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Lesson 52
The Real Numbers
This text began with the natural numbers, the numbers 1, 2, 3, 4, and so on that we use for counting. We have gradually
included fractions, negative numbers, and decimals. In this section we examine how all these kinds of numbers fit together.
Thinking about the numbers will also help with understanding the difference between mathematics as itself and mathematics
as it is used practically.
The Natural Numbers: {1, 2, 3, …}
Even their name, the natural numbers, makes these numbers feel comfortable and right. It seems easy, primitive, and
natural to count how many goats in the flock, or how many nights since the last full moon.
After counting, people thought about measuring.
Example: Mark and label the number line with the natural numbers. The first unit is marked.
Use the first unit to measure the rest.
If we mark units on a ruler, it quickly becomes unsatisfying to round to the nearest cubit or inch or whatever all the time.
Dividing up the units, or the operation of division with the natural numbers, give us fractions, and these were the next kind of
number in general use. Instead of moving historically through the development of the number system, though, here we’ll
move logically, gradually expanding it so that each group is included in the one that follows. After the natural numbers, then,
we expand the number system by including only one new number – the number zero.
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Lesson 52 ! page 2
The Whole Numbers: {0, 1, 2, 3, …}
Our number system has grown by only one new number, but it’s a new idea as well. If you used to watch Sesame Street as
a kid, you may remember the Count. Occasionally the Count would run out of things to count, so he would count things that
weren’t there, like cabbages, and find that there were “ZERO! Zeerrro cabbages!” This is funny because counting to zero is
not really what we think of as counting. Including “none” as a number is a good idea, though, because it gives us the answer
to the subtraction problem 3 – 3 right there in our number system.
Example: Mark and label the number line with the whole numbers.
It’s much the same as before, except that now we label the zero point.
The arrow at the right hand side of the number line shows that it doesn’t really end at 5. The numbers go on forever,
because however high we count, we can always count one more. So the arrow represents that the line goes on to infinity,
just as do the three dots (ellipses) in the list {0, 1, 2, 3, …}.
Example: Answer true or false, and give a reason for your answer.
True or False? The number zero (0) is not a natural number.
True. The natural numbers begin at 1.
True or False? The number zero (0) is not a whole number.
False. The whole numbers include the number zero.
True or False? The number one (1) is not a whole number.
False. The whole numbers include all the natural numbers.
True or False? The number one-half (1/2) is not a whole number.
True. The whole numbers do not include fractions.
True or False? The number negative one (–1) is not a whole number.
True. The whole numbers do not include negatives.
True or False? All natural numbers are also whole numbers.
True. The whole numbers include all the natural numbers.
True or False? There is no end to the natural numbers, they go on forever to infinity.
True. The number line or a list cannot show all the natural numbers.
Whatever number you count up to, you can always count higher.
True or False? The number 6.5777 x 1015 is a natural number.
True. 6.5777 x 1015 = 6,577,700,000,000,000 which is a natural number.
© 2010 Cheryl Wilcox
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Lesson 52 ! page 3
The Integers: {… –3, –2, –1, 0, 1, 2, 3, …}
The integers include all the whole numbers and their negatives. The great thing about including the negatives is that they
give us the answers to every subtraction problem with whole numbers. We can now subtract 6 – 106 and have an answer,
–100, that is part of our number system. The negative numbers expand the number line itself – it now has two directions.
Note that the term integer does not include all the negative numbers that you know about. Only the negatives of whole
numbers are included. Zero is the only number whose negative is itself, and therefore it sits at the center of the number line
(though it is otherwise hard to find the center of an infinite line), separating the positives from the negatives.
Example: Mark and label the number line with the integers. The first unit is marked for you.
Example: Answer true or false, and give a reason for your answer.
True or False? The whole numbers are all integers.
True. The integers include the whole numbers and their negatives.
True or False? The negative numbers and the integers are the same thing.
False. Not all negative numbers are integers. For example, –1/2 is not an integer.
Also, not all integers are negative. For example, 3 is an integer.
True or False? The number zero (0) is an integer.
True. The integers include the whole numbers.
Right now our number line seems pretty empty, even though there are infinitely many numbers on it. There are wide spaces
between the integers. Can we fill them with fractions?
The Rational Numbers: {p/q such that p and q are integers and q ! 0}
The kinds of numbers we’ve talked about so far had lists with their names, marked with ellipses to show the lists continued
to infinity. But if you look in the title’s set brackets here, the rational numbers have instead a kind of definition. The rational
numbers are the answers to all the division problems with integers (except that division by zero is never, ever allowed). Do
you remember that there is a sense in which the fraction 3/4 is both the division problem 3 ÷ 4 and the answer to that
division problem? That’s why 3/4 is a rational number. So is –155/3481, and so is 6/(–2) = –3. Even 0 is the answer to a
division problem, for example, 0/16 = 0. So the rational numbers include all the integers but also the positive and negative
fractions.
Example: Circle the rational numbers that are labeled on the number line.
All the integers are included in the rational numbers, and all the fractions, both proper and improper. Different forms of
fractions, such as mixed numbers and decimals, are also rational numbers.
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Lesson 52 ! page 4
We should feel pretty good at this point. It seems as if the number line is completely filled with numbers. We can add,
subtract, multiply, and divide, and all those problems (except dividing by zero) have answers in our number system. If we’re
marking up a ruler, we can zoom in really tight and mark it with tiny, tiny subdivisions. Here’s the space between 0 and 1
enlarged and marked in hundredths:
If each hundredth were divided into ten equal parts, those would mark thousandths, and you can see how tiny those would
be even in this rather large unit. And don’t forget, we’re not limited to tenths and hundredths and so forth. If we decided to
use fractions, we could divide the space between 0 and 1 into 45,689 equal parts, or into 9999999 equal parts. So it seems
as if we could measure any length and find an exact fraction to label it. Doesn’t it?
Fractions and Decimals Briefly Reviewed
To write a fraction as a decimal, you divide numerator by denominator, and continue the decimal places in the division.
There are two possible results when you convert a fraction to a decimal: the decimal can either terminate, or repeat.
Terminating
When the denominator includes only factors of 2 and/or 5,
the decimal representation of the fraction will terminate.
Example: Find the decimal representation of 3/8.
On the calculator, we simply press 3 ÷ 8 = 0.375
The division by hand is shown below:
0.375
8)3.0000
2.4
60
56
40
40
0
0.375
.
Repeating
8)3.0000
If the denominator has factors
2.4 other than 2 or 5, the decimal
representation of the fraction
60will eventually consist of a
repeating pattern of digits. 56
40
40representation of 11/60.
Example: Find the decimal
0
On the calculator, we simply press 11 ÷ 60 = 0.18333333.
The division by hand is shown below:
0.18333
.
60)11.00000
6.0
5.00
4.80
200
180
200
180
200
180
20
You can see that you become locked in an endless loop
which creates the repeating 3s at the end of the decimal.
0.18333
60)11.00000
6.0
5.00
4.80
200
Two examples do not prove that this
180is always true, but it has been proved, and so mathematicians know that the decimal
representation of any fraction (that 200
is, any rational number) either terminates or eventually repeats one or more digits in
sequence to inifinity.
180
200
180
20
© 2010 Cheryl Wilcox
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Lesson 52 ! page 5
But if the decimal representation of every rational number either terminates or repeats, what about decimals that do neither?
We learned that the decimal representation of ! continues to infinity with no repeating pattern. The same is true of the
square roots that are not perfect squares. Greek mathematicians since the 5th century BC knew (with mathematical proof)
that 2 could not be written as a fraction. The numbers ! and 2 , as well as the other non-perfect square roots, are not
rational numbers.
What does this mean? The rational numbers, a seemingly complete number system we
have built by beginning with counting and finding answers for all addition, subtraction,
multiplication, and division problems, are not really enough.
The fractions are not enough to fill the spaces between the
integers, because we could measure the circumference of a
circle with diameter one unit and not find ! as a fraction on the
number line, no matter how small or cleverly we choose the
fraction divisions. We can draw a square that measures one
unit on each side, and there will be no possible fraction mark
on the ruler to measure the length of the diagonal.
This is a little shocking, when you think about it.
The Irrational Numbers
The number ! and the non-perfect square roots are examples of what we call irrational numbers. This is the first of our
number categories that doesn’t include the previous categories. The irrational numbers are the other numbers, the leftovers, the ones that aren’t rational. We can place them on the number line, since we know they measure lengths. We can
approximate them with rational numbers as closely as we like, zooming in, and narrowing down, writing decimal place after
decimal place. But we cannot represent these numbers exactly with a fraction.
Furthermore, there are not just a few of these numbers. If you were to match each irrational number up with a rational
number partner, no matter how you rearranged and manipulated, there would be irrational numbers to spare. In this sense
there are actually more lengths on the number line that are measured by irrational numbers than by rational numbers,
although there are infinitely many of both. Don’t let anyone tell you that mathematics is cut-and-dried, devoid of mystery. We
deal with infinity here.
The Real Numbers
Since the irrational numbers are simply all the numbers on the number line that are not rational, together with the rationals
they cover the number line. The set of all the numbers on the number line is called the real numbers.
Example: Answer true or false, and give a reason for your answer.
True or False? The decimal 1.414213562 is equal to the square root of 2.
False. The square root of 2 is irrational, so its decimal representation does not terminate.
True or False? The square root of 2 will fall between the rational numbers 1.4 and 1.5 on the number line.
True. The square root of 2 is more than 1.4 and less than 1.5.
True or False? The whole numbers are included in the irrational numbers.
False. The whole numbers are rational numbers.
Irrational numbers are numbers that are not rational numbers.
True or False? The square root of 2 is a real number.
True. It is a number on the number line.
© 2010 Cheryl Wilcox
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Lesson 52 ! page 6
Digital or Analog?
If you paint or draw a picture, your line is continuous. When you copy it digitally, it becomes pixelated, made up of tiny
squares. If there are enough pixels, your eye is fooled, and sees the picture as continuous. A sound in the air is made by a
continuous vibration. If you record that sound with analog equipment, the continuous vibration in the air is translated into a
continuous electronic wave. If you record that sound with digital equipment, it is recorded as a sequence of tiny, discrete
steps.
You know from experience how good digital recording or digital photography can be. If we use small enough pixels or steps
the quality is amazing. In a similar way, scientists and engineers and business people who use numbers make decimal
approximations to irrational numbers, just as your calculator does, and those approximations are more than good enough for
their purposes..
But mathematicians want to use all the real numbers, and work in the analog, continuous world. That’s why we insist on the
distinction between the equals ( = ) and approximately equals ( ! ) signs. That’s why mathematicians prefer fraction to
decimal answers, so we can see right away which numbers are rational and which irrational. That’s why a mathematician
would rather write the circumference of a circle with diameter 23 inches as 23! inches, rather than approximating it by
writing 72 inches, or 72.26 inches or even 72.256631 inches. And that’s why a mathematician considers 2 both the
problem and the answer to “How long is the hypotenuse of a right triangle with legs of length one?”
Although mathematics has extensive practical uses in every technical field, it is a separate discipline and has its own
concerns, questions, conventions, and methods. It’s a big world, and often surprising and beautiful.
!
© 2010 Cheryl Wilcox
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Lesson 52 ! page 7a
Lesson 52: The Real Numbers
Worksheet
Name______________________________________
1. Number Sorting: Write each number in the list in the smallest appropriate bin.
!195,
3,
7
1
, 18 , 0.9999, " ,
17
5
25, 0, 8, ! 8, 7.85 # 1099
Why are some bins inside each other?
What is the name of the entire big bin containing both the rational and irrational bins?
2. Number line.
a. Circle the irrational numbers.
b. Circle the rational numbers.
c. Circle the integers.
d. Circle the whole numbers.
© 2010 Cheryl Wilcox
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Lesson 52 ! page 8a
3. Answer true or false, and give a reason for your answer.
a. True or False? A rational number must be positive.
b. True or False? An integer is always negative.
c. True or False? The real numbers do not include !.
d. True or False? Zero is a natural number.
e. True or False? The irrational numbers include all the rational numbers as well as numbers like ! and the square root of 2.
f. True or False? Every integer is a real number.
g. True or False? The square root of three is between the rational numbers 1.73 and 1.74 on the number line.
h. True or False? For most practical applications of mathematics, rational approximations for irrational numbers are fine.
i. True or False? Mathematicians are a little weird about accuracy.
© 2010 Cheryl Wilcox
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Lesson 52 ! page 9
Lesson 52: The Real Numbers
Homework 52A
Name_________________________________________
1. A 20-yard roll of red duct tape costs $6.99. A 60-yard roll
of gray duct tape costs $7.95. Find the price per yard of
each roll.
2. Solve the equation 0.7a ! 1.4 = 2.1
3. What is the length of the line?
4. Find the shaded area.
1
2
3
5. 55 people, or 22% of those interviewed, were selected as
finalists. How many people were interviewed?
6. If your workday is eight hours, and you have worked one
and a half hours, what percent of your work day is
completed?
.
30 + 6
7. Evaluate 7 !
.
2
2
© 2010 Cheryl Wilcox
8. A ten pound bag of potting soil contains 20% sand. How
much sand is in the bag?
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Lesson 52 ! page 10
9. Evaluate
10. Evaluate.
a.
9 • 100
a.
(!8)2
b.
9 • 100
b.
82
c.
9 ! 100
d.
9 ! 100
c. ! 82
d.
11. Find the length of the hypotenuse. Round to the nearest
tenth if rounding is necessary.
!82
12. If the pole is 18 feet and the stabilizing wire is 20 feet,
how far from the base of the pole should the wire be
fastened so that the pole makes a right angle with the
ground?
13. Answer true or false and give a reason for your answer.
a. All the natural numbers are rational numbers.
b. The length of the diagonal of a square with sides 1 cm cannot be measured exactly with a fraction.
.
c. Irrational numbers are not real numbers.
d. The number –18 is an integer.
T
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 52 ! page 11
Lesson 52: The Real Numbers
Homework 52A Answers
1. A 20-yard roll of red duct tape costs $6.99. A 60-yard roll
of gray duct tape costs $7.95. Find the price per yard of
each roll.
$6.99
= $0.3495 / yd
20 yd
Red duct tape is about $0.35 per yard.
$7.95
= $0.1325 / yd
60 yd
2. Solve the equation 0.7a ! 1.4 = 2.1
0.7a ! 1.4 = 2.1
0.7a ! 1.4 + 1.4 = 2.1+ 1.4
0.7a = 3.5
0.7a / 0.7 = 3.5 / 0.7
a=5
Gray duct tape is about $0.13 per yard.
3. What is the length of the line?
4. Find the shaded area.
The line is 1 and 1/8 inch long.
1
2
3
Rectangle + Triangle =
(18)(10) + (18)(10)/2 =
180
+
90 =
270 square inches
5. 55 people, or 22% of those interviewed, were selected as
finalists. How many people were interviewed?
22% of those interviewed is 55 people.
0.22x = 55
x = 55 / 0.22 = 250
6. If your workday is eight hours, and you have worked one
and a half hours, what percent of your work day is
completed?
1.5 hours / 8 hours = 0.1875
18.75% of your work day Is complete.
250 people were interviewed.
30 + 6
7. Evaluate 7 !
.
2
2
36
6
= 49 !
2
2
= 49 ! 3 = 46
= 72 !
© 2010 Cheryl Wilcox
8. A ten pound bag of potting soil contains 20% sand. How
much sand is in the bag?
20% of 10 lb is
0.2 • 10 = 2 lb sand
The bag contains 2 lbs sand.
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Lesson 52 ! page 12
9. Evaluate
10. Evaluate.
a.
9 • 100 = 900 = 30
a.
(!8)2 = 64 = 8
b.
9 • 100 = 3 • 10 = 30
b.
82 = 64 = 8
c.
9 ! 100 = 3 ! 10 = !7
d.
9 ! 100 = !93 not a real number
11. Find the length of the hypotenuse. Round to the nearest
tenth if rounding is necessary.
c. ! 82 = ! 64 = !8
d.
!82 = !64 not a real number
12. If the pole is 18 feet and the stabilizing wire is 20 feet,
how far from the base of the pole should the wire be
fastened so that the pole makes a right angle with the
ground?
a2 + b2 = c2
a2 + b2 = c2
11.72 + 4.4 2 = 156.25
c 2 = 156.25
c = 156.25 = 12.5
The hypotenuse is 12.5 cm.
a 2 + 182 = 202
a 2 + 324 = 400
a 2 = 400 ! 324 = 76
a = 76 " 8.718
The wire should be fastened about
8.7 feet from the base of the pole.
13. Answer true or false and give a reason for your answer.
a. All the natural numbers are rational numbers.
True. The rational numbers include the natural numbers.
b. The length of the diagonal of a square with sides 1 cm cannot be measured exactly with a fraction.
True. The length of the diagonal is the square root of 2, which is irrational.
c. Irrational numbers are not real numbers.
False. The real numbers include both the rational and the irrational numbers.
d. The number –18 is an integer.
True. –18 is the negative of 18, which is a whole number.
The integers include the whole numbers and their negatives.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 52 ! page 13
Lesson 52: The Real Numbers Name
Homework 52B
Name______________________________________
1. Find the price per track for downloading a 30-track album
for $5.99 and an 18-track album for $6.49.
2. Solve the equation 8.5a ! 2.3 = 6.2
3. What is the length of the line?
4. Find the shaded area.
1
2
3
5. 52 people, or 40% of those interviewed, were selected as
finalists. How many people were interviewed?
6. If your workday is seven hours, and you have worked five
and a half hours, what percent of your work day is
completed?
7. Evaluate 2 • 7 ! 2 • 49 .
8. A twenty pound bag of potting soil contains 30% sand.
How much sand is in the bag?
2
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 52 ! page 14
9. Evaluate
a.
b.
10. Evaluate.
36 • 144
36 • 144
a.
102
b.
(!10)2
( 100 )
d. ( ! 100 )
2
c.
36 ! 144
d.
36 ! 144
c.
2
11. Find the length of the side. Round to the nearest tenth if
rounding is necessary.
12. If the 12.5 foot long stabilizing wire is fastened 11.7 feet
up the pole, how far from the base of the pole should the it
be fastened so that the pole makes a right angle with the
ground?
13. Answer true or false and give a reason for your answer.
a. None of the natural numbers are irrational.
.
b. Zero is its own negative.
c. The result of dividing 16 ÷ 0 is a rational number.
d. Zero is not a rational number.
.
© 2010 Cheryl Wilcox