0

1, 2, 3, 4,

5, 27, 45


Relationships Between Sets of Numbers

1.1 Basic Concepts

Real numbers

Irrational numbers

Rational numbers

Integers

Noninteger rational numbers

Positive integers

Zero

Negative integers


Sets of Numbers

Natural numbers or Whole numbers

Integers

Rational numbers

Irrational numbers

Real numbers

{1, 2, 3, 4, 5, 6, … }

{0, 1, 2, 3, 4, 5, 6, … }

{…,–3, –2, –1, 0, 1, 2, 3, … }

1.1 Basic Concepts

and are integers, 0

pp q q

q

| is represented by a point on the number linex x

| is a real number that is not rationalx x


Relationships Between Sets of Numbers

Decide whether each statement is true or false.

a) All natural numbers are integers.

b) Zero is an irrational number.

c) Every integer is a rational number.

d) The square root of 9 is an irrational number.

e) is an irrational number.

1.1 Basic Concepts

3

p-

True

False

True

False

False


Additive Inverse

Additive InverseFor any real number a, the number –a is the additive inverse of a.

1.1 Basic Concepts

–4 units from zero 4 units from zero

0 5–5 –4 –3 –2 –1 1 2 3 4

The number –4 is the additive inverse of 4, and the number 4 is the additive inverse of –4.


The Minus Sign

The symbol “−” can be used to indicate

any of the following:

1. a negative number, such as –13 or –121;

2. the additive inverse of a number, as in

“ –7 is the additive inverse of 7”.

3. subtraction, as in 19 – 7.

1.1 Basic Concepts


Signed Numbers / Additive Inverses

• The sum of a number and its additive

inverse is always zero.

4 + (–4) = 0 or –16 + 16 = 0

• For any real number a,

–(–a) = a.

1.1 Basic Concepts


Absolute Value

Geometrically, the absolute value of a number,

a, written |a| is the distance on the number line

from 0 to a.

1.1 Basic Concepts

Distance is 4,

so |–4| = 4.

Distance is 4,

so |4| = 4.

0 5–5 –4 –3 –2 –1 1 2 3 4

Absolute value is always positive.


Formal Definition of Absolute Value

if a is positive or zero

if a is negative

aa

a

1.1 Basic Concepts

Evaluate the following absolute value expressions:

|–14| |0| –|9| –|–13|

|14| + |–7| –|6–3| –|8–8|


Equality vs. Inequality

An equation is a statement that two quantities are

equal.

8 + 3 = 11 19 – 12 = 7

An inequality is a statement that two quantities

are not equal. One must be less than the other.

9 < 12 This means that 9 is less than 12.

–7 > – 10 This means that –7 is greater than –10.

1.1 Basic Concepts


Inequalities on the Number Line

On the number line, a < b if a is to the left of b;

a > b if a is to the right of b.

0 5–5 –4 –3 –2 –1 1 2 3 4

1.1 Basic Concepts

–2 < 3

1 > –4

The inequality symbol always points to the smaller number.


Inequality Symbols

1.1 Basic Concepts

Symbol Meaning Example

is not equal to –6 10

is less than –9 –3

is greater than 8 –2

is less than or equal to –8 –8

is greater than or equal to –2 –7

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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.1 - 1