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Basic Properties of Real Numbers: The Real Number Line and the Real Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret. Objectives of this module: be familiar with the real number line and the real numbers, understand the ordering of the real numbers.

Overview

  • The Real Number Line
  • The Real Numbers
  • Ordering the Real Numbers

The Real Number Line

Real Number Line

In our study of algebra, we will use several collections of numbers. The real number line allows us to visually display the numbers in which we are interested.

A line is composed of infinitely many points. To each point we can associate a unique number, and with each number we can associate a particular point.

Coordinate

The number associated with a point on the number line is called the coordinate of the point.

Graph

The point on a line that is associated with a particular number is called the graph of that number.

We construct the real number line as follows:

Construction of the Real Number Line

  1. Draw a horizontal line.

    A horizontal line with arrows on both the ends.
  2. Choose any point on the line and label it 0. This point is called the origin.

    A horizontal line with arrows on both the ends, and a mark labeled as zero.
  3. Choose a convenient length. This length is called "1 unit." Starting at 0, mark this length off in both directions, being careful to have the lengths look like they are about the same.

    A horizontal line with arrows on both the ends, and a mark labeled as zero. There are equidistant marks on both sides of zero.

    We now define a real number.

Real Number

A real number is any number that is the coordinate of a point on the real number line.

Positive and Negative Real Numbers

The collection of these infinitely many numbers is called the collection of real numbers. The real numbers whose graphs are to the right of 0 are called the positive real numbers. The real numbers whose graphs appear to the left of 0 are called the negative real numbers.
 The real numbers having graphs on the right side of the origin are positive numbers, and those having graphs on the left side of the origin are negative numbers.

The number 0 is neither positive nor negative.

The Real Numbers

The collection of real numbers has many subcollections. The subcollections that are of most interest to us are listed below along with their notations and graphs.

Natural Numbers

The natural numbers (N) (N) :   {1, 2,3, …} {1, 2,3, …}

Graphs of natural numbers one to six plotted on a number line. The numberline has arrows on each sides, and is labeled from zero to six in increments of one. There are three dots after six indicating that the graph continues indefinitely.

Whole Numbers

The whole numbers (W) (W) :   {0, 1, 2,3, …} {0, 1, 2,3, …}

Graphs of whole numbers zero to six plotted on a number line. The number line has arrows on each side, and is labeled from zero to six in increments of one. There are three dots after six indicating that the graph continues indefinitely.

Notice that every natural number is a whole number.

Integers

The integers (Z) (Z) :   {…, −3,− 2,−1,0,1, 2, 3, …} {…, −3,− 2,−1,0,1, 2, 3, …}

Graphs of integers negative five to five plotted on a number line. The number line has arrows on each side, and is labeled from negative five to five in increments of one. There are three dots after five indicating that the graph continues indefinitely.

Notice that every whole number is an integer.

Rational Numbers

The rational numbers (Q) (Q) : Rational numbers are real numbers that can be written in the form a/b a/b , where a a and b b are integers, and b≠0 b≠0 .

Fractions

Rational numbers are commonly called fractions.

Division by 1

Since b b can equal 1, every integer is a rational number: a 1 =a a 1 =a .

Division by 0

Recall that 10 /2 =5 10 /2 =5 since 2⋅5=10 2⋅5=10 . However, if 10 /0 =x 10 /0 =x , then 0⋅x=10 0⋅x=10 . But 0⋅x=0 0⋅x=0 , not 10. This suggests that no quotient exists.

Now consider 0/0 =x 0/0 =x . If 0/0 =x 0/0 =x , then 0⋅x=0 0⋅x=0 . But this means that x x could be any number, that is, 0/0 =4 0/0 =4 since 0⋅4=0 0⋅4=0 , or 0/0 =28 0/0 =28 since 0⋅28=0 0⋅28=0 . This suggests that the quotient is indeterminant.

x/0x/0 Is Undefined or Indeterminant

Division by 0 is undefined or indeterminant.

Do not divide by 0.

Rational numbers have decimal representations that either terminate or do not terminate but contain a repeating block of digits. Some examples are:

3 4 =0.75 ︸ Terminating 15 11 =1.36363636… ︸ Nonterminating,​ but ​repeating​ 3 4 =0.75 ︸ Terminating 15 11 =1.36363636… ︸ Nonterminating,​ but ​repeating​

Some rational numbers are graphed below.

Graphs of rational numbers negative nine over two, negative five over three, negative one over eight, zero, two, and two and one fourth plotted on a number line.

Irrational Numbers

The irrational numbers (Ir) (Ir) : Irrational numbers are numbers that cannot be written as the quotient of two integers. They are numbers whose decimal representations are nonterminating and nonrepeating. Some examples are

4.01001000100001… π=3.1415927… 4.01001000100001… π=3.1415927…

Notice that the collections of rational numbers and irrational numbers have no numbers in common.

When graphed on the number line, the rational and irrational numbers account for every point on the number line. Thus each point on the number line has a coordinate that is either a rational or an irrational number.

In summary, we have

Sample Set A

The summaray chart illustrates that

A rectangle labeled as Real numbers is divided into two parts, labeled as Rational numbers, and Irrational numbers, respectively. The part labeled as Rational number has three more rectangles placed one inside the other. These rectangles are labeled: the outermost as integers, the innermost as natural numbers, and the middle one as whole numbers. This illustrates that all real numbers are primarily classified as rational and irrational numbers. And that all the natural numbers are whole numbers, all the whole numbers are integers, and all the integers are rational numbers. But the vice versa is not true.

Example 1

Every natural number is a real number.

Example 2

Every whole number is a real number.

Example 3

No integer is an irrational number.

Practice Set A

Exercise 1

Is every natural number a whole number?

Solution

yes

Exercise 2

Is every whole number an integer?

Solution

yes

Exercise 3

Is every integer a rational number?

Solution

yes

Exercise 4

Is every rational number a real number?

Solution

yes

Exercise 5

Is every integer a natural number?

Solution

no

Exercise 6

Is there an integer that is a natural number?

Solution

yes

Ordering the Real Numbers

Ordering the Real Numbers

A real number b b is said to be greater than a real number a a , denoted b>a b>a , if the graph of b b is to the right of the graph of a a on the number line.

Sample Set B

As we would expect, 5>2 5>2 since 5 is to the right of 2 on the number line. Also, −2>−5 −2>−5 since −2 −2 is to the right of −5 −5 on the number line.

Graphs of numbers negative five, negative two, five, and two plotted on a number line. The number line has arrows on each side, and is labeled from negative five to five in increments of one. The number line explains that negative two is greater than negative five, and five is greater than two.

Practice Set B

Exercise 7

Are all positive numbers greater than 0?

Solution

yes

Exercise 8

Are all positive numbers greater than all negative numbers?

Solution

yes

Exercise 9

Is 0 greater than all negative numbers?

Solution

yes

Exercise 10

Is there a largest positive number? Is there a smallest negative number?

Solution

no, no

Exercise 11

How many real numbers are there? How many real numbers are there between 0 and 1?

Solution

infinitely many, infinitely many

Sample Set C

Example 4

What integers can replace x x so that the following statement is true?

−4≤x<2 −4≤x<2

This statement indicates that the number represented by x x is between −4 −4 and 2. Specifically, −4 −4 is less than or equal to x x , and at the same time, x x is strictly less than 2. This statement is an example of a compound inequality.

Graphs of integers negative five to one plotted on a number line. The number line has arrows on each side, and is labeled from negative five to five in increments of one.

The integers are −4, −3, −2, −1, 0, 1 −4, −3, −2, −1, 0, 1 .

Example 5

Draw a number line that extends from −3 −3 to 7. Place points at all whole numbers between and including −2 −2 and 6.

Example 6

Draw a number line that extends from −4 −4 to 6 and place points at all real numbers greater than or equal to 3 but strictly less than 5.

Graphs of whole numbers between and including negative two and six plotted on a number line. The number line has arrows on each side, and is labeled from negative three to seven in increments of one. Negative two and negative one are not whole numbers, therefore they are not included in the graph.

It is customary to use a closed circle to indicate that a point is included in the graph and an open circle to indicate that a point is not included.

A number line with arrows on each end, and labeled from negative four to six in increments of one. There is a closed circle at three, and an open circle at five. These two circles are connected by a black line.

Practice Set C

Exercise 12

What whole numbers can replace x x so that the following statement is true?

 −3≤x<3 −3≤x<3

Solution

0, 1, 2

Exercise 13

Draw a number line that extends from −5 −5 to 3 and place points at all numbers greater than or equal to −4 −4 but strictly less than 2.

A horizontal line with arrows on both the ends.

Solution

A number line with arrows on each end, and labeled from negative five to three in increments of one. There is a closed circle at negative four and an open circle at two. These two circles are connected by a black line.

Exercises

For the following problems, next to each real number, note all collections to which it belongs by writing N N for natural numbers, W W for whole numbers, Z Z for integers, Q Q for rational numbers, Ir Ir for irrational numbers, and R R for real numbers. Some numbers may require more than one letter.

Exercise 14

1 2 1 2

Solution

Q, R Q, R

Exercise 15

−12 −12

Exercise 16

0

Solution

W, Z, Q, R W, Z, Q, R

Exercise 17

−24 7 8 −24 7 8

Exercise 18

86.3333… 86.3333…

Solution

Q, R Q, R

Exercise 19

49.125125125… 49.125125125…

Exercise 20

−15.07 −15.07

Solution

Q, R Q, R

For the following problems, draw a number line that extends from −3 −3 to 3. Locate each real number on the number line by placing a point (closed circle) at its approximate location.

Exercise 21

1 1 2 1 1 2

Exercise 22

−2 −2

Solution

A number line with arrows on each end, labeled from negative three to two in increments of one. There is a closed circle at negative two.

Exercise 23

− 1 8 − 1 8

Exercise 24

Is 0 a positive number, negative number, neither, or both?

Solution

neither

Exercise 25

An integer is an even integer if it can be divided by 2 without a remainder; otherwise the number is odd. Draw a number line that extends from −5 −5 to 5 and place points at all negative even integers and at all positive odd integers.

Exercise 26

Draw a number line that extends from −5 −5 to 5. Place points at all integers strictly greater than −3 −3 but strictly less than 4.

Solution

A number line with arrows on each side, labeled from negative five to five in increments of one. The graphs of the integers negative two to three are plotted on the number line.

For the following problems, draw a number line that extends from −5 −5 to 5. Place points at all real numbers between and including each pair of numbers.

Exercise 27

−5 −5 and −2 −2

Exercise 28

−3 −3 and 4

Solution

A number line with arrows on each end, labeled from negative five to five in increments of one. There are closed circles at negative three and four. These two circles are connected by a black line.

Exercise 29

−4 −4 and 0

Exercise 30

Draw a number line that extends from −5 −5 to 5. Is it possible to locate any numbers that are strictly greater than 3 but also strictly less than −2 −2 ?

Solution

A number line with arrows on each end, labeled from negative five to five, in increments of one. There are open circles at negative two and three with a dark shaded arrow to the left of negative two and right of three.; no

For the pairs of real numbers shown in the following problems, write the appropriate relation symbol (<, >, =) (<, >, =) in place of the ∗ ∗ .

Exercise 31

−5∗−1 −5∗−1

Exercise 32

−3∗0 −3∗0

Solution

< <

Exercise 33

−4∗7 −4∗7

Exercise 34

6∗−1 6∗−1

Solution

> >

Exercise 35

− 1 4 ∗− 3 4 − 1 4 ∗− 3 4

Exercise 36

Is there a largest real number? If so, what is it?

Solution

no

Exercise 37

Is there a largest integer? If so, what is it?

Exercise 38

Is there a largest two-digit integer? If so, what is it?

Solution

99

Exercise 39

Is there a smallest integer? If so, what is it?

Exercise 40

Is there a smallest whole number? If so, what is it?

Solution

yes, 0

For the following problems, what numbers can replace x x so that the following statements are true?

Exercise 41

−1≤x≤5 x an integer −1≤x≤5 x an integer

Exercise 42

−7<x<−1, x an integer −7<x<−1, x an integer

Solution

−6, −5, −4, −3, −2 −6, −5, −4, −3, −2

Exercise 43

−3≤x≤2, x a natural number −3≤x≤2, x a natural number

Exercise 44

−15<x≤−1, x a natural number −15<x≤−1, x a natural number

Solution

There are no natural numbers between −15 and −1.

Exercise 45

−5≤x<5, x a whole number −5≤x<5, x a whole number

Exercise 46

The temperature in the desert today was ninety-five degrees. Represent this temperature by a rational number.

Solution

( 95 1 ) ° ( 95 1 ) °

Exercise 47

The temperature today in Colorado Springs was eight degrees below zero. Represent this temperature with a real number.

Exercise 48

Is every integer a rational number?

Solution

Yes, every integer is a rational number.

Exercise 49

Is every rational number an integer?

Exercise 50

Can two rational numbers be added together to yield an integer? If so, give an example.

Solution

Yes. 1 2 + 1 2 =1 or 1+1=2 1 2 + 1 2 =1 or 1+1=2

For the following problems, on the number line, how many units (intervals) are there between?

Exercise 51

0 and 2?

Exercise 52

−5 −5 and 0?

Solution

5 units

Exercise 53

0 and 6?

Exercise 54

−8 −8 and 0?

Solution

8 units

Exercise 55

−3 −3 and 4?

Exercise 56

m m and n n , m>n m>n ?

Solution

m−n units m−n units

Exercise 57

−a −a and −b −b , −b>−a −b>−a ?

Exercises for Review

Exercise 58

((Reference)) Find the value of 6+3(15−8)−4 6+3(15−8)−4 .

Solution

23

Exercise 59

((Reference)) Find the value of 5(8−6)+3(5+2⋅3) 5(8−6)+3(5+2⋅3) .

Exercise 60

((Reference)) Are the statements y<4 y<4 and y≥4 y≥4 the same or different?

Solution

different

Exercise 61

((Reference)) Use algebraic notation to write the statement "six times a number is less than or equal to eleven."

Exercise 62

((Reference)) Is the statement 8(15−3⋅4)−3⋅7≥3 8(15−3⋅4)−3⋅7≥3 true or false?

Solution

true

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