In Lesson 3–1, you learned about rational numbers. Natural numbers, whole numbers, and integers are all rational numbers. These sets are listed below.
Recall that repeating or terminating decimals are also
rational numbers because they can be expressed as 
, where a and b are integers and b ¹ 0. The square roots of perfect squares are also
rational numbers. For example, 
is a rational number
since 
. However, 
is irrational because
21 is not a perfect square.
The Venn diagram shows the relationship among the different
types of rational numbers. For example, the set of whole numbers is a subset of
the integers. This means that all whole numbers are integers. Similarly, all rational
numbers are real numbers.
In Lesson 8–6, you learned about irrational numbers. A
few examples of irrational numbers are shown below.
The
set of rational numbers and the set of irrational numbers together form the set of real numbers. Numbers such as 
and 
are
called complex
numbers. The set of complex numbers includes all
of the real numbers as well as numbers involving
square roots of negative numbers. We will not deal with complex numbers in this text.
Name the set or sets of numbers to which each real number
belongs.
If you graph all of the rational numbers, you will still have
some “holes” in the number line. The irrational numbers “fill in” the number line.
The graph of all real numbers is the entire number line without any “holes.
This property of real numbers is called the Completeness
Property.
You have learned how to graph rational numbers. Irrational
numbers can also be graphed. Therefore, every real number can be graphed. Use a
calculator or a table of squares and square roots to find approximate values of
square roots that are irrational. These values can be used to approximate the
graphs of square roots.
Example
Find an approximation, to the nearest tenth, for each square
root. Then graph
the square root on a number line.
Alternative Solutions:
Alternative Solutions:
Alternative Solutions:
Alternative
Solutions:
.PNG)
You have solved equations with
rational number solutions. Some equations have solutions that are irrational
numbers.
Science Link
10. The
time t in seconds it takes for a pendulum to complete one full swing (back and
forth) is given by the equation 
, where l is the length of the pendulum in meters. Suppose
a pendulum has a length of 4.9 meters. How long does it take the pendulum to
complete one full swing?
Alternative
Solutions:
The pendulum will complete one full
swing in about 4.4 seconds.
Sumber
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