copyright © cengage learning. all rights reserved. chapter 5 extending the number system

Copyright © Cengage Learning. All rights reserved.

CHAPTER 5

Extending theNumber System

Copyright © Cengage Learning. All rights reserved.

SECTION 5.4

Beyond Integers and Fractions: Decimals, Exponents, and Real Numbers

3

What Do You Think?

• What would Alphabitian decimals look like?

• Why is there no “oneths” place?

4

Decimals

5

Decimals

Technically, the set of decimals is not an extension of the set of fractions.

Rather, our use of decimal numbers comes out of an alternative way of representing fractions, using the advantages of our base ten system.

In fact, the word decimal comes from the Latin word decem, which means “ten.”

Thus decimal numbers are any numbers that are written interms of our base ten place value.

6

Decimals

Conventionally, we refer to decimals as fractions whose denominator has been converted into a power of ten.

The history of decimals: The use of decimals was spurred primarily by a desire to make computation easier.

For example, which computation would be easier to do: 37 6 or 37.75 6.8?

You may want to do both. How much faster was the decimal computation?

7

Decimals

Even today there is not one universal notation. For example:

No decimals? Imagine our world without decimals!

What would life in the United States be like if no one had invented decimals—that is, if we expressed amounts using only whole numbers and fractions?

8

Decimals

Would things just look different (for example, 43 dollars and 26 cents instead of $43.26), or would some things be impossible?

As you may have realized as you thought about these questions, much of our use of decimals involves measurement:

• We measure mass: for example, the package weighs 2.4 ounces.

• We measure distance: for example, Joan lives 3.7 miles from school.

9

Decimals

• We measure time: for example, the world record for the 100-meter dash is 9.58 seconds.

Decimals are used in many aspects of daily life to communicate. For example,

• The U.S. birth rate is 14.1 per 1000 population.

• The average life expectancy in the United States is 77.9 years.

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Connecting Decimals and Integers

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There are many important similarities and differences between the set of decimals and the set of integers.

What similarities and differences can you see?

From one perspective, the set of decimals can be seen as an extension of base ten counting numbers.

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That is, the system of decimals and the system of integers are both base ten place value systems.

The value of each digit depends on its place, and the value of each place to the right is the value of the previous place.

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Investigation A – Base Ten Blocks and Decimals

Let’s say that a sample of gold weighs 3.6 ounces. How would you use base ten blocks (Figure 5.38) to represent this amount? Note that the “small cube” was formerly called a “unit.”

Figure 5.38

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Investigation A – Discussion

Any solution requires us to designate one of the pieces as the unit.

For example, if we designate the big cube as having a value of 1, then we would represent 3.6 ounces as shown in Figure 5.39.

Figure 5.39

15

Investigation A – Discussion

However, we could also have selected the flat to be the unit.

If we had, then we would represent 3.6 ounces as shown in Figure 5.40.

cont’d

Figure 5.40

16

Expanded Form

17

Expanded Form

Like whole numbers, decimal numbers can be expressed in expanded form:

As with whole numbers, zeros often prove challenging to make sense of. However, the same principles still apply:

18

Zero and Decimals

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Investigation B – When Two Decimals Are Equal

What if we added one or two zeros at the end of 0.2? Would that change its value?

That is, do 0.2, 0.20, and 0.200 have the same value? Many sixth- and seventh-graders (and a surprising number of adults) believe that 0.2, 0.20, and 0.200 not only look different but also have different values because of the added zeros.

How would you convince such a person that the value is not changed?

20

Investigation B – Discussion

There are several ways to justify the equality. We will consider two below.

Strategy 1: Connect Decimals To Fractions

• 0.2 is equivalent to the fraction

• 0.20 is equivalent to the fraction which can be shown to be equivalent to • 0.200 is equivalent to the fraction which can be

shown to be equivalent to

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Applying the transitive property of equality, we say that because 0.2, 0.20, and 0.200 are all equal to they are equal to each other.

Strategy 2: Use Manipulatives

In this case, let a square represent a value of 1.

Such manipulatives are sold commercially as Decimal Squares.

cont’d

22


We can divide this square into 10 equal pieces, 100 equal pieces, or 1000 equal pieces, as in Figure 5.41.

Shade in 0.2 of the first square, 0.20 of the second square, and 0.200 of the third square.

We find that in each case, the same area is shaded. Thus, 0.2, 0.20, and 0.200 are all equivalent decimals.

cont’d

Figure 5.41

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Investigation C – When Is the Zero Necessary and When Is It Optional?

An important difference between decimals and whole numbers has to do with our old friend zero.

Examine each number below. In each case, explain whether you think the zero in the number is necessary, optional, or incorrect.

A. 2.08 B. 0.56 C. .507 D. 20.6 E. 3.60

Discussion:

A. With 2.08, the zero is necessary: 2 ones, 0 tenths, and 8 hundredths.

24

Investigation C – Discussion

B. With 0.56, the use of the zero is optional; it is a convention, like shaking hands with the right hand instead of the left.

C. With .507, the zero is necessary: 5 tenths, 0 hundredths, and 7 thousandths.

D. With 20.6, the zero is necessary: 2 tens, 0 ones, and 6 tenths.

E. In one sense, the zero here is optional. Mathematically, 3.6 = 3.60; you can verify this using the strategies in Investigation B.

cont’d

25


In another sense, however, it depends on how thenumber is being used.

For example, if we ask how long the room is, whatis the difference between a response of 3.6 meters and aresponse of 3.60 meters?

A response of 3.6 meters implies that the room has beenmeasured to the nearest tenth of a meter; that is, thelength of the room is closer to 3.6 meters than to 3.5 or3.7 meters.

cont’d

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If the person doing the measuring has used aninstrument that measures only tenths but reports thelength as 3.60.

cont’d

27

Zero and Decimals

Some differences between decimals and integers:With decimals, when we add a zero at the right end ofthe number, the value is unchanged; for example,48.6 = 48.60.

As you know, this is not true with integers; for example,48 480.

Two other differences between the two systems areworth noting.

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Zero and Decimals

One has to do with language: With whole numbers, thesuffix for each place is -s; with decimals, it is -ths—forexample, the hundreds place versus the hundredthsplace.

Another difference is that there is a ones place but nota oneths place; this lack of symmetry has been noted bymore than one youngster.

How would you explain it?

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Investigation D – Connecting Decimals and Fractions

A. How many of the following common fractions can you convert to decimals immediately? How can you determine the decimal equivalent of others without using a calculator?

Discussion:

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Investigation D – Discussion

Either by hand or with a calculator, all but and can be readily converted into decimal form.

When we convert into a decimal (by dividing 1 by 6), we

find that it repeats; that is, the value is 0.1666666. . . . There are two ways to write repeating decimals.

We can write in decimal form as 0.166... or as

The bar shows the part that repeats. For example, we

would write as 0.348348348... or as

cont’d

31

Investigation D – Connecting Decimals and Fractions

B. Now let us examine translating decimals into fractions. Translate the decimals 0.1, 0.007, and into fractions.

Discussion:Most people have no trouble with the first two:

The last decimal is equivalent to

Someone (unknown to history) discovered the process that enables us to conclude that there is a unique fraction associated with every repeating decimal.

cont’d

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Investigation D – Discussion

The process works like this:

Let x = 0.18181818 . . .

That is, x = 0.18

Multiply both sides of this equation by 100: 100x = 18.18

Subtract the first equation from the second: 99x = 18

Solve for x:

cont’d

33

Decimals, Fractions, and Precision

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We have used number lines as a model to represent whole numbers and fractions.

The number line can also be used to represent decimals, using the analogy of a zoom lens.

Suppose you had never heard of decimals and had to determine the length of the wire in Figure 5.42 as accurately as you could.

Figure 5.42

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A visual inspection of Figure 5.42 shows that the wire appears to be closer to inches than to inches.

With decimals, we do not look at the most appropriate unitfraction; instead, we look at the fractional length in terms of tenths, hundredths, thousandths, and so forth.

Now imagine using a zoom lens (Figure 5.43).

Figure 5.43

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To the nearest tenth of an inch, how long is the wire? Because the length is closer to 4.3 inches than to 4.4 inches, we say that the length is 4.3 inches.

If we want more precision, we can keep zooming in, in which case we move more decimal places to the right.

Theoretically, we can continue this magnification processindefinitely.

Realistically, electron microscopes are able to measure distances of about one ten-millionth of an inch, or0.0000001 inch.

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Investigation E – Ordering Decimals

Just as young children need time to develop proper ordering relationships with whole numbers (for example, that 30, not 31, comes after 29), older children need time to develop ordering relationships with decimals.

Order the following decimals, from smallest to largest: 0.39, 0.046, and 0.4. Justify your solution.

Discussion:Strategy 1: Use Equivalent Fractions With CommonDenominators

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Investigation E – Discussion

Thus the order is 0.046, 0.39, 0.4.

Strategy 2: Line Up The Decimal Points And Add Zeros

0.39 = 0.390

0.046 = 0.046

0.4 = 0.400

cont’d

39

Investigation E – Discussion

Strategy 3: Represent Them On A Number Line

cont’d

40

Investigation F – Rounding with Decimals

The process of rounding with decimals is very similar to the process of rounding with whole numbers.

In fact, if you really understand place value, then rounding with decimals is very straightforward.

As you may have found with the Right Bucket game being able to round decimals is important when estimating.

Round each of the numbers to the nearest hundredth and to the nearest tenth:

A. 3.623 B. 76.199 C. 36.215 D. 2.0368

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Investigation F – Rounding with Decimals

E. As you work on these, take mental stock of your confidence level. If you do not feel 100 percent confident, what models (discussed earlier) might you use to increase your confidence?

Discussion:

A. 3.623 to the nearest hundredth is 3.62, and to the nearest tenth is 3.6.

B. 76.199 to the nearest hundredth is 76.20, and to the nearest tenth is 76.2.

cont’d

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Investigation F – Discussion

C. 36.215 to the nearest hundredth is 36.22, and to the nearest tenth is 36.2.

D. 2.0368 to the nearest hundredth is 2.04, and to the nearest tenth is 2.0.

E. Some students find the number line or physical models useful in determining how to round. One other common strategy goes like this: Look at 76.199.

If we look at only two decimal places, we have 76.19. Therefore, the question can be seen as: Is 76.199 closer to 76.19 or to 76.20?

cont’d

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Investigation F – Discussion

If we insert zeros so that all three decimals are given in thousandths, we have

76.200

76.199

76.190

cont’d

44

Investigation G – Decimals and Language

“The School Board has proposed a budget of $24.06 million.” Express the School Board’s proposed budget as a whole number.

Discussion:

Strategy 1: Connect Decimals To Fractions

Some students who have successfully used this strategy did not know how to represent 24.06 million at first, and so they began with decimal / fraction connections that they didknow.

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Investigation G – Discussion

24.5 million = 24,500,000 because



[This is often a hard place for many students to start, but it

makes sense now, in light of the first two steps. Does it make sense to you?]

cont’d

46


24.06 million = 24,060,000 because it must be less than 24,100,000

Strategy 2: Connect Decimals To Expanded Form

Some students found the solution by analyzing themeaning of 0.06 million. 24.06 million means 24 million and six hundredths of a million.

Now we have to find the value of six hundredths of a million. Using a calculator or computing by hand yields

0.06 1,000,000 = 60,000

cont’d

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Thus,

24.06 million = 24,000,000 + 60,000 = 24,060,000

Strategy 3: Connect Decimals To Place Value

If your understanding of place value is powerful, you know that the places of the digits will not change.

Thus, when we represent 24.06 million as a whole number, we simply “add zeros” and we have 24,060,000.

The place of the 6 does not change!

cont’d

48

Operations with Decimals: Addition and Subtraction

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Consider the addition problem 7.8 + 0.46.

We can justify the lining up of decimal places in terms of whole-number computation.

When adding whole numbers, we begin with the ones place and move to the left, regrouping as necessary.

Adapting this procedure to decimals, we begin with the smallest place—in this case, the hundredths place.

We have 6 hundredths in 0.46 and 0 hundredths in 7.8.

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Moving to the tenths place, we have 12 tenths (regrouped to 1 and 2 tenths).

Moving to the ones place, we have 7 plus the regrouped 1, giving us a sum of 8 ones, 2 tenths, and 6 hundredths—that is, 8.26.

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Multiplication with Decimals

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Children often find multiplication with decimals confusing. Let us examine two problems that arise:

• How do we know where to put the decimal point?

• How can you multiply tenths by tenths and get hundredths—for example, 3.2 2.6 = 8.32?

Let us examine the problem 3.2 meters 2.6 meters.

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Before reading on, please work on the following questions:

• How can you determine the area of the rectangle directly from the diagram shown below?

• What are the connections between the diagram and the standard algorithm for multiplying decimals?

Figure 5.44

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In the diagram, we have four distinct regions thatcorrespond to the four partial products [see Figures 5.44(b) and (c)].

The top left region consists of 6 squares (each of whichmeasures 1 meter by 1 meter). Thus the top left region has a value of 6 square meters.

The top right region consists of 4 longs, each of which measures 1 meter by 0.1 meter. Thus each long has a value of 0.1 square meter.

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The value of the top right region, then, is 0.4 square meter.

The bottom left region consists of 3 6 = 18 longs and has a value of 1.8 square meters.

The bottom right region consists of 12 little squares, each of which is 0.1 meter by 0.1 meter. Thus the value of each square is 0.1 0.1, or 0.01 square meter.

Therefore, the value of this region is 0.12 square meter,because we have 2 6 = 12 of these little squares.

Adding up the value of the four regions, we have 6 + 0.4 + 1.8 + 0.12 = 8.32 square meters.

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The calculation on the left below shows the sums of the four regions, and the calculation on the right shows the standard multiplication procedure.

Do you see the connections?

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We are still applying the distributive property. Going back to the meaning of whole-number multiplication, the multiplication algorithm simply tells us the sums of the fourregions.

The beauty of the procedure is more evident when we realize that 3.2 .6 = 1.92 and 3.2 2 = 6.4 don’t “line up.”

However, when we move 64 over one place, the two partial sums line up just right.

We are not generally aware of this lining up; we just remember how many decimal places to move over in the answer column.

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Division with Decimals

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For example, let us say that you travel 247.9 miles on 7.4 gallons of gas. To determine the miles per gallon, you divide 247.9 by 7.4.

One key is the idea of multiple representations, which has been presented many times.

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An equivalent representation of this division problem is

This form connects to the idea of equivalent fractions. If we multiply this decimal fraction by we have

In other words, has the same value as

By moving the decimal point over, we create an equivalent computation in which the divisor (denominator) is a whole number.

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Investigation H – Decimal Sense: Grocery Store Estimates

Let’s say that it is Friday afternoon. You drop by the supermarket to get a few items. Suddenly you realize that you didn’t go to the bank, and you find that you have only $11.

The milk is $2.95. The ice cream is $1.87. Grapes are 89¢ a pound, and you have just over a pound.

The bag of potato chips is $2.69. The loaf of bread is $3.29. Do you have enough money? How would you estimate this amount in your head?

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Investigation H – Discussion

Strategy 1: Round Up And Use Compatible Numbers

$2.95 is almost $3

$2.70 + 3.30 = $6

$1.87 + $.89 $1.90 + $.90 = $2.80

$3 + $6 + $2.80 = $11.80

We know that the actual sum is lower than this estimate.

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In situations like this, we intentionally overestimate because we don’t want to find out at the checkout counter that we don’t have enough money.

This next investigation offers several ways to apply ideas from different concepts we have studied.

Strategy 2: Use Leading Digit And Compatible Numbers

Looking at the “dollars column” (that is, the ones column), we have 4.

Next, we look at the “dimes” (tenths) column.

cont’d

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This is what that strategy sounds like:

• 8(dimes) + 2(dimes) is one dollar.

cont’d

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• 9(dimes) + 6(dimes) is a dollar fifty, so that’s two fifty.

• 8(dimes) more makes three thirty, plus the 8(dollars). That’s eleven thirty plus the cents. You don’t have enough money!

cont’d

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Investigation I – Decimal Sense: How Much Will the Project Cost?

Let’s say a contractor needs 308 sheets of plywood for a project. He can get them for $9.55 each at one store.

He could also get them for $9.39 at another store, but he doesn’t particularly like the owners, and more paperwork would be required.

How much money would he save if he went to the second store?

Estimate the answer as closely as you can without using a calculator or pencil and paper.

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Investigation I – Discussion

We could try to estimate 308 9.55 and 308 9.39, but this would be quite tedious.

With a bit of reflection, we realize that what we need to know is not the approximate total cost but the difference.

Because the difference is 16¢, or $.16, per sheet, we findthat we really have to estimate 308 $.16 or 308 16¢.

Strategy 1: Converting To Fractions

0.16 308 300 = $50

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Investigation I – Discussion

Strategy 2: Use Substitution and The Associative Property

0.16 308 0.16 (100 3)

= (0.16 100) 3

= 16 3

= $48

cont’d

69

Investigation J – How Long Will She Run?

A runner is running at 7.6 meters per second. At this rate, about how long will it take her to run 100 meters?

Discussion:

This problem is a classic example of how decimals can obscure simple relationships.

The problem-solving strategy of using simpler numbers is powerful here.

For example, suppose the runner was running at 5 meters per second.

How long would it take her to run 100 meters?

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Investigation J – Discussion

With these simpler numbers, it is almost immediately apparent that it will take 20 seconds; that is, divide 100 by 5.

We can now apply this understanding to this problem and

see that it will take her or about seconds, which is

about 12 or 13 seconds.

cont’d

71

Investigation K – Exponents and Bacteria

With the development of negative numbers and decimals, we are now poised to understand better the operation of exponentiation, which is an operation that we have alreadyreferred to (for example, in describing place value relationships).

If a kind of bacteria doubles every hour, how many bacteria will there be after 24 hours if we begin with 1 bacterium?

Discussion:

There are several ways to address this problem. For example, we can make a table to represent the growth.

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Investigation K – Discussion

The third column simply represents the number of bacteria, using exponents.

The number of bacteria after 24 hours will be 2 to the 24th power.

cont’d

73


Exponents and multiplication: Just as multiplication can be seen as repeated addition, exponentiation can be seen as repeated multiplication, as in Figure 5.45.

In mathematical language, b is called the base and x is called the exponent. We say

b2 = b squared

b3 = b cubed

Figure 5.45

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b4 = b to the fourth power

bx = b to the xth power

Computing exponents: This then brings us to another problem: How do we determine the value of

What do you think? If you remember anything about exponents, explore alternatives for a while before reading on.

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In one sense, we want to know how to make the following computation less tedious:

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

One (of several) possibilities is to translate this computation into an equivalent computation, using the associative property of multiplication:

(2 2 2 2) (2 2 2 2) (2 2 2 2) (2 2 2 2)

(2 2 2 2) (2 2 2 2)

76


That is, it is much quicker to press 16 16 16 16 16 16 on a calculator than to press 2 twenty-four times.

If you have a scientific calculator, pressing the following entries yields the answer:

that is, 16,777,216, andalso.

Negative exponents: When the exponent is 2 or greater, we can interpret it as repeated multiplication.

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However, what if the exponent is less than 2? For example, what are the meanings and value of the following amounts:21, 20, 2–1, 2–2?

One way to verbalize the pattern is to say that each time the exponent decreases, the value of the expression is divided by 2.

It is reasonable to assume that this pattern will continue,and thus it is reasonable to conclude that the value of 21

is 4 2 = 2.

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This line of reasoning enables us to continue the progression. In this manner, we deduce that the valueof 20 is 2 2 = 1.

Continuing this pattern, 2–1 = 1 2 = and so 2–2 = 2 =

Once again, we rely on multiple representations. We can represent 2–2 as or we can represent 2–2 as because4 = 22.

This now leads us to the generalization that

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Investigation L – Scientific Notation: How Far Is a Light-Year?

One way in which astronomers address the hugeness of interstellar distance is to measure astronomical distances not in miles or kilometers but rather in a unit called“light-years”—that is, the distance light travels in one year.

The speed of light is 186,000 miles per second. Determine the length of a light-year.

Discussion:

In this problem, dimensional analysis can greatly simplifythe computations.

80

Investigation L – Discussion

Do the multiplication now on a calculator.

Most of you will have something like this:

5.8697 12 or 5.8697136 12

which is the calculator’s code for 5.8697 1012 .

Formally, we say that a number is in scientific notation if it is in the form a 10b, where a is a number between 1 and 10, 1 a < 10, and b is an integer.

The reasoning behind restricting a to a number between 1 and 10 is based on convention rather than mathematical structure.

cont’d

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Investigation L – Discussion

This convention makes it easier for us to compare amounts.

One strategy is to write the decimal, and then move the decimal point 12 times:

When we insert the commas, we have 5,869,700,000,000. How would we say this amount?

We would say 5 trillion, 869 billion, 700 million miles or we would round further to 5.9 trillion miles.

cont’d

82

Irrational Numbers

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Irrational Numbers

Consider a right triangle in which the length of each of the two sides is one inch, as shown in Figure 5.46.

If we call the hypotenuse x and apply the Pythagorean theorem, we have 12 + 12 = x2

x2 = 2

That is, x is the number that, when multiplied by itself, equals 2.

Figure 5.46

84

Irrational Numbers

This problem was perplexing to the Pythagoreans, because try as they might, they could not find a rational number that, when multiplied by itself, came to exactly 2.

Using modern notation and base ten arithmetic, we know that we can get very close:

1.4 1.4 = 1.96

1.41421 1.41421 = 1.9999899

However, there is no rational number (that is, a number that can be expressed as the quotient of two integers) that will produce a product of exactly 2.

85

Irrational Numbers

In modern language, we say x =

Finally, one member of the sect proved that there is no rational number that will solve this problem.

86

Investigation M – Square Roots

We will focus our explorations on the positive square roots of numbers, also called the principal square root.

Without using the square root button on your calculator, determine the square root of 800 to the nearest hundredth.

Discussion:

To determine a little estimating and mental math help.

For example, 30 30 = 900

Therefore, a reasonable starting point would be 29.

87

Investigation M – Discussion

We can represent 29 29 as (30 – 1)(30 – 1) and apply the distributive property 900 – 30 – 30 + 1 = 841.

In any case, because we know that 302 = 900 and 292 = 841, a reasonable first guess is 28.

cont’d

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The Real-Number System

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With the set of irrational numbers, we can now extend our system of numbers to the set of real numbers, which is defined as the union of the sets of rational and irrational numbers.

We began our study of numbers with the set of natural numbers, which is the first set of numbers young children encounter.

Our first extension of the number line was to add zero. The union of the set of natural numbers and zero is called theset of whole numbers.

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We then examined two kinds of numbers that children will encounter in elementary school: integers and rational numbers.

Finally, we encountered numbers that cannot be represented as the ratio of two whole numbers—that is, irrational numbers.

91


Figure 5.47 is a visual representation of these different sets of numbers and their relationships.

Figure 5.47

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Properties of Real Numbers

93


The operations of addition, subtraction, and multiplication are closed for real numbers.

That is, for any two real numbers, a and b, a + b is a real number, a – b is a real number, and a b is a real number.

If we were to say that division is closed, we would have to make one quali-fication.

If we say that for any two real numbers a and b, is a real number, we would simply have to note that b cannot equal zero.

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Another property that is important for you to know about has to do with the density property of rational numbers, which represents a qualitative difference between the set of rational numbers and the three previous sets.

However, as dense as the rational numbers were, we found that the number line was not yet complete, because there are points on the real number line that have no rational name.

When we add the set of irrational numbers to the set ofrational numbers, our number line is complete.

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This is known as the completeness property:There is a real-number name for every single point on thenumber line.

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