Download Lesson 4: The Number System

Lesson 4: The Number System
Introduction
The next part of your course covers math skills. These lessons have been designed to provide an
overview of some basic mathematical concepts. This part of your course includes six lessons, as follows:
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Lesson 4: The Number System
Lesson 5: Common Operations
Lesson 6: Exponents
Lesson 7: Expressions
Lesson 8: Working with Fractions
Lesson 9: Working with Decimals
Each lesson is designed to provide you with an overview of some fundamentals that you should know as
you embark on your high school studies. Each lesson will review the basic concepts and skills of the
topic. This part of your course should help to prepare you for the math work you will complete later on
in your high school program.
You will complete three examinations. These examinations are designed to check your learning as you
move through the course. You will take these examinations after the following lesson groupings:
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Lessons 4 and 5: Complete Examination
Lessons 6 and 7: Complete Examination
Lessons 8 and 9: Complete Examination
In this first lesson on math skills, we will review the basic terminology and concepts of the real number
system. Pay close attention, because these concepts are some of the most fundamental principles of
mathematics.
Lesson Objectives
After completing this lesson, you will be able to:
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Describe the real number system.
Identify the subsets of the real number system.
Compare rational numbers and irrational numbers.
Explain the uses of a number line and identify points along the line.
The Real Number System
The real number system represents one of the most basic principles of mathematics. Real numbers
appear in all areas of mathematics, including algebra, functions, geometry, measurement, probability,
and statistics.
“Real numbers” are the numbers that you already know. They are called real numbers to distinguish
them from imaginary numbers, which are a concept you will learn about in a more advanced math
course.
Real numbers have several subsets, as shown in the diagram (Figure 1). This diagram is an example of a
Venn diagram, which is a type of graphic organizer that shows how things are related to each other. In
this case, the Venn diagram shows how the subsets of numbers in the real number system are related to
each other.
Rational
Numbers
Integers
REALNUMBER
SYSTEM
Irrational
Numbers
Whole
Numbers
Natural
Numbers
Figure 1—The Real Number System
Real numbers have two major classifications: rational and irrational. On the left of the Venn diagram you
can find all of the subsets of numbers that are considered rational numbers. Refer to the Venn diagram
as you review the below; this will help you to understand how the sets of rational numbers relate to
each other.
Rational numbers include all of the following:
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Natural numbers are the counting numbers. These are written as {1, 2, 3, 4, 5, …}. You will
notice that natural numbers make up the very heart of the rational number set.
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Whole numbers include all of the counting numbers plus 0 (zero). Whole numbers are written
as {0, 1, 2, 3, 4, 5, …}.
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Integers are positive and negative whole numbers. They include the counting numbers (whole
numbers) as well as the opposite of the counting numbers (negative numbers), plus 0 (zero).
Integers are written as {…, −3, −2, −1, 0, 1, 2, 3, …}.
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Other rational numbers include decimals and fractions. A few examples of these include −5.45,
2.75, 3.8,
4
, and so on.
5
On the right side of the diagram is a second set of numbers known as irrational numbers. The set of
irrational numbers and set of rational numbers are separate, which means they do not overlap.
Irrational numbers are numbers that do not terminate or repeat. A few examples of irrational numbers
include 2 ,  ,
fractions.
5 , 3.96123…., and −5.21345… Also, irrational numbers cannot be written as
You will learn more about irrational numbers in a more advanced math course. For now, it is enough to
know that irrational numbers, as illustrated in the diagram, are not contained in the set of rational
numbers. However, they are part of the set of real numbers. They also can be found on the real number
line.
The Real Number Line
The real number line is useful for visualizing the location of real numbers. A number line is exactly what
it sounds like: a horizontal line that extends forever in both directions. Mathematicians say that a
number line extends from the point that’s called the origin or point 0. This is the point at which you can
find the zero (0). Figure 2 shows an example of the real number line.
Figure 2—The Real Number Line
In Figure 2, the point of origin is 0, which appears in the middle of the number line. Negative numbers
are shown in red and extend to the left of the point 0. Positive numbers are shown in blue. They extend
to the right of point 0. The line itself extends forever in both directions, as represented by the arrows.
The points on the number line correspond to real numbers. Every point corresponds to a real number,
such as 1, 2, 3, 4, –1, –2, –3, –4, etc. It is important to note that a number line includes all of the real
numbers, which means it includes both rational and irrational numbers. (Remember, irrational numbers
are not contained in the sets of rational numbers, but they are part of the real number system.)
Irrational numbers can be approximated along the number line using rational numbers. As a matter of
fact, between any two rational numbers, there are an infinite, or endless, number of irrational numbers.
The number line in Figure 3 has several rational numbers marked. The number line also shows the
approximate location where the irrational number
2 can be found.
2
Figure 3—A Second Number Line
This particular irrational number is located between the rational numbers of 1.4 and 1.5; as shown, it is
closer to 1.4.
The number line in Figure 4 shows the location of the following irrational numbers:
and 1.9345…
2
3
2,
3 , 1.8693…,
1.8693… 1.9345…
…
Figure 4—Another Sample Number Line
Notice that all four of these irrational numbers fall between the numbers 1 and 2. As already stated, the
irrational number
2 falls between the rational numbers 1.4 and 1.5, while being closer to 1.4. The
irrational number
3 falls between the rational numbers 1.7 and 1.8, while being closer to 1.7.
Additionally, the irrational number 1.8693… falls between the rational numbers 1.8 and 1.9, while being
closer to 1.9. A better way to approximate the location of 1.8693…. would be to describe it as “falling
between 1.85 and 1.9.” Similarly, the irrational number 1.9345… falls between the rational numbers 1.9
and 2, while being closer to 1.9. A better approximation of 1.9345… would describe it as “falling
between 1.9 and 1.95.”
Irrational numbers are an advanced topic. You may encounter them in a later course.
Now it is time to move on to Lesson 5.