Download The Real Number System

The Real Number
System
1.1
So Many Numbers, So Little Time
Number Sort................................................................................ 3
1.2 Is It a Bird or a Plane?
Rational Numbers. ......................................................... 11
1.3 Sew What?
Irrational Numbers........................................................ 19
1.4 Worth 1000 Words
Real Numbers and Their Properties. .............................. 29
© Carnegie Learning
Pi is
probably one of
the most famous
numbers in all of history.
As a decimal, it goes on and
on forever without repeating.
Mathematicians have already
calculated trillions of the
decimal digits of pi. It
really is a fascinating
number. And it's
delicious!
1
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© Carnegie Learning
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So Many Numbers,
So Little Time
Number Sort
Learning Goals
In this lesson, you will:
 Review and analyze numbers.
 Determine similarities and differences among various numbers.
 Sort numbers by their similarities and rationalize the differences between the groups of numbers.
I
f someone were to ask you to define the word “number”, could you do it? Could
you then think of the different types of numbers and try to organize them? It’s
actually not as easy as it sounds.
Throughout history, the task of categorizing numbers has caused a lot of
headaches. For example, the Greeks didn’t use negative numbers, and for a long
time people didn’t recognize 0 as a number. Even if mathematicians could agree
on which numbers should be included, they struggled with how to group them.
How many types of numbers are there? Should we define them as positive or
negative? What about zero? Should fractions be separate? What about decimals?
These are the types of questions mathematicians have taken very seriously over
© Carnegie Learning
the years. In fact, the history of numbers probably has caused more anger and
drama than a reality TV show. If you doubt this, consider the Greek mathematician
Hippasus. He was the first to claim that some numbers exist which go on forever and
never repeat (gasp!). He was allegedly drowned as punishment for this statement!
We have a system now for classifying numbers. It isn’t necessarily the “right” way,
or even the only way—it’s just a way the mathematical community has agreed to
group numbers.
Can you think of other systems that people have developed for classifying things?
1.1 Number Sort • 3
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Problem 1 Ready, Set, Sort!
Mathematics is the science of patterns and relationships. Looking for patterns and sorting
objects into different groups can provide valuable insights. In this lesson, you will analyze
many different numbers and sort them into various groups.
1. Cut out the thirty numbers on the following page. Then, analyze and sort the numbers
into different groups. You may group the numbers in any way you feel is appropriate.
However, you must sort the numbers into more than one group.
In the space provided, record the information for each of your groups.
• Name each group of numbers.
• List the numbers in each group.
© Carnegie Learning
• Provide a rationale for why you created each group.
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p
0.25
3
2​ __  ​
8
2101
20%
|23|
26.41
0.​91 ​ 
√​ 100 ​ 
627,513
0.001
2
2​ __ ​ 
3
0
√​ 2 ​ 
3.21 3 1012
1,000,872.0245
42
0.5%
____
___
__
__
2​√9 ​ 
__
2​√2 ​ 
__
20.​3 ​ 
1.523232323. . .
21
___
1.0205 3 10
1
6​ __  ​ 
4
223
√
9
​ ​ ___  ​ ​  
16
_____
√​ 0.25 ​ 
© Carnegie Learning
212%
100
​ ____
 
 ​ 
11
|2|
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© Carnegie Learning
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2. Compare your groupings with your classmates’ groupings. Create a list of the
different types of numbers you noticed.
© Carnegie Learning
Are any of the types
of numbers shared
among your groups? Or,
are they unique to each
group?
1.1 Number Sort • 7
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Problem 2 Let’s Take a Closer Look
1. Lauren grouped these numbers together.
__
__
__2 ___
100
0.​91 ​ , 2​    ​, ​   ​ , 1.523232323. . ., 20.​3 ​ 
3 11
Why do you think Lauren put these numbers in the same group?
2. Zane and Tanya provided the same rationale for one of their groups of numbers.
However, the numbers in their groups were different.
Zane
√
|23|, 
1 0 0, 627,513, 3.21 3 1012,
Tanya
√
20%, 
100,
627,513, 3.21 3 1012,
42, |2|
42, |2|, 212%
When I simplify each number,
Each of these numbers
it is a positive integer.
represents a positive integer.
Who is correct? Explain your reasoning.
© Carnegie Learning
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3. Tim grouped these numbers together.
__
__
3
2
2​ __8 ​ , 2101, 26.41, 2​__ 3  ​, 2​√9 ​ , 21, 20.​3 ​ 
What rationale could Tim provide?
4. Isaac used the reasoning shown when creating one of his groups of numbers.
The numbers are between 0 and 1.
Identify all of the numbers that satisfy Isaac’s reasoning.
5. Lezlee grouped these numbers together.
100 ​ , 1.523232323. . ., 212%, 6​ _1  ​ 
26.41, ​ ___
11
4
What could Lezlee name the group? Explain your reasoning.
© Carnegie Learning
Clip all your
numbers together and
keep them. You’ll need
them later in
this chapter.
Be prepared to share your solutions and methods.
1.1 Number Sort • 9
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© Carnegie Learning
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Is It a Bird
or a Plane?
Rational Numbers
Learning Goals
Key Terms
In this lesson, you will:





 Use a number line to compare and order
rational numbers.
 Learn about types of numbers and their
properties.
natural numbers (counting numbers)
whole numbers
integers
closed
rational numbers
 Perform operations with rational numbers.
T
here are all kinds of different numbers, which have been given some strange
names. It seems that mathematicians can come up with an infinite number of
different kinds of numbers.
A repunit number is an integer with all 1’s as digits. So, 11, 111, and so on, are all
repunit numbers. Pronic numbers are numbers that are the products of two
consecutive numbers. The numbers 2, 6, and 12 are the first pronic numbers
(1 3 2 5 2, 2 3 3 5 6, and 3 3 4 5 12).
To find the numbers that some call “lucky” numbers, first start with all the
© Carnegie Learning
counting numbers (1, 2, 3, 4, and so on). Delete every second number. This will
give you 1, 3, 5, 7, 9, 11, and so on. The second number in that list is 3, so cross off
every third number remaining. Now you have 1, 3, 7, 9, 13, 15, 19, 21, and so on. The
next number that is left is 7, so cross off every seventh number remaining.
Can you list all the “lucky” numbers less than 50?
1.2 Rational Numbers • 11
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Problem 1 A Science Experiment
Your science class is conducting an experiment to
see how the weight of a paper airplane affects the
distance that it can fly. Your class is divided into two
groups. Group 1 uses a yard stick to measure the
distances that an airplane flies, and Group 2 uses a
meter stick. Group 2 then takes their measurements in
Because paper is
typically sold in
500-sheet quantities, a
paper's weight is determined
by the weight of 500 sheets
of the paper. So, 500
sheets of 20-pound paper
weighs 20 pounds.
meters and converts them to feet. The results of the
experiment are shown in the table.
Type of Paper
Group 1
Measurements
Group 2 Converted
Measurements
20-pound paper
7
13 __
​    ​feet
8
13.9 feet
28-pound paper
3
14 ​ __  ​feet
8
14.4 feet
1. Your science class needs to compare the Group 1 measurement to
the Group 2 converted measurement for each type of paper.
3  ​as a decimal.
7  ​as a decimal.
b. Write 14 ​ __
a. Write 13 ​ __
8
8
2. On the number line shown, graph the Group 1 measurements
13.5
13.6
13.7
13.8
13.9
14.0
14.1
14.2
14.3
14.4
14.5
3. Use the number line to determine which group’s flight traveled farther for the
20-pound paper and for the 28-pound paper. Write your answers using
© Carnegie Learning
written as decimals and the Group 2 converted measurements.
complete sentences.
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Problem 2 Natural Numbers, Whole Numbers, and Integers
1. The first set of numbers that you learned when you
were very young was the set of counting numbers,
or natural numbers. Natural numbers consists
of the numbers that you use to count objects:
{1, 2, 3, 4, 5, …}.
a. How many counting numbers are there?
In the set
{1, 2, 3, 4, 5, . . .} the dots
at the end of the list mean
that the list of numbers goes
on without end.
b. Does it make sense to ask which counting number is the
greatest? Explain why or why not.
c. Why do you think this set of numbers is called the natural numbers?
You have also used the set of whole numbers. Whole numbers are made
up of the set of natural numbers and the number 0, the additive identity.
2. Why is zero the additive identity?
3. Other than being used as the additive identity, how else is zero used
© Carnegie Learning
in the set of whole numbers?
4. Explain why having zero makes the set of whole numbers more useful than the set of
natural numbers.
1.2 Rational Numbers • 13
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Another set of numbers is the set of integers, which is a set that includes all of the whole
numbers and their additive inverses.
5. What is the additive inverse of a number?
6. Represent the set of integers. Remember to use three dots to
show that the numbers go on without end in both directions.
Use brackets
to represent
sets.
7. Does it make sense to ask which integer is the least or which integer
is the greatest? Explain why or why not.
When you perform operations such as addition or multiplication on the
numbers in a set, the operations could produce a defined value that is also in the set.
When this happens, the set is said to be closed under the operation.
The set of integers is said to be closed under the operation of addition. This means that
for every two integers a and b, the sum a 1 b is also an integer.
8. Are the natural numbers closed under addition? Write an example to support
9. Are the whole numbers closed under addition? Write an example to support your answer.
© Carnegie Learning
your answer.
10. Consider the operation of subtraction. Are the natural numbers closed under
subtraction? Write an example to support your answer.
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11. Are the whole numbers closed under subtraction? Write an example to support
your answer.
12. Are the integers closed under subtraction? Write an example to support your answer.
13. Are any of these sets closed under multiplication? Write examples to support
your answers.
14. Are any of these sets closed under division? Write examples to support your answer.
You have learned about the additive inverse, the multiplicative inverse, the additive
identity, and the multiplicative identity.
© Carnegie Learning
15. Which of these does the set of natural numbers have, if any? Explain your reasoning.
16. Which of these does the set of whole numbers have, if any? Explain your reasoning.
1.2 Rational Numbers • 15
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17. Which of these does the set of integers have, if any? Explain your reasoning.
Problem 3 Rational Numbers
a ​, where a and b are
1. A rational number is a number that can be written in the form ​ __
b
both integers and b is not equal to 0.
a. Does the set of rational numbers include the set of whole numbers? Write an
example to support your answer. b. Does the set of rational numbers include the set of integers? Write an example to
support your answer. c. Does the set of rational numbers include all fractions? Write an example to
support your answer. d. Does the set of rational numbers include all decimals? Write an example to
2. Is the set of rational numbers closed under addition? Write an example to support
your answer.
© Carnegie Learning
support your answer.
3. Is the set of rational numbers closed under subtraction? Write an example to support
your answer.
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4. Is the set of rational numbers closed under multiplication? Write an example to
support your answer.
5. Is the set of rational numbers closed under division? Write an example to support
your answer.
6. Does the set of rational numbers have an additive identity? Write an example to
support your answer.
7. Does the set of rational numbers have a multiplicative identity? Write an example to
support your answer.
8. Does the set of rational numbers have an additive inverse? Write an example to
© Carnegie Learning
support your answer.
9. Does the set of rational numbers have a multiplicative inverse? Write an example to
support your answer.
1.2 Rational Numbers • 17
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10. You can add, subtract, multiply, and divide rational numbers in much the same way
that you did using integers. Perform the indicated operation.
a. 1.5 1 (28.3) 5 b. 212.5 2 8.3 5
1 3
​   ​ 5
c. 2​ __  ​2 __
2 4
7 ​   ​5
d. 2 __
​ 1 ​ 1 ​23​ __
2
8
e. 22.0 3 (23.6) 5 f. 6.75 3 (24.2) 5
2  ​3 __
​ 3 ​ 5
g. 2 ​ __
3 8
3 ​   ​5
3 ​ 3 ​22​ __
h. 23​ __
5
4
i. 21.5 4 4.5 5 j. 22.1 4 (23.5) 5
2
k. 2​__
    ​4 ___
​ 3  ​ 5
5 10
3 ​ 4 ​22​ __
2 ​   ​5
l. 21​ __
5
8
Remember,
it is a good idea
to estimate
first!
(  )
(  )
© Carnegie Learning
(  )
Be prepared to share your solutions and methods.
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Sew What?
Irrational Numbers
Learning Goals
Key Terms
In this lesson, you will:




 Identify decimals as terminating or repeating.
 Write repeating decimals as fractions.
 Identify irrational numbers.
irrational number
terminating decimal
repeating decimal
bar notation
I
n 2006, a 60-year-old Japanese man named Akira Haraguchi publicly recited
the first 100,000 decimal places of p from memory.
The feat took him 16 hours to accomplish—from 9 a.m. on a Tuesday morning to
1:30 a.m. the next day.
Every one to two hours, Haraguchi took a break to use the restroom and have a
snack. And he was videotaped throughout the entire process—to make sure he
© Carnegie Learning
didn’t cheat!
1.3 Irrational Numbers • 19
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Problem 1 Repeating Decimals
You have worked with some numbers like p that are not rational numbers. For example, ​
__
__
√ 2 ​ and √
​ 5 ​ are not the square roots of perfect squares and cannot be written in the
form __
​ a ​, where a and b are both integers.
b
Even though you often approximate square roots using a decimal, most square roots are
irrational numbers. Because all rational numbers can be written as __
​ a ​where a and b are
b
integers, they can be written as terminating decimals (e.g. __
​ 1 ​ 5 0.25) or repeating decimals
4
1  ​5 0.1666...). Therefore, all other decimals are irrational numbers because these
(e.g., ​ __
6
decimals cannot be written as fractions in the form __
​ a ​where a and b are integers and b is
b
not equal to 0.
1. Convert the fraction to a decimal by dividing the numerator by the denominator.
Continue to divide until you see a pattern.
_________
1  ​5 3) 1  ​
​
 
​ __
3
2. Describe the pattern that you observed in Question 1.
3. Order the fractions from least to greatest. Then, convert each fraction to a decimal by
dividing the numerator by the denominator. Continue to divide until you see a pattern.
________
__________
5 ​ 56 5
​)
 
 ​
a. ​ __
6
b. __
​ 2 ​ 59 ​) 2
9
________
 
 ​
​)
c. ___
​ 9  ​ 511 9
11
_
​ __________
d. ___
​ 3  ​ 5 22 ) 3
    ​
22
 
 ​
© Carnegie Learning
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4. Explain why these decimal representations are called repeating decimals.
A terminating decimal is a decimal that has a last digit. For instance, the decimal
5 __
​ 1 ​ . 1 divided by 8 is equal to 0.125.
0.125 is a terminating decimal because _____
​  125  ​ 
1000 8
A repeating decimal is a decimal with digits that repeat in sets of one or more. You can
use two different notations to represent repeating decimals. One notation shows one set
of digits that repeat with a bar over the repeating digits. This is called bar notation.
__
___
7  ​ 5 0.3​18 ​
1  ​5 0.​3 ​ ​ 
___
 
 
​ __
3
22
Another notation shows two sets of the digits that repeat with dots to indicate repetition.
You saw these dots as well when describing the number sets in the previous lesson.
7  ​ 5 0.31818…
1  ​5 0.33… ​ ___
​ __
3
22
5. Write each repeating decimal from Question 2 using both notations.
5  ​5 a.​ __
6
2  ​5
b. ​ __
9
9  ​ 5 c.​ ___
11
3  ​ 5
d. ​ ___
22
​ 2 ​ , and __
​ 1 ​ , and are used
Some repeating decimals represent common fractions, such as __
​ 1 ​ , __
3 3
6
often enough that you can recognize the fraction by its decimal representation. For most
repeating decimals, though, you cannot recognize the fraction that the decimal represents.
For example, can you tell which fraction is represented by the repeating decimal
___
© Carnegie Learning
 
0.44… or 0.​09 ​?
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You can use algebra to determine the fraction that is represented by the repeating
decimal 0.44… . First, write an equation by setting the decimal equal to a variable that
will represent the fraction.
w 5 0.44…
Next, write another equation by multiplying both sides of the equation by a power of 10.
The exponent on the power of 10 is equal to the number of decimal places until the
decimal begins to repeat. In this case, the decimal begins repeating after 1 decimal
place, so the exponent on the power of 10 is 1. Because 1​01​ ​5 10, multiply both
sides by 10.
10w 5 4.4…
Then, subtract the first equation from the second equation.
10w 5 4.44…
2w 5 0.44…
9w 5 4
Finally, solve the equation by dividing both sides by 9.
6. What fraction is represented by the repeating decimal 0.44...? ​ 
___
8. Repeat the procedure above to write the fraction that represents each repeating decimal.
a. 0.55… 5 ​  ___
c. 0.​12 ​ 5 ​ 
b. 0.0505… 5 ​ 
___
© Carnegie Learning
7. Complete the steps shown to determine the fraction that is represented by 0.​09 ​. 
d. 0.​36 ​ 5 ​ 
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Problem 2 Nobody’s Perfect . . . Unless
They’re a Perfect Square
Recall that a square root is one of two equal factors of a given number. Every positive
number has two square roots: a positive square root and a negative square root.
For instance, 5 is a square root of 25 because (5)(5) 5 25. Also, 25 is a square root of 25
because (25)(25) 5 25. The positive square root is called the principal square root. In this
course, you will only use the principal square root.
The symbol, , is called a radical and it is used to indicate square roots. The radicand
is the quantity under a radical sign.
radical
√25
radicand
This is read as “the square root of 25,” or as “radical 25.”
Remember that a perfect square is a number that is equal to the product of a distinct
factor multiplied by itself. In the example above, 25 is a perfect square because it is equal
to the product of 5 multiplied by itself.
1. Write the square root for each perfect square.
__
a.​√1 ​ 5
___
d.​√ 16 ​ 5
___
g.​√ 49 ​ 5
____
j.​√100 ​ 5
____
m.​√169 ​ 5
__
b.​√4 ​ 5
___
e.​√ 25 ​ 5
h.​√64 ​ 5
___
____
k.​√121 ​ 5
n.​√196 ​ 5
____
__
c.​√9 ​ 5
___
f.​√ 36 ​ 5
i.​√ 81 ​ 5
___
____
l.​√144 ​ 5
o.​√225 ​ 5
____
__
© Carnegie Learning
2. What do you think is the value of √
​ 0 ​?
  Explain your reasoning.
1.3 Irrational Numbers • 23
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3. Notice that the square root of each expression in Question 1 resulted in a rational
number. Do you think that the square root of every number will result in a rational
number? Explain your reasoning.
4. Use a calculator to evaluate each square root. Show each answer to the hundredthousandth.
___
​√25 ​ 5
__
​√5 ​ 5
_____
​√0.25 ​ 5
_____
​√ 225 ​ 5
_____
​√ 2500 ​ 
5
_____
​√ 6.76 ​ 5
____
​√ 676 ​ 5
_____
​√ 67.6 ​ 5
____
​√250 ​ 5
___
​√2.5 ​ 5
_____
5
​√ 6760 ​ 
______
5
​√26.76 ​ 
5. What do you notice about the square roots of rational numbers?
7. Is the square root of a decimal always an irrational number?
© Carnegie Learning
6. Is the square root of a whole number always a rational number?
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8. Consider Penelope and Martin’s statements and reasoning, which are shown.
Penelope
___
I know that 144 is a perfect square, and so √
​ 144  ​ is a rational
____
​ and ​
number. I can move the decimal point to the left and √
​ 14.4  
____
√ 1.44 ​ will also be rational numbers.
____
Likewise, I can move the decimal point to the right so ​√ 1440  
​ and ​
______
√14,400  
​ will also be rational numbers.
Martin
___
I know that 144 is a perfect square, and so √
​ 144  ​ is a rational
number. I can move the decimal point two places to the right or
left to get another perfect square rational number. For instance, ​
____
______
√ 1.44 ​ and √
​ 14,400  
​ will also be rational numbers.
Moving the decimal two places at a time is like multiplying or
dividing by 100. The square root of 100 is 10, which is also a
rational number.
Who is correct? Explain your reasoning.
© Carnegie Learning
1.3 Irrational Numbers • 25
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The square root of most numbers is not an integer. You can estimate the square root of a
number that is not a perfect square. Begin by determining the two perfect squares closest
to the radicand so that one perfect square is less than the radicand, and one perfect
square is greater than the radicand. Then, use trial and error to determine the best
estimate for the square root of the number.
“It might be
helpful to use the
grid you created in
Question 1 to identify
the perfect squares.”
___
To estimate √
​ 10 ​ to the nearest tenth, identify
the closest perfect square less than 10 and
the closest perfect square greater than 10.
The closest The closest
perfect square
The square root
perfect square
less than 10:
you are estimating:
greater than 10:
___
 
9 ​√10 ​ You know:
__
16
___
√
​ 9 ​  3 ​√16 ​  4
___
___
​√ 10 ​ 
So, 10 __
is___
between √​ 90 ​ 
and √​ 10 ​
16  . Why can't
I say__ it's between
___
√
​ 0 ​
10 ​ ?
1  and √​ 25
This means the estimate of √
​ 10 ​ is between 3 and 4.
Next, choose decimals between 3 and 4, and calculate the square
of each number to determine which one is the best estimate.
Consider:
(3.1)(3.1) 5 9.61
(3.2)(3.2) 5 10.24
___
So, ​√10 ​  3.2
The symbol < means approximately equal to.
___
The location of √
​ 10 ​ is closer to 3 than 4 when plotted on a
√10
0
1
2
3
4
5
6
7
8
9
10
© Carnegie Learning
number line.
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9. Identify the two closest perfect squares, one greater than the radicand and one less
than the radicand.
__
a.​√8 ​ 
___
b.​√ 45 ​ 
___
c.​√ 70 ​ 
___
d.​√ 91 ​ 
10. Estimate the location of each square root in Question 9 on the number line.
Then, plot and label a point for your estimate.
0
1
2
3
4
5
6
7
8
9
10
© Carnegie Learning
1.3 Irrational Numbers • 27
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11. Estimate each radical in Question 9 to the nearest tenth. Explain your reasoning.
__
a. ​√8 ​ 
___
b. ​√45 ​ 
___
c. ​√70 ​ 
___
© Carnegie Learning
d. ​√91 ​ 
Be prepared to share your solutions and methods.
28 • Chapter 1 The Real Number System
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Worth 1000 Words
Real Numbers and
Their Properties
Learning Goals
Key Terms
In this lesson, you will:
 real number
 Venn diagram
 closure
 Classify numbers in the real number system.
 Understand the properties of real numbers.
T
he word zero has had a long and interesting history so far. The word comes
from the Hindu word sunya, which meant "void" or "emptiness." In Arabic, this
word became sifr, which is also where the word cipher comes from. In Latin, it was
changed to cephirum, and finally, in Italian it became zevero or zefiro, which was
shortened to zero.
The ancient Greeks, who were responsible for creating much of modern formal
© Carnegie Learning
mathematics, did not even believe zero was a number!
1.4 Real Numbers and Their Properties • 29
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Problem 1 Picturing the Real Numbers
1. A wrestler uses a scale at home and at work to monitor his weight for a week. He
records the following weights:
3 ​ lbs, 152.0 lbs, 151​ __
1 ​ lbs, 151.6 lbs
1 ​ lbs, 151.8 lbs, 152.1 lbs, 151​ __
152​ __
2
4
2
Write his weights in order from least to greatest.
In the first lesson of this chapter, you cut out 30 real numbers and sorted them. Now, let’s
create a Venn diagram to organize this set of numbers.
2. Write the 30 numbers from the first lesson in order from least to greatest.
The Venn
diagram was introduced
in 1881 by John Venn,
British philosopher and
mathematician.
Combining the set of rational numbers and the set of irrational numbers
produces the set of real numbers. You can use a Venn diagram to
represent how the sets within the set of real numbers are related.
3. On the next page, create a Venn diagram to show the relationship
between the six sets of numbers shown. Then, write each of the
the Venn diagram.
integers
irrational numbers
natural numbers
rational numbers
real numbers
whole numbers
© Carnegie Learning
30 numbers from Question 2 in the appropriate section of
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4. Use your Venn diagram to decide whether each statement is true or false. Explain
your reasoning.
a. A whole number is sometimes an irrational number.
© Carnegie Learning
b. A real number is sometimes a rational number.
c. A whole number is always an integer.
1.4 Real Numbers and Their Properties • 31
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d. A negative integer is always a whole number.
e. A rational number is sometimes an integer.
f. A decimal is sometimes an irrational number.
5.
Omar
A square root is always an
irrational number.
Explain to Omar why he is incorrect in his statement.
6.
A fraction is never an
irrational number.
© Carnegie Learning
Robin
Explain why Robin’s statement is correct.
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Problem 2 Properties of Real Numbers
The real numbers, together with their operations and properties, form the real number
system. You have already encountered many of the properties of the real number system
in various lessons. Let’s review these properties.
Closure: A set of numbers is said to be closed under an operation if the result of the
operation on two numbers in the set is a defined value also in the set. For instance, the set
of integers is closed under addition. This means that for every two integers a and b, the
sum a 1 b is also an integer.
1. Is the set of real numbers closed under addition? Write an example to support
your answer.
2. Is the set of real numbers closed under subtraction? Write an example to support
your answer.
3. Is the set of real numbers closed under multiplication? Write an example to support
your answer.
4. Is the set of real numbers closed under division? Write an example to support
your answer.
Additive Identity: An additive identity is a number such that when you add it to a second
© Carnegie Learning
number, the sum is equal to the second number.
5. For any real number a, is there a real number such that a 1 (the number) 5 a?
What is the number?
6. Does the set of real numbers have an additive identity? Write an example to
support your answer.
1.4 Real Numbers and Their Properties • 33
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Multiplicative Identity: A multiplicative identity is a number such that when you multiply it
by a second number, the product is equal to the second number.
7. For any real number a, is there a real number such that a 3 (the number) 5 a?
What is the number?
8. Does the set of real numbers have a multiplicative identity? Write an example to
support your answer.
Additive Inverse: Two numbers are additive inverses if their sum is the additive identity.
9. For any real number a, is there a real number such that a 1 (the number) 5 0? What
is the number?
10. Does the set of real numbers have an additive
inverse? Write an example to support
your answer.
Multiplicative Inverse: Two numbers are
multiplicative inverses if their product is the
multiplicative identity.
You have been
using these properties
for a long time, moving
forward you now know
that they hold true for
the set of real
numbers.
11. For any real number a, is there a real number
such that a 3 (the number) 5 1? What is
12. Does the set of real numbers have a multiplicative
inverse? Write an example to support your answer.
© Carnegie Learning
the number?
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Commutative Property of Addition: Changing the order of two or more addends in an
addition problem does not change the sum.
For any real numbers a and b, a 1 b 5 b 1 a.
13. Write an example of the property.
Commutative Property of Multiplication: Changing the order of two or more factors in a
multiplication problem does not change the product.
For any real numbers a and b, a 3 b 5 b 3 a.
14. Write an example of the property.
Associative Property of Addition: Changing the grouping of the addends in an addition
problem does not change the sum.
For any real numbers a, b and c, (a 1 b) 1 c 5 a 1 (b 1 c).
15. Write an example of the property.
Associative Property of Multiplication: Changing the grouping of the factors in a
multiplication problem does not change the product.
For any real numbers a, b, and c, (a 3 b) 3 c 5 a 3 (b 3 c).
© Carnegie Learning
16. Write an example of the property.
Reflexive Property of Equality:
Symmetric Property of Equality:
For any real number a, a 5 a.
For any real numbers a and b, if a 5 b,
17. Write an example of the property.
then b 5 a.
18. Write an example of the property.
Transitive Property of Equality:
For any real numbers a, b, and c, if a 5 b and b 5 c, then a 5 c.
19. Write an example of the property.
1.4 Real Numbers and Their Properties • 35
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Talk the Talk
For each problem, identify the property that is represented.
1. 234 1 (2234) 5 0
2. 24 3 (3 3 5) 5 (24 3 3) 3 5
3. 224 3 15 224
4. 267 3 56 5 56 3 (267)
5. 2456 1 34 5 34 1 (2456)
6. 4 3 0.25 5 1
7. If 5 5 (21)(25) then (21)(25) 5 5.
8. If c 5 5 3 7 and 35 5 70 4 2,
then c 5 70 4 2.
(  )(  )
9. a 1 (4 1 c) 5 (a 1 4) 1 c
3 4
10. ​2__
 ​  ​   ​​2__
​   ​   ​5 1
4 3
3 ​ 
3 ​ 3 1 5 22​ __
11. 22​ __
4
4
3
3 4
​   ​   ​1 ​__
​ 4  ​1 5 ​5 ​2__
​   ​ 1 __
​   ​  ​1 5
12.​2__
4
4 3
3
(  ) ( 
) ( 
)
© Carnegie Learning
13. Order each set of real numbers from least to greatest.
__
__
4  ​, 22.8, √
a.​ __
​ 3 ​,  2​√5 ​,  22%
b. A baker measures the following amounts
3
of whole wheat flour for several recipes:
​ 1 ​ cup, 2.5 cups, 0.75 cup,
__
​ 2 ​ cup, __
3
4
1
__
1​    ​cups, 0.2 cup
3
Be prepared to share your solutions and methods.
36 • Chapter 1 The Real Number System
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Chapter 1 Summary
Key Terms
Properties
 natural numbers
 Additive Identity (1.4)
 Multiplicative
(counting numbers) (1.2)






whole numbers (1.2)
integers (1.2)
closed (1.2)
rational numbers (1.2)
irrational number (1.3)
terminating decimal
Identity (1.4)
 Additive Inverse (1.4)
 Multiplicative
 repeating decimal (1.3)
 bar notation (1.3)
Addition (1.4)
 Associative Property of
Multiplication (1.4)
 Reflexive Property of
Equality (1.4)
Inverse (1.4)
 Commutative Property of
 Symmetric Property of
Equality (1.4)
Addition (1.4)
 Commutative Property of
(1.3)
 Associative Property of
 Transitive Property of
Equality (1.4)
Multiplication (1.4)
 real number (1.4)
 Venn diagram (1.4)
 closure (1.4)
Providing Rationale for Groupings of Numbers
Numbers can be grouped in a variety of ways according to their characteristics. Numbers
can be identified as whole numbers or integers, fractions or decimals, rational or irrational.
Sometimes a number may fit into multiple groupings. When providing a rationale, all of the
numbers in the group must fit that rationale.
© Carnegie Learning
Example
3
p, 3.25, 2​__
   ​ , 65%, 215, 2.52, |28|, ___
​ 20 ​ , 3.9 3 102
7
5
The numbers 215, |28|, ___
​ 20 ​ , and 3.9 3 102 can be grouped together and identified as
5
integers because each of these numbers can be written as an integer.
Chapter 1 Summary • 37
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Comparing and Ordering Rational Numbers Using a Number Line
Fractions and decimals can be compared and ordered by converting fractions to decimals
and plotting on a number line.
Example
5  ​ . 5.3. The fraction 5​ ___
5  ​ is equal to 5.3125.
The number line is used to show 5​ ___
16
16
5.3
5.28
5.29
5.30
5.3125
5.31
5.32
5.33
5.34
5.35
Performing Operations with Rational Numbers
A rational number is a number that can be written in the form __
​ a ​, where a and b are both
b
integers and b is not equal to 0. You can add, subtract, multiply, and divide rational
numbers in much the same way that you do using integers.
Example
19 ___
3 ​ 3 6​ __
4 ​ 5 2​ ___
22​ __
 ​ 3 ​ 58 ​ 
8
8
9
9
1102
5 2​ _____
 
 ​ 
72
22 ​ 
5 215​ ___
72
11 ​ 
5 215​ ___
36
Identifying Terminating and Repeating Decimals
A terminating decimal is a decimal that has a last digit. A repeating decimal is a decimal
with digits that repeat in sets of one or more. Two different notations are used to represent
repeat, and place a bar over the repeating digits. Another notation is to write the decimal,
including two sets of the digits that repeat, and using dots to indicate repetition.
Examples
__
​ 7 ​ is a terminating decimal:
8
4  ​is a repeating decimal:
and​ __
9
0.875
______
© Carnegie Learning
repeating decimals. One notation is to write the decimal, including one set of digits that
0.444
______
4  ​5 9​)4.000 ​ 
__
__
 
​ 7 ​ 5 8​)7.000 ​​ 
8
9
__
4  ​5 0.​4 ​ 
__
​ 7 ​ 5 0.875​ __
8
9
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Writing Repeating Decimals as Fractions
Some repeating decimals represent common fractions, such as 0.33… 5 __
​ 1 ​ , and are
3
used often enough that we recognize the fraction by its decimal representation.
However, there are decimals in which it is difficult to determine the fractional equivalent. To
determine the fraction for the decimal, first, write an equation by setting the decimal equal
to a variable that will represent the fraction. Next, write another equation by multiplying
each side of the equation by a power of 10. The exponent on the power of 10 is equal to
the number of decimal places until the decimal begins to repeat. Then, subtract the first
equation from the second equation. Finally, solve the equation.
Example
___
The repeating decimal 0.​15 ​ is equal to the fraction ___
​ 5  ​ .
33
___
100w 5 15.​15 ​ 
___
15 ​
5   0.​
____________
  
 ​  
​ 2w
99w
99w
55
1515
w 5 ___
​ 15 ​ 
99
w 5 ___
​ 5  ​ 
33
Identifying Irrational Numbers
Decimals that do not repeat and do not terminate are said to be irrational numbers. An
irrational number is a number that cannot be written in the form __
​ a ​, where a and b are both
b
integers and b fi 0.
Example
___
An example of an irrational number is √
​ 11 ​ because it is a square root that is not a perfect
© Carnegie Learning
square and therefore has no repeating patterns of digits.
Chapter 1 Summary • 39
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Estimating Square Roots
A square root is one of two equal factors of a nonnegative number. Every positive number
has two square roots, a positive square root (called the principal square root) and a
negative square root. To determine a square root that is not a whole number, identify the
two closest perfect squares, one greater than and one less than the radicand. Then,
estimate the square root to the nearest tenth.
Example
___
Estimate √
​ 23 ​ to the nearest tenth.
___
Twenty-three is between the two perfect squares 16 and 25. This means that √
​ 23 ​ 
is between 4 and 5, but closer to 5.
___
​ 23 ​ is approximately 4.8.
Because 4.72 5 22.09 and 4.82 5 23.04, √
___
The location of √
​ 23 ​ is closer to 5 than 4 when plotted on a number line.
√23
1
2
3
4
5
6
7
8
9
10
© Carnegie Learning
0
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Classifying Numbers in the Real Number System
Combining the set of rational numbers and the set of irrational numbers produces the set
of real numbers. Within the set of rational numbers, a number can be or not be an integer,
whole number, natural number, or some combination.
You can use a Venn diagram to represent how the sets within the set of real numbers
are related.
Real Numbers
Rational Numbers
2
3
, – , – 0.3, 1.0205 10–23,
3
8
9
,
0.001, 0.5%, 20%, 0.25, 0.25,
16
100
, 1,000,872.0245
0.91, 1.523232323…, 212%, 6 1 ,
4 11
– 6.41, –
Irrational Numbers
– 2,
2, Integers
– 101, – 9 , –1
Whole Numbers
0
Natural
Numbers
|2|, |–3|, 100, 42,
627,513,
3.21 1012
Examples
π is an irrational number.
28 is a rational number and an integer.
23 is a natural number, whole number, integer, and rational number.
__
​ 1  ​is a rational number.
© Carnegie Learning
4
Chapter 1 Summary • 41
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Understanding the Properties of Real Numbers
The real numbers, together with their operations and properties, form the real number
system. The properties of real numbers include:
Closure: A set of numbers is said to be closed under an operation if the result of the
operation on two numbers in the set is another member of the set.
Additive Identity: An additive identity is a number such that when you add it to a second
number, the sum is equal to the second number.
Multiplicative Identity: A multiplicative identity is a number such that when you multiply it
by a second number, the product is equal to the second number.
Additive Inverse: Two numbers are additive inverses if their sum is the additive identity.
Multiplicative Inverse: Two numbers are multiplicative inverses if their product is the
multiplicative identity.
Commutative Property of Addition: Changing the order of two or more addends in an
addition problem does not change the sum.
Commutative Property of Multiplication: Changing the
order of two or more factors in a multiplication problem
does not change the product.
Associative Property of Addition: Changing the
grouping of the addends in an addition problem does
Hopefully
you didn't become
irrational during this
chapter! Remember keep
a positive attitude_it makes
a difference!
not change the sum.
Associative Property of Multiplication: Changing the grouping of the
factors in a multiplication problem does not change the product.
Reflexive Property of Equality: For any real number a, a 5 a.
Symmetric Property of Equality: For any real numbers a and b,
if a 5 b, then b 5 a.
Transitive Property of Equality: For any real numbers a, b, and c,
Examples
128 1 (2128) 5 0 shows the additive inverse.
13 3 (27) 5 27 3 13 shows the commutative property of multiplication.
© Carnegie Learning
if a 5 b and b 5 c, then a 5 c.
89 3 1 5 89 shows the multiplicative identity.
(31 3 x) 1 y 5 31 1 (x 1 y) shows the associative property of addition.
If x 5 7 1 y and 7 1 y 5 21, then x 5 21 shows the transitive property of equality.
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