- The Real Number Line
- The Real Numbers
- Ordering the Real Numbers

Inside Collection (Textbook): Basic Mathematics Review

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.

- The Real Number Line
- The Real Numbers
- Ordering the Real Numbers

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.

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

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:

- Draw a horizontal line.

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

- 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.
We now define a real number.

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

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 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.

The natural numbers

The whole numbers

Notice that every natural number is a whole number.

The integers

Notice that every whole number is an integer.

The rational numbers

Rational numbers are commonly called fractions.

Since

Recall that

Now consider

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.

The irrational numbers

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

The summaray chart illustrates that

Every natural number is a real number.

Every whole number is a real number.

No integer is an irrational number.

Is every natural number a whole number?

yes

Is every whole number an integer?

yes

Is every integer a rational number?

yes

Is every rational number a real number?

yes

Is every integer a natural number?

no

Is there an integer that is a natural number?

yes

A real number

As we would expect,

Are all positive numbers greater than 0?

yes

Are all positive numbers greater than all negative numbers?

yes

Is 0 greater than all negative numbers?

yes

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

no, no

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

infinitely many, infinitely many

What integers can replace

This statement indicates that the number represented by

The integers are

Draw a number line that extends from

Draw a number line that extends from

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.

What whole numbers can replace

0, 1, 2

Draw a number line that extends from

For the following problems, next to each real number, note all collections to which it belongs by writing

0

For the following problems, draw a number line that extends from

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

neither

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

Draw a number line that extends from

For the following problems, draw a number line that extends from

Draw a number line that extends from

; no

For the pairs of real numbers shown in the following problems, write the appropriate relation symbol

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

no

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

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

99

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

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

yes, 0

For the following problems, what numbers can replace

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

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

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

Is every integer a rational number?

Yes, every integer is a rational number.

Is every rational number an integer?

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

Yes.

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

0 and 2?

5 units

0 and 6?

8 units

*((Reference))* Find the value of

23

*((Reference))* Find the value of

*((Reference))* Are the statements

different

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

*((Reference))* Is the statement

true

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Comments:"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"