2.3: The Real Number Line and the Real Numbers

The 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.

2. Choose any point on the line and label it 0. This point is called the origin.

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.

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 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 ($N$): ${1, 2, 3, ...}$

Whole Numbers:

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

Notice that every natural number is a whole number.

Integers:

The integers ($Z$): ${..., -3, -2, -1, 0, 1, 2, 3, ...}$

Notice that every whole number is an integer.

Rational Numbers

The rational numbers (Q): Rational numbers are real numbers that can be written in the form $\dfrac{a}{b}$, where $a$ and $b$ are integers, and $b \not= 0$.

Fractions

Rational numbers are commonly called fractions.

Division by 1

Since $b$ can equal 1, every integer is a rational number: $\dfrac{a}{1} = a$.

Division by 0

Recall that $\dfrac{10}{2} = 5$ since $2 \cdot 5 = 10$. However, if $\dfrac{10}{0} = x$, then $0 \cdot x = 10$. But $0 \cdot x = 0$, not 10. This suggests that no quotient exists.

Now consider $\dfrac{0}{0} = x$. If $\dfrac{0}{0} = x$, then $0 \cdot x = 0$. But this means that $x$ could be any number, that is $\dfrac{0}{0} = 4$ since $0 \cdot 4 = 0$, or $\dfrac{0}{0} = 28$ since $0 \cdot 28 = 0$. This suggests that the quotient is indeterminant.

$\dfrac{x}{0}$ 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:

Some rational numbers are graphed below.

Irrational Numbers

The irrational numbers (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

$4.01001000100001... \pi = 3.1415927...$

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 summary chart illustrates that

• Every natural number is a real number
• Every whole number is a real number.
• No integer is an irrational number.

Practice Set A

Practice Problem $\PageIndex{1}$

Is every natural number a whole number?

Answer

Yes

Practice Problem $\PageIndex{2}$

Is every whole number an integer?

Answer

Yes

Practice Problem $\PageIndex{3}$

Is every integer a rational number?

Answer

Yes

Practice Problem $\PageIndex{4}$

Is every rational number a real number?

Answer

Yes

Practice Problem $\PageIndex{5}$

Is every integer a natural number?

Answer

No

Practice Problem $\PageIndex{6}$

Is there an integer that is a natural number?

Answer

Yes

Ordering the Real Numbers

Ordering the Real Numbers

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

Sample Set B

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

Practice Set B

Practice Problem $\PageIndex{7}$

Are all positive numbers greater than 0?

Answer

Yes

Practice Problem $\PageIndex{8}$

Are all positive numbers greater than all negative numbers?

Answer

Yes

Practice Problem $\PageIndex{9}$

Is 0 greater than all negative numbers?

Answer

Yes

Practice Problem $\PageIndex{10}$

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

Answer

No, No.

Practice Problem $\PageIndex{11}$

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

Answer

infinitely many, infinitely many

Sample Set C

Example $\PageIndex{1}$

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

$$-4 \le x < 2$$

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

The integers are $-4, -3, -2, -1, 0, 1$.

Example $\PageIndex{2}$

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

Example $\PageIndex{3}$

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

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.

Practice Set C

Practice Problem $\PageIndex{12}$

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

$-3 \le x < 3$

Answer

0, 1, 2

Practice Problem $\PageIndex{12}$

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

Answer

Exercises

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

Exercise $\PageIndex{1}$

$\dfrac{1}{2}$

Answer

$Q, R$

Exercise $\PageIndex{2}$

$-12$

Exercise $\PageIndex{3}$

0

Answer

$W, Z, Q, R$

Exercise $\PageIndex{4}$

$-24\dfrac{7}{8}$

Exercise $\PageIndex{5}$

$86.3333...$

Answer

$Q, R$

Exercise $\PageIndex{6}$

$49.125125125...$

Exercise $\PageIndex{7}$

$-15.07$

Answer

$Q, R$

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

Exercise $\PageIndex{8}$

$1 \dfrac{1}{2}$

Exercise $\PageIndex{9}$

$-2$

Answer

Exercise $\PageIndex{10}$

$- \dfrac{1}{8}$

Exercise $\PageIndex{11}$

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

Answer

neither

Exercise $\PageIndex{12}$

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 to 5 and place points at all negative even integers and at all positive odd integers.

Exercise $\PageIndex{13}$

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

Answer

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

Exercise $\PageIndex{14}$

$-5$ and $-2$

Exercise $\PageIndex{15}$

$-3$ and $4$

Answer

Exercise $\PageIndex{16}$

−4 and 0

Exercise $\PageIndex{17}$

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

Answer

no

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

Exercise $\PageIndex{18}$

−5∗−1

Exercise $\PageIndex{19}$

−3∗0

Answer

<

Exercise $\PageIndex{20}$

−4∗7

Exercise $\PageIndex{21}$

6∗−1

Answer

>

Exercise $\PageIndex{22}$

$\dfrac{1}{4}$ * $-\dfrac{3}{4}$

Exercise $\PageIndex{23}$

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

Answer

no

Exercise $\PageIndex{24}$

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

Exercise $\PageIndex{25}$

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

Answer

99

Exercise $\PageIndex{26}$

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

Exercise $\PageIndex{27}$

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

Answer

yes, 0

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

Exercise $\PageIndex{28}$

$-1 \le x \le 5$, $x$ an integer

Exercise $\PageIndex{29}$

$-7 < x < -1$, $x$ an integer

Answer

$-6, -5, -4, -3, -2$

Exercise $\PageIndex{30}$

$-3 \le x \le 2$, $x$ a natural number

Exercise $\PageIndex{31}$

$-15 < x \le -1$, $x$ a natural number

Answer

There are no natural numbers between $-15$ and $-1$.

Exercise $\PageIndex{32}$

$-5 \le x < 5$, $x$ a whole number

Exercise $\PageIndex{33}$

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

Answer

$\dfrac{95}{1}$

Exercise $\PageIndex{34}$

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

Exercise $\PageIndex{35}$

Is every integer a rational number?

Answer

Yes

Exercise $\PageIndex{36}$

Is every rational number an integer?

Exercise $\PageIndex{37}$

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

Answer

Yes. $\dfrac{1}{2} + \dfrac{1}{2} = 1$ or $1 + 1 = 2$

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

Exercise $\PageIndex{38}$

0 and 2

Exercise $\PageIndex{39}$

-5 and 0

Answer

5 units

Exercise $\PageIndex{40}$

0 and 6

Exercise $\PageIndex{41}$

-8 and 0

Answer

8 Units

Exercise $\PageIndex{42}$

-3 and 4

Exercise $\PageIndex{43}$

$m$ and $n$, $m > n$

Answer

$m - n$ units

Exercise $\PageIndex{44}$

$-a$ and $-b$, $-b > -a$

Exercises for Review

Exercise $\PageIndex{45}$

Find the value of $6 + 3(15 - 8) - 4$

Answer

23

Exercise $\PageIndex{46}$

Find the value of $5(8 - 6) + 3(5 + 2 \cdot 3$

Exercise $\PageIndex{47}$

Are the statements $y < 4$ and $y \ge 4$ the same or different?

Answer

Different

Exercise $\PageIndex{48}$

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

Exercise $\PageIndex{49}$

Is the statement $8(15 - 3 \cdot 4) - 3 \cdot 7 \ge 3$ true or false?

Answer

true