the real numbers - diablo valley collegevoyager.dvc.edu/~lmonth/prealg/lesson52student.pdf · the...

Free Pre-Algebra Lesson 52 ! page 1

© 2010 Cheryl Wilcox

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.



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.



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.


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.



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.

Repeating If the denominator has factors other than 2 or 5, the decimal

representation of the fraction will eventually consist of a

repeating pattern of digits.

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:

Example: Find the decimal representation of 11/60.

On the calculator, we simply press 11 ÷ 60 = 0.18333333. The division by hand is shown below:

You can see that you become locked in an endless loop which creates the repeating 3s at the end of the decimal.

Two examples do not prove that this is always true, but it has been proved, and so mathematicians know that the decimal representation of any fraction (that is, any rational number) either terminates or eventually repeats one or more digits in

sequence to inifinity.

0.375 .

8)3.0000 2.4

60 56

40

40 0

0.18333

60)11.00000 6.0

5.00 4.80

200

180 200

180 200

180

20

0.375 .

8)3.0000 2.4

60 56

40

40 0

0.18333

60)11.00000 6.0

5.00 4.80

200

180 200

180 200

180

20



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 left-

overs, 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.


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.



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.

!

Free Pre-Algebra 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

17, 18

1

5, 0.9999, ", 25, 0, 8, ! 8, 7.85 # 10

99

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.

Free Pre-Algebra 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.




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.

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?

.

7. Evaluate

72!

30 + 6

2.

8. A ten pound bag of potting soil contains 20% sand. How much sand is in the bag?

1 32



9. Evaluate

a. 9 • 100

b. 9 • 100

c. 9 ! 100

d. 9 ! 100

10. Evaluate.

a.

(!8)2

b. 82

c. ! 82

d. !82

11. Find the length of the hypotenuse. Round to the nearest tenth if rounding is necessary.

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




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

20 yd= $0.3495 / yd

Red duct tape is about $0.35 per yard.

$7.95

60 yd= $0.1325 / yd

Gray duct tape is about $0.13 per yard.


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


The line is 1 and 1/8 inch long.


Rectangle + Triangle = (18)(10) + (18)(10)/2 =

180 + 90 =

270 square inches


22% of those interviewed is 55 people. 0.22x = 55

x = 55 / 0.22 = 250

250 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?

1.5 hours / 8 hours = 0.1875

18.75% of your work day Is complete.

7. Evaluate

72!

30 + 6

2.

= 72!

36

2= 49 !

6

2

= 49 ! 3 = 46

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.

1 32



9. Evaluate

a. 9 • 100 = 900 = 30

b. 9 • 100 = 3 • 10 = 30

c. 9 ! 100 = 3 ! 10 = !7

d. 9 ! 100 = !93 not a real number

10. Evaluate.

a.

(!8)2

= 64 = 8

b. 82

= 64 = 8

c. ! 82

= ! 64 = !8

d. !82

= !64 not a real number

11. Find the length of the hypotenuse. Round to the nearest tenth if rounding is necessary.

a2+b

2= c

2

11.72+ 4.4

2= 156.25

c2= 156.25

c = 156.25 = 12.5

The hypotenuse is 12.5 cm.

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+b

2= c

2

a2+ 18

2= 20

2

a2+ 324 = 400

a2= 400 ! 324 = 76

a = 76 " 8.718

The wire should be fastened about 8.7 feet from the base of the pole.


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.



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.





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 • 72! 2 • 49 .

8. A twenty pound bag of potting soil contains 30% sand. How much sand is in the bag?

1 32



9. Evaluate

a. 36 • 144

b. 36 • 144

c. 36 ! 144

d. 36 ! 144

10. Evaluate.

a. 102

b.

(!10)2

c.

100( )2

d.

! 100( )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?


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.

.

the real numbers - diablo valley collegevoyager.dvc.edu/~lmonth/prealg/lesson52student.pdf · the...

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