UNIT 4: Prealgebra in a Technical World
4.1 Decimal Fractions and Scientific Notation
SWBAT 1. Read and write decimal fractions.
2. Add and subtract signed decimal fractions.
3. Read and write scientific notation for whole numbers.
4. Read and write scientific notation for decimal fractions.
While Europe was sinking
into the Dark Ages, by 500 CE
scholars in India were writing
symbols for numbers that
morphed into the ten digits we
. Damascus
use today. Europe still used
Roman numerals at this time,
and Roman numerals do not
have a zero.
By this time, all of Asia
used a number for zero, but
Europeans still did not have a
symbol, or even the idea, of zero.
By 980 CE Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi, most probably from Damascus,
Syria, (see map1), had introduced the decimal fractions. A decimal fraction is a fraction that has
a power of 10 for its denominator. The brilliant idea of Abu'l Hasan Ahmad was to write these
fractions by extending the decimal place value system. No longer did we need to write
denominators for fractions; we could simply put numerators in their correct decimal places.
In this section, we study naming decimal fractions and using scientific notation for decimals.
1
http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Uqlidisi.html accessed 6/26/10
285
SECTION 4.1 Decimals and Scientific Notation
Read and Write Decimal Fractions
Using the table below, we can see Abu'l Hasan Ahmad’s idea. Instead of writing or
1
saying 2, we write 0.5 and say “five tenths.” Instead of
7
1,000
, we write 0.007 and we say
1/10
1/100
1/1,000
1/10,000
1/100,000
1/1,000,000
Hundredths
Thousandths
Ten-thousandths
Hundred-thousandths
Millionths
1/1
Ones
Tenths
10
Tens
1,000
Thousands
,
100
10,000
Ten thousands
,
Hundreds
100,000
Hundred thousands
1,000,000
“seven thousandths.”
Millions
286
.
RULE: Read and say decimal fractions in English, using these three steps in
order:
1. If there is an integer before the decimal point, name the integer part. Next
read “and” for the decimal point. Otherwise skip this step.
2. For the decimal fraction, ignore zeros and say the number after the decimal
as if it were a second integer.
3. Say the decimal place value for the last digit given after the decimal point.
Reading and writing of decimals accurately is not just about money. In medicine,
accurate decimals keep patients alive. Accurate decimals keep your car running, keep natural
gas and propane in the pipes, and keep all of your electronics working.
UNIT 4: Prealgebra in a Technical World
Example 1: A typical hourly dosage of morphine depends upon the patient’s age. For each of
the dosages below, write how to say the decimal fraction using words.
a. A 16-year old would receive 0.006 grams per hour.
b. A 40-year old would receive 0.0025 grams per hour.
c. A 25-year old would receive 0.003125 grams per hour.
Think it through: Follow the steps in order for each.
ANSWER: a. 0.006 is read, “six thousandths.”
b. 0.0025 is read, “twenty-five ten-thousandths.”
c. 0.003125 is read “ three thousand one hundred twenty-five millionths.”
Even when lives are not at stake, decimals are used in measurement in many fields.
Example 2: Write the underlined decimals in English.
a. The new Ford Scorpion, sold now in the United Kingdom, has a cylinder core diameter of
93.530 mm.
b. The total property tax rate for Williams, Oregon, property owners was $7.6037 per
thousand dollars of assessed property value in 2007.
c. A certain extension cord has 1.277 ohms of resistance.
Think it through: Follow the steps in order for each.
ANSWER:
a. 93.530 is read, “ninety-three and five hundred thirty thousandths.”
b. 7.6037 is read, “seven and six thousand thirty-seven ten-thousandths.”
c. 1.277 is read, “one and two hundred seventy-seven thousandths.”
 Check Point 1
Write the underlined decimals in English.
a. Applegate residents were assessed $0.5128 per thousand dollars of assessed property value
in 2007 to support Rogue Community College.
__________________________________________________________________________
b. One meter is 39.370079 inches.
__________________________________________________________________________
287
288
SECTION 4.1 Decimals and Scientific Notation
c. Digoxin delivers digitalis to the heart. A tablet contains 0.125 mg of digoxin.
__________________________________________________________________________
Add and Subtract Signed Decimals
From the very beginning of recorded civilization, people have paid taxes and tithes for
the protective, organizational, and civilizing services that governments and religious orders
provide. The words decimal and tithe both have their roots in paying money in tenths. Decimal
and dime have the same root word, while tithe and ten share the same root word.
In our democratic republic we have the right to vote, or have our representatives vote,
on our own taxation. The property taxes we pay are actually a sum of several assessed taxes,
and these amounts are reported using decimals.
Example 3: Chesney lives in Grants Pass, and her property taxes were assessed to pay for the
services listed in the table on the right. What was the
total property tax rate in Grants Pass for 2007?
Think it through: Estimate by rounding to the
nearest whole number:
1+6+6+0+0+0+1=
$14 per 1,000 of assessed value.
The table has already lined up the
place values correctly, and we add
2007 City of Grants Pass Tax Rate per
$1,000 of Assessed Property Value
Josephine County
0.7458
City of Grants Pass
5.8339
School District #7
5.9030
Rogue Community College
0.4757
Southern Oregon ESD
0.3269
4H/Extension Service
0.0426
GP Redevelopment Agency
0.9332
as if we were adding whole
numbers. In the end, we keep the decimal in exactly the same place because
the place values are still the same.
ANSWER: The total property tax rate in Grants Pass for 2007 was $14.2611 per $1000
of assessed value.
UNIT 4: Prealgebra in a Technical World
 Check Point 2
Fredrico lives in the Kerby Water District in Josephine County. His property taxes are assessed
to pay for certain services. What was the total tax
rate for people living in the Kerby Water District in
Josephine County in 2007?
2012 Kerby Water District Tax Rate per
$1,000 of Assessed Property Value 1
Josephine County
0.7532
Three Rivers School District
4.2460
Rogue Community College
0.5128
add and subtract signed decimals, we “line up the
Southern Oregon ESD
0.3524
decimal points” because this keeps the place
4H/Extension Service
0.0459
Most applications of decimals do not
involve signed numbers, but some do. When we
values in the correct order. We subtract, or we change subtracting to adding the opposite, when
this makes sense.
Example 4: Sheryl kept a running list of her income and expenses.
balance
1,845.78
rent
- 450.00
positives and negatives separately. Finally she
bookstore
- 291.32
subtracts these sums.
pay check
880.32
groceries
- 127.02
tuition & fees
-1342.43
What is the balance in her account?
Think it through: Sheryl makes an estimate and then adds the
Sheryl’s estimated income is:
(1800 + 900) = 2700.
Her estimated expenses are:
(500 + 300 + 100 + 1300) = 2200.
Her estimated balance is ≈$500.
Sheryl adds her income and her expenses.
She subtracts the sums.
ANSWER: The balance in Sheryl’s account is $515.33.
1
http://www.co.josephine.or.us/Files/Statement%20of%20Taxes%209x12.pdf
1845.78
+880.32
2726.10
450
291.32
127.02
+1342.43
2210.77
2726.10 − 2210.77 =
$515.33
289
290
SECTION 4.1 Decimals and Scientific Notation
 Check Point 3
After a car accident, a 16-year-old and a 40-year-old are given morphine for pain control. The
dosage is age-dependent. The 16-year-old receives 0.006 grams per hour and the 40-year-old
receives 0.0025 grams per hour. How much more morphine does the 16-year-old receive each
hour than the 40-year-old when they are both in pain?
No matter what the application, when we calculate sums and differences, we estimate
first. This check step is an efficient way to check our work.
Example 5: Estimate and then calculate these sums and differences.
a. −25.4 − 983.2
Think it through: Change subtraction to adding the opposite. The answer is negative.
ANSWER: Estimate -20 + -980 ≈-1000. Exact −𝟐𝟓. 𝟒 + (−𝟗𝟖𝟑. 𝟐) = −𝟏𝟎𝟎𝟖. 𝟔
b. −3.08 + 0.905 − 3.42
Think it through: Round each to the nearest integer and add: −𝟑 + 𝟏 + (−𝟑) = −𝟓.
ANSWER: Estimate ≈ −𝟓. Exact: Use the commutative property here:
𝟎. 𝟗𝟎𝟓 + (−𝟑. 𝟎𝟖 + −𝟑. 𝟒𝟐) = −𝟓. 𝟓𝟗𝟓
c. −0.47 + 3.95 + (−0.112)
Think it through: Round and add the negatives first:
−0.47 + (−0.112) ≈ −0.5 + −0.1 ≈ −0.6
Round the positive decimal and add to the sum of the negatives:
−0.6 = 3.4
ANSWER: Estimate ≈ 𝟑. 𝟒. Exact: +𝟑. 𝟗𝟓 + (−𝟎. 𝟒𝟕 + −𝟎. 𝟏𝟏𝟐) = 𝟑. 𝟑𝟔𝟖
4+
UNIT 4: Prealgebra in a Technical World
 Check Point 4
Estimate and then calculate these sums and differences.
a. −3.82 + 55.01 ______________________________________________________________
b. −9.96 + 8.21 − 100.79 _______________________________________________________
c. 506.11 − 62.5 + 0.859 _______________________________________________________
Positive Powers of 10 and Scientific Notation
In the first chapter of this book, we studied just how to take advantage of our decimal
system to make estimates. We rounded to the leading digit and then performed calculations.
For any number, the place value of its leading digit is the most important part in determining
,
,
103 102
101
,
Ones
104
Thousands
106 105
Tens
107
Hundreds
108
Ten thousands
109
Hundred thousands
1010
Millions
Ten billions
1011
Ten millions
Hundred billions
1012
Hundred millions
Trillions
1013
Billions
Ten trillions
the value of the number, and all place values can be written as powers of 10.
100
,
Scientists invented a method of writing numbers that makes reading the leading digit
and its place value easier than reading its decimal. Scientific notation is this method. Using
scientific notation we write numbers using decimals
between 1 and 10 multiplied by a power of ten. Powers of
Decimal
Scientific
Notation
First Second
Part
Part
23,000,000
2.3 x 107
ten determine every place in our place value system, so
translating between decimals and scientific notation is a
mental calculation.
Scientific notation has two parts. In the first part,
we write the leading digit in the ones place. The
34,050,000,000
3.405 x 1010
25.89
2.589 x 101
420,000
4.2 x 105
291
292
SECTION 4.1 Decimals and Scientific Notation
remaining digits of the number follow in order, but below the decimal point. In the second part
we multiply by the power of ten of the leading digit. The power of ten of the leading digit is
called the magnitude of a number written in scientific notation.
RULE: To write a number using scientific notation,
1. Write the original leading digit in the ones place.
2. Write remaining digits, in the original order, after the decimal point.
3. Multiply this new decimal by the power of ten of the original leading digit
and use the symbol “x” for multiply.
Example 6: Write using scientific notation.
a. The average distance from the Earth to the sun: 92,900,000 miles.
b. The average distance from the Earth to the moon: 239,000 miles
c. In August, 20122 the approximate population of the world was 7,063,000,000.
d. The approximate number of barrels of oil left in the world3 was 1,266,000,000,000.
Think it through: For each, write the leading digit, a decimal point and the remaining nonzero digits.
Then use the x for multiply and write the place value of the original leading
digit using the correct power of ten.
ANSWER: a. 92,900,000 = 9.29 x 107 miles
b. 239,000 = 2.39 x 105 miles
c. 7,063,000,000 = 7.063 x 109 people
d. 1,266,000,000,000 = 1.266 x 1012 barrels
RULE: To write a number given in scientific notation as its equivalent decimal,
multiply the two factors in the scientific notation.
2
3
http://www.worldometers.info/ accessed 8/30/12.
http://www.worldometers.info/ accessed 8/30/12.
UNIT 4: Prealgebra in a Technical World
Example 7: Rewrite each number in standard notation.
a. In one single day, over 2.5 x 107 dollars are spent on weight loss programs in the U.S.4
b. In one day over 2.86 x 1011 kilowatt hours (kWh) of solar energy strike the Earth.5
c. About 1.8 x 1011 e-mails will be sent worldwide today.6
d. Over 5.07 x 105 television sets will be sold worldwide today.7
Think it through: Write the decimal part of the number that is in scientific notation.
Starting with the digit in the tenths place as “1,” count the number of powers
of ten shown in the exponent. Use zeros to hold each place that does not
have another value.
ANSWER: a. 2.5 x 107
= 25,000,000 dollars.
b. 2.86 x 1011 = 286,000,000,000 kWh of solar energy.
c. 1.8 x 1011 = 180,000,000,000 e-mails today.
d. 5.07 x 105 = 507,000 televisions.
 Check Point 5
a. Write the 2010population of Jackson County, 203,000 people, in scientific notation.8
b. Write the 2010 population of Josephine County, 82,000 people, in scientific notation.9
c. Write the land area of Josephine County, 1.049 x 106 acres, in standard notation.10
d. Write the land area of Jackson County, 2.785 x 103 square miles, in standard notation.11
4 http://www.worldometers.info/
accessed 8/18/09.
accessed 8/18/09.
6 http://www.worldometers.info/ accessed 8/18/09.
7 http://www.worldometers.info/ accessed 8/18/09.
8 http://quickfacts.census.gov/qfd/states/41/41029.html
9 http://quickfacts.census.gov/qfd/states/41/41029.html
10 http://quickfacts.census.gov/qfd/states/41/41029.html
11 http://quickfacts.census.gov/qfd/states/41/41029.html
5 http://www.worldometers.info/
293
294
SECTION 4.1 Decimals and Scientific Notation
Scientific Notation for Decimal Fractions
Besides the large numbers in our technical world, developments in biology, chemistry,
and electronics demand that we take measurements so small that their decimal fractions are
hard to comprehend.
For example, the length of the Ebola virus is 0.000000002 meters12. In the press it
would be reported as “20 nanometers.” In most of the world, this decimal fraction is written
0.000 000 002 meters, using spaces to make it easier to read. In technical fields, we write this
number using scientific notation so we can both read it easily and calculate with it quickly.
To use scientific notation for decimal fractions, we must first understand just what
powers of ten to use for decimal fractions.
10−2
One Less Power of Ten
1 1
∙
10 10
1 1 1
∙
∙
10 10 10
1 1 1 1
∙
∙
∙
10 10 10 10
10−3


10−1


12
1
10


0.0001
100


0.001
10


0.01
10


0.1
1=


1


Ten Times Smaller Decimal
Decimal Tens and Tenths Powers of Ten
10,000 10 ∙ 10 ∙ 10 ∙ 10
104
1,000
10 ∙ 10 ∙ 10
103
100
10 ∙ 10
102
10
10
101
10−4
“The Ebola virus is named after the Ebola River in Congo on the African continent. Largely limited to Africa at this time, the Ebola virus
attacks humans and other animals causing internal bleeding and often death.” http://mayoclinic.com/health/ebola-virus/DS00996 accessed
8/19/2009.”
UNIT 4: Prealgebra in a Technical World
In the table, we start with the pattern for larger place values and powers of ten. As each
decimal decreases by a place value, the power of ten decreases by one. Extending our
exponent pattern to 1, we find that the 1 must equal 10 0.
As we continue to decrease the decimal place value, we arrive at decimal fractions. For
each decimal fraction, we have a power of ten equal to that decimal fraction (see Power of Ten
column). For decimal fractions, all of the powers of ten are negative.
Also notice that the power of ten for the thousands place (103 ) has the same absolute
value as the power of ten for the thousandths place (10−3 ), that is |3| = |-3|. This makes
,
,
10-4 10-5
Millionths
10-3
Hundred
thousandths
Ten
thousandths
10-1 10-2
Thousandths
100
Hundredths
101
Tenths
103 102
Ones
Thousands
Tens
104
Hundreds
106 105
Ten
thousands
Millions
Hundred
thousands
learning powers of ten easier than we might expect.
10-6
.
With the practice that comes in applied business, technical, and science classes, many of
us learn to read these powers of ten more easily than their word names! For now, we
concentrate on reading and writing numbers in scientific notation.
The rules for scientific notation stay the same for decimal fractions. Only the place
values are negative powers of ten.
Example 8: Write each value using scientific notation.
a. The amount of caffeine in a cup of hot chocolate or cocoa is 0.005 grams.13
b. The amount of caffeine in a brewed cup of coffee is 0.135 grams.14
c. The length of the Ebola virus is 0.000 000 002 meters.
13
http://web.archive.org/web/20070614144016/http://www.cspinet.org/nah/caffeine/caffeine_content.htm
14
http://web.archive.org/web/20070614144016/http://www.cspinet.org/nah/caffeine/caffeine_content.htm
295
296
SECTION 4.1 Decimals and Scientific Notation
Think it through: 0.1 = 10-1. Powers of 10 decrease by 1 for each place value, so we count
backwards, using negative integers, to the place value of the leading digit.
We do not have to count as far as we did in Part a.
ANSWER: a. 0.005 = 5 x 10-3 grams of caffeine
b. 0.135 = 1.35 x 10-1 grams of caffeine (The leading digit is at 10-1, not 10-3.)
c. 0.000 000 002 = 2 x 10-9 meters
Example 9: Write each value in standard notation.
a. Airborne pollutants below 2.5 x 10-9 meters in diameter are called PM2.5 (PM stands for
particulate matter).
b. The width of your hair is based on your genetics. The widest human hair is about 1.8 x 10-7
meters wide.
Think it through: To translate from scientific notation to standard notation, we write the
leading digit followed by the remaining digits first.
Next we count backwards the number of remaining place values and hold
each of these with a 0.
For Part a, write the “25” first. Then, because the 2 is in the -9 place, write
zeros and the decimal point to the left.
ANSWER: a. 2.5 x 10-9 = 0.000 000 002 5 meters
b. 1.8 x 10-7 = 0.000 000 18 meters
UNIT 4: Prealgebra in a Technical World
 Check Point 6
a. Write in scientific notation. The time a home computer will take to complete one software
instruction is about 0.000 000 000 003 seconds.15
b. Write in scientific notation. The shutter speed of the camera is set to 0.001 seconds.
c. Write in scientific notation. Every 0.0625 seconds a Twinkie™ is “born.”16
d. Write using decimal notation. Nanotechnology is a new field that produces microscopic
components for electronic devices. For instance a carbon nanotube is 2 x 10-12 meters wide.
Carbon nanotubes move electricity much faster than anything we use today.17
Cell phone signals reach satellites in nanoseconds (10-9 seconds). Computer drive
storage is measured in terabytes (1012 bytes). In this century, sciences and technologies
depend on very small and very large numbers. In our technological world it is essential to read,
write and calculate with fractions, with decimals and with scientific notation.
15
http://science.howstuffworks.com/time1.htm
16
http://recipes.howstuffworks.com/10-quirky-facts-about-mass-produced-food1.htm
17
http://science.howstuffworks.com/nanotechnology.htm
297
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SECTION 4.1 Decimals and Scientific Notation
UNIT 4: Prealgebra in a Technical World
4.1 Exercise Set
Name _______________________________
Skills
Write the decimals in English.
1. 0.32 ______________________________________________________________________
2. 4.069______________________________________________________________________
3. 0.2________________________________________________________________________
4. 182.3______________________________________________________________________
5. 508.00684__________________________________________________________________
6. 250.856____________________________________________________________________
7. 0.0381_____________________________________________________________________
8. 0.00000907_________________________________________________________________
Write the following using decimal notation.
9. forty-eight hundredths
_____________________________
10. five hundred and seventy-nine thousandths
_____________________________
11. three and forty-five ten thousandths
_____________________________
12. four hundred and twenty-five ten thousandths
_____________________________
13. Six thousand forty-one and two hundredths
_____________________________
14. Three thousand five hundred twenty and forty-five
ten thousandths
_____________________________
15. One hundred ninety-nine thousand six hundred
three and seventeen hundredths
_____________________________
16. Seven thousand and four hundred five
ten thousandths
_____________________________
299
300
SECTION 4.1 Decimals and Scientific Notation
Write the underlined numbers in English.
17.
One centimeter is approximately 0.3937 inches.
_________________________________________________________________________
18.
A child's dosage for Zyrtec is 0.0025 mg per day.
_________________________________________________________________________
19.
A weight of one carat is 0.2 grams, so 1/3 carat is approximately 0.067 grams.
_________________________________________________________________________
20.
The heaviest rainfall ever recorded in Grants Pass was 5.27 inches in one day.
_________________________________________________________________________
Add or subtract as indicated without using a calculator. Remember to determine the sign first!
21.
5.04 + 6.01
22. 6.01 + (−4.02)
23.
−7.3 − 7.7
24. 0.44 + 0.077
25.
0.073 − 8.01
26. −53.1 − 1.07
27.
−12.03 + 60.84
28. −95.39 − (−4.33)
Write using decimal notation.
29. 102
_________________________
31. 9 x 102
______________________
33. −9.349 x 102
35. 6.356 x 105
________________
__________________
30. 10𝟗
__________________________
32. 5.331 x 102
___________________
34. 4.84 x 103
_____________________
36. −2.039 x 104
__________________
Write in scientific notation. To write signed numbers in scientific notation, keep the sign.
37. 539
________________________
39. −3,415,000
41. 367,000
_________________
____________________
38. 393.2
40. −52.52
________________________
______________________
42. 59,700,000
___________________
43.
At this writing, the United Nations gives the total population of the Western Hemisphere,
(the Americas) as 858,000,000. Write this number in scientific notation.
44.
At this writing, the world population is 6.79 billion. Using your answer in question 43, find
the population of the Eastern Hemisphere. Write this population in scientific notation.
UNIT 4: Prealgebra in a Technical World
Write in decimal notation. When writing signed numbers in scientific notation, keep the sign.
45.
10−1
47.
6 x 10−1
49.
−9.9 x 10−2
51.
7.368 x 10−10
_________________________
______________________
___________________
__________________
46. 10−4
___________________________
48. 3.962 x 10−3
____________________
50. 7.36 x 10−4
_____________________
52. −4.002 x 10−2
__________________
Write in scientific notation. To write signed numbers in scientific notation, keep the sign.
53.
0.334
55.
−0.00170
_____________________
57.
0.0000286
____________________
_________________________
54. 0.08
__________________________
56. −0.00047
_____________________
58. 0.000000870
__________________
Applications UPS
59.
The radius of a hydrogen atom is 2.5 x 10−11 meters. Write this in decimal notation.
60.
The speed of light is 300,000,000 meters per second. Write this in scientific notation.
61.
A plant cell is 12.76 x 10−6 meters wide. Express this in decimal notation.
62.
The time it takes light to travel one meter (about the distance from your nose to the
fingertips of your outstretched arm) is approximately 0.000000003 seconds. Write this in
scientific notation.
63.
There are 8,360,000,000,000,000,000,000,000 molecules in a cup of tea.
a. Write this in scientific notation.
b. Write this number in English. (Use any source to find the name of this number!)
c. Which of the notations (decimal, scientific, or English) is easier to understand? Why?
301
302
SECTION 4.1 Decimals and Scientific Notation
64.
The following table gives the medalists' times for two swimming events at the 2012 Summer
Olympics. Times are in minutes, seconds. For example: 3:42.15 would be three minutes,
forty-two and fifteen one-hundredth seconds. Fill in the charts for the difference of medalists'
times. (Hint: 1 minute = 60 seconds.)
Men's 100 m
Backstroke
Medal
Time
Gold
52:16
Silver
Bronze
Difference from Gold
Medal time:
52: 16 − 52.16 = 0 sec
Women's 200 m
Breaststroke
Medal
Time
Gold
2:19:59
52:92
Silver
2:20:72
52:97
Bronze
2:20:92
Difference from Gold Medal
time:
2: 19: 59 − 2: 19: 59 = 0
65.
The period of a 100 MHz FM radio wave is 1 x 108 . Write this in decimal notation.
66.
The following table gives the property tax breakdown for San Francisco. Find the total
property tax rate in San Francisco for 2011 and complete the table.
2011 City and County of San Francisco Tax Rates for
$100 of Assessed Property Value
SF Community College
SF Unified School District
City and County of SF
Library Preservation Fund
SF Children's Fund
Open Space Acquisition
Bond Interest and Redemption Fund
Superintendent of Schools
General Obligation Bonds
Bay Area Air Quality District
0.0740
0.3237
0.565900123
0.0250000
0.03
0.025
0.114700
0.0010000
0.0063
0.00208539
Total Tax Assessment (per $100)
Review and Extend
The leading digit of a decimal is the first non-zero digit. Round each to the place value given.
0.361
67. up to the leading digit
68. down to the leading digit
69. to the nearest tenth
70. up to the hundredth
0.0849
0.928
0.0035
$2.384