Date: 4-18/19-12
Topic: 6-6 Rational
and Irrational
Numbers
Objective:
Essential Question: What is a rational number, and what is
an irrational number?
To find and use decimal representations of real numbers.
The 37set of real numbers includes both rational numbers,
such as and and irrational numbers, such as √ and .
One way to learn about real numbers is to look at their decimal
representations. We accept as an axiom the following property
of real numbers.
Completeness Property Every real number has a decimal representation, and every
of Real Numbers: decimal represents a real number.
Rational Number: Recall that a rational number is any number that can be
expressed as the ratio, or quotient, of two integers. To find the
decimal representation of a rational number, you can use the
division process.
Find a decimal representation for each rational number.
a.
Using a calculator,
Terminating
decimal:
b.
Using a calculator,
Repeating decimal:
Summary
1
A terminating decimal is also called a finite decimal.
A repeating decimal is also called an infinite decimal.
You can use a bar to indicate the block of digits that repeats.
̅̅̅̅
̅̅̅̅̅̅̅̅̅̅
Decimal Properties
of Rational
Numbers:
1. The decimal representation of any rational number is either
terminating or repeating.
2. Every terminating or repeating decimal represents a
rational number.
Every terminating or repeating decimal can be written in the
form,
,
where p and q are integers and
Example 1
Terminating
Decimal as a Ratio
.
Write each terminating decimal as a common fraction in lowest
terms.
a.
b.
2
Exercise 1
Write each terminating decimal as a common fraction in lowest
terms.
a.
b.
c.
Example 2
Repeating Decimal
as a Ratio
Write each repeating decimal as a common fraction in lowest
terms.
a.
̅
̅
̅
̅
b.
̅̅̅̅̅
̅̅̅̅̅
̅̅̅̅̅
̅̅̅̅̅
3
Exercise 2
Write each repeating decimal as a common fraction in lowest
terms.
a.
̅̅̅̅
b.
̅̅̅̅
c.
̅̅̅̅̅̅̅̅̅̅
4
Irrational Numbers:
An irrational number is a real number that is not rational. An
irrational number cannot be represented as a ratio of two
integers. Therefore, the decimal representation of an irrational
number is neither terminating nor repeating.
Two Irrational
Numbers:
√
1. The decimal representation of any irrational number is
infinite and nonrepeating.
2. Every infinite and nonrepeating decimal represents an
irrational number.
This fact is significant when you use a calculator to obtain the
decimal representation of an irrational number. Because the
calculator's display is finite, you can get only a rational
approximation of the irrational number.
Example 3
Classify each real number as either rational or irrational.
a. √
Infinite,
nonrepeating
decimal
√
b. √
√
Ratio of two integers
c.
Repeating decimal
5
d.
Infinite,
nonrepeating
decimal
Exercise 3
Classify each real number as either rational or irrational.
a.
b.
̅̅̅̅̅
c. √
d. √
e. √
√
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Definition
A set S of real numbers is dense if between any two numbers
in the set there is a member of S. Both the set of rational
numbers and the set of irrational numbers are dense.
Example 4
Find a rational number r and an irrational number s between
1.51287 and 1.51288.
There are an infinite number of possible answers for r and s.
Repeating
For example,
Nonrepeating
and
Exercise 4
Find a rational number r and an irrational number s between
0.3725 and 0.3726.
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