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Lesson 36 ! page 1
Lesson 36
Scientific Notation
There’s more than one way to write any number. Depending on the circumstances, we might write the quantity “three” as III
(Roman numerals), $3.00, or 3.0 cm, or 3 burritos, or “a few” friends. The fractions 2/3, 4/6, 6/9, etc are all equal quantities
expressed differently, and we could also write a repeating decimal 0.666… to represent the same amount. Scientific notation
is yet another way to write numbers, especially helpful when numbers are very large.
You’ve probably seen scientific notation in a science textbook. Your calculator uses scientific notation when the number of
digits in an answer exceeds the number of digits in the display.
Example: The radius of the earth is about 4000 miles. What is the approximate volume of the earth?
V=
(
4 3 4
!r = ! 4000
3
3
)
3
When you enter this in your calculator, you will see something like:
You may wonder why a number you expect to be very large has a
decimal point after the ones place. And what is over in the right-hand corner? (Your calculator may only display the
exponent 11 instead of the entire expression x 1011.) Because the answer is so large, the calculator has switched
automatically to scientific notation. The answer 2.68082572 x 1011 is written in scientific notation.
A number written in scientific notation has two parts, a decimal part multiplied by a power of ten:
trillions
100
10
billions
1
100
10
millions
thousands
11
2.68082572
x1 10100
1
100
10
10
1
100
10
1
0
4,
5
2
5
In this section, you will learn
• to convert numbers written
notation to ordinary
place valuethousands
notation;
trillionsin scientificbillions
millions
100
10
1
100
10
1
100
10
1
100
10
1
• to convert ordinary numbers to scientific notation;
• some contexts in which scientific notation is helpful;
• how we use an adaptation of scientific notation in everyday speech.
100
10
1
trillions
billions
millions
thousands
Powers of Ten
100
10
1
100
10
1
100
10
1
100
10
1
Tens are easy to multiply, because our number system is based on 10s.
100
10
1
2
1
8
The decimal part is a number
2 between
3, 0one and
7 ten.
4,
1
8
1,
3,
3
4
8
5,
So 102 = (10 )(10 )= 100, 103 = (10 )(10 )(10 ) = 1000, 104 = 10,000 and so forth.
You can notice quickly that the exponent tells you how many zeros to write after the 1.
In fact, the powers of 10 are just the place value groups: 103 is one thousand, 104 is ten thousand, 106 is one million, etc.
109 108 107 106 105 104 103 102 101
trillions
100
10
billions
1
100
10
millions
10
thousands
1
100
1
100
10
of10ten. 1
100
10
4
0 name
9, in words.
0 0
1
100
10
1
Example: Write the numbers as indicated.
trillions
a. Write the number
100 trillion
10
ten
as 1a
billions
100
power
millions
As the place values increase, we increase the exponent of
ten by 1. In the ten trillions place, the exponent will be 13.
Ten trillion = 1013
© 2010 Cheryl Wilcox
1
Write
thousands
10
the100
number
100
10 notation
1
10121 in standard
and write its
0,
0
0
0
The exponent 12 is the number of zeros after the digit 1.
1012 = 1,000,000,000,000 = one trillion
Free Pre-Algebra
Lesson 36 ! page 2
Check the multiplication table below using your calculator. Then read the observations to the right.
! 10
! 102
1.0
10
100
1,000
1,000,000
3.0
30
300
3,000
3,000,000
43.0
430
0.2
02
020
0200
3.2
32
320
3,200
! 103
! 106
This row is just the powers of ten written in everyday place value
notation.
Multiplying by 3 replaces the 1 in the power of ten by a 3.
Multiplying by 43 replaces the 1 in the power of ten by the 3 in the
ones place of 43. The 4 remains in front of the 3.
The ghostly 0 that was in the ones place replaces the digit 1 in
0,200,000
the power of 10. The 2 that was in the tenths keeps its relative
position to the 0.
The 3 in the ones place replaces the digit 1 in the power of 10.
3,200,000
The 2 in the tenths place keeps its relative position to the 3.
4300 43,000 43,000,000
Example: Multiply 67.08 x 1013 without using a calculator (a calculator cannot display all the digits).
1 Write out the power of ten (1013) in
everyday place value notation. The
exponent 13 is the number of zeros
after the digit 1 in the place value
expression.
2 Line up the ones digit of the
number you are multiplying with the
digit 1 in the power of ten. Keep other
digits in their same relative positions.
3 Drop the decimal point and fill in
any missing place values with zeros,
inserting commas to match the power
of ten.
10,000,000,000,000
670,800,000,000,000
1 0,000,000,000,000
67.08
1013 = 10,000,000,000,000
1013 is ten trillion.
67.08 x 1013 = 670,800,000,000,000
The multiplication 67.08 x 1013 is not a number written in scientific notation, because the decimal number 67.08 is not
between 1 and 10. If you enter this multiplication on your calculator, you will see the result 6.708 x 1014, which is correct
scientific notation for the result 670,800,000,000,000.
Scientific Notation to Standard Notation
Example: Write the approximate volume of the earth, 2.68082572 x 1011 cubic miles, in everyday place
value notation.
1. Write 1011 out in place value notation.
2. Align the 2 in the ones place with the digit 1.
3. Drop the decimal point and fill in any empty places with 0s.
1 00000000000
2.68082572
2 68082572000
The approximate volume of the earth is 268,082,572,000 cubic miles.
The number is automatically rounded because the calculator can’t display all the digits. However, since the radius of the
earth as given originally was rounded to one significant figure, we really ought to write the volume approximation with one
significant figure as well: about 300 billion cubic miles. (You are not expected to calculate significant figures in this section.)
You can see that converting scientific notation to place value notation is just a multiplication by the power of 10.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 36 ! page 3
Standard Notation to Scientific Notation
To change a number written in standard notation to scientific notation, follow the steps below:
Example: Write the number 45,900,000,000,000,000,000 in scientific notation.
1 Write out the number without
commas and put a decimal point after
the first digit.
2 Count the number of digits after the
(new) decimal point. This will be the
exponent for 10.
4.5900000000000000000
4.5900000000000000000
3 Drop any zeros to the right of the
last non-zero digit, and write the
decimal times the power of 10.
4.59 x 1019
19 digits after the decimal point.
x 1019
This procedure always puts the leading digit of any number in the ones place of the decimal part of the scientific notation,
resulting in a decimal part that is between 1 and 10.
Example: Write the number 96 billion in scientific notation.
Write 96 billion in standard place value notation.
1. Write a decimal point after the first digit and drop the commas.
2. Count the digits after the decimal point and
use that number for the exponent of 10.
3. Drop extra zeros from the right of the number and write
as a multiplication with the power of ten.
96,000,000,000
9.6000000000
1010
9.6 x 1010
Everyday Speech
Sometimes numbers are written with a decimal point and a place value name. You see this in news articles and financial
statements. For example, “That penthouse cost $3.5 million!” Interpret this as 3.5 x 1,000,000 and you see that it means
$3,500,000 (three million, five hundred thousand dollars). Unlike true scientific notation, you are not limited to putting the
decimal point after the first digit. Instead, use the largest place value group name, and use a decimal point for digits in the
group below.
Example: Convert the underlined numbers as requested.
In 2009, a total of 33.3 million pounds of toxic chemicals
were released into the air in the state of New York.
The amount of financial aid increased by over
$1,500,000.
Write in standard notation:
Write using the largest place value name.
33,300,000
© 2010 Cheryl Wilcox
$1.5 million
Free Pre-Algebra
Lesson 36 ! page 4
Why Use Scientific Notation?
There are about 25 trillion red blood cells in our body at one time. About 12,500,000,000,000 locusts appeared in Nebraska
in 1875. There are about 1018 molecules in a snowflake. (The Sizeasaurus, Strauss, 1995) When working with large
numbers, it helps to have a condensed notation to express their size in a way that is easily comparable.
25 trillion red blood cells
2.5 x 1013
12,500,000,000,000 locusts 1.25 x 1013
1018 molecules in a snowflake
1 x 1018
By writing all the numbers in scientific notation, it is much easier to see that the number of red blood cells in your body is
close to the same as the the number of locusts that appeared in Nebraska, (Is that significant? creepy? meaningless?)
whereas the number of molecules in a snowflake is much greater.
Comparing Orders of Magnitude
An increase of one place value, called an order of magnitude, is the same as increasing the exponent by 1, say from 1013
to 1014. It makes the number 10 times greater. Since the exponent 18 is five more than the exponent 13, the number of
molecules is “5 orders of magnitude” greater than the number of locusts. That means it is 105 = 100,000 times greater.
Comparing numbers by order of magnitude is a kind of ratio comparison because we are telling about how many times
greater one number is than another.
Example: Write the numbers in scientific notation and compare the orders of magnitude.
The volume of the earth is about 1,100,000,000,000,000,000,000 m3 and the volume of the sun is about
1,400,000,000,000,000,000,000,000,000 m3.
volume of the earth, cubic meters 1.1 x 1021
volume of the sun, cubic meters 1.4 x 1027
The volume of the sun is 6 orders of magnitude greater than that of the earth.
The sun is 106 = 1,000,000 = one million times larger than the earth.
Another reason scientists use scientific notation is that in many cases it indicates the accuracy of the measurement or
calculation. Conventionally scientists use significant figures in the decimal part and the power of ten to indicate the
magnitude.
Using scientific notation, you could indicate that the mountain measured 29,000 feet exactly by including the zeros in the
decimal part: 2.9000 x 104. (By the way, in 1999 geographers using GPS technology measured the height of Mt. Everest to
be 29,035 feet.)
!
© 2010 Cheryl Wilcox
Free Pre-Algebra
55
00
11
100
100
trillions
trillions
100
100
10
10
00
00
00
11
100
100
billions
billions
10
10
00
00
55
11
100
100
millions
millions
3, 44
3,
Worksheet
0
0
0
0
100
100
10
10
11
8 5, 2
1
8
thousands
thousands
10
10
Lesson 36: Scientific Notation
3
10
10
11
Lesson 36 ! page 5
Name __________________________________________
1. Write the power of 10 above each place value. The first few are done for you.
100
100
trillions
trillions
10
10
1099 10
1088 10
1077 10
1066 10
1055 10
1044 10
1033 1022 1011
10
11
100
100
billions
billions
10
10
11
1. Write the number one hundred trillion in standard
notation and as a power of ten.
trillions
billions
trillions
billions
100
100
10
10
11
100
100
10
10
11
3. Change scientific notation to standard notation.
100
100
millions
millions
10
10
11
thousands
thousands
100
100
10
10
11
100
100
10
10
11
2. Write the number 1011 in standard notation and in words.
100
100
millions
millions
44
10
10
11
thousands
thousands
100
100
9, 00
00 9,
10
10
11
100
100
0, 0
0
00 0,
10
10
00
11
00
4. Change standard notation to scientific notation.
a. 9.6 x 1020
a. 87,909,000,000,000,000
b. 4.006621 x 1016
b. 807,000,000,000,000
c. 7.15 x 108
c. 1,709,110,000,000,000,000,000
5. Write the underlined number in standard notation.
6. Write a place value name with a decimal point to convey
the underlined numeric information.
The real estate listings show the asking price is $1.8 million.
Total costs of $40,500,000,000 for clean up.
The company showed profits of $46.8 billion in 2010.
7. Write the numbers in scientific notation.
About 4,900,000 barrels of oil.
8. Compare the orders of magnitude of the distances in #7.
At that time, the moon was about 399,000 km from earth,
and Mars was 356 million km from earth.
About how many times greater is the distance to Mars than
the distance to the moon?
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 36 ! page 6
Lesson 36: Scientific Notation
Homework 36A
Name __________________________________________
1. Use the formula C = (F – 32) / 1.8 to find the Celsius
temperature equivalent to 98.6ºF.
2. Find the height of an object shot into the air at 180 ft/sec
when t = 11.25 seconds using the equation
h = !16t 2 + 180t
What is special about the time t = 11.25?
What is special about 98.6ºF?
3. Mark the number line with the following values: 0.5, 0.05, 0.49, 0.51.
4. Fill in the missing values in the chart.
Fraction in
Lowest Terms
Sixteenths
5. Use the chart and the inch ruler to answer the questions.
Decimal
Equivalent
1/16
a. Between which two decimals in the chart in #4 would 0.2
be located?
2/16
3/16
b. Between which two fractions on the ruler would 0.2 inches
be located?
4/16
5/16
c. Between which two fractions on the ruler would 0.3 inches
be located?
6/16
7/16
8/16
Fraction in
Lowest Terms
Sixteenths
1/16
1/16
0.0625
2/16
0.1250
3/16
0.1875
4/16
0.2500
1/8
© 2010 Cheryl Wilcox
3/16
1/4
Decimal
Equivalent
Free Pre-Algebra
Lesson 36 ! page 7
2010 Giants Season Statistics
Player
At Bats
Hits
Huff
569
165
Fontenot
240
68
7. What is the ratio of Huff’s At Bats to Fontenot’s?
Write a sentence comparing the number of At Bats.
6. Write a sentence to compare the data.
8. If you divide Hits by At Bats, then round to three decimal
places, you have calculated the rate called the batting
average. Compute the batting average for each player.
9. Write the numbers in scientific notation.
a. 888,800,000,000
b. 5,009,000,000,000,000,000
Write a sentence comparing the player’s batting averages.
c. 11,780,000,000
10. Write the numbers in standard notation.
11. a. Re-write with a place value name.
a. 9.05 x 108
Human stone age cultures arose about 2,500,000 years ago.
b. 7.992 x 1012
b. Translate the written name to standard notation.
c. 5.0 x 107
Evidence from radiometric dating indicates that the Earth is
about 4.570 billion years old.
12. Write the numbers from #11 in scientific notation:
13. Simplify, then solve the equation.
a. Human stone age culture:
0.04 10,000 ! x + 0.07x = 475
b. Age of Earth:
c. What is the difference in order of magnitude of the two
numbers?
d. How many times greater is the age of the earth than the
time since stone age culture began?
© 2010 Cheryl Wilcox
(
)
Free Pre-Algebra
Lesson 36 ! page 8
Lesson 36: Scientific Notation
Homework 36A Answers
1. Use the formula C = (F – 32) / 1.8 to find the Celsius
temperature equivalent to 98.6ºF.
(
2. Find the height of an object shot into the air at 180 ft/sec
when t = 11.25 seconds using the equation
)
Fraction in C = 98.6 ! 32 Decimal
/ 1.8
Sixteenths
Lowest Terms
Equivalent
= 37ºC
h = !16t 2 + 180t
(
)
2
(
h = !16 11.25 + 180 11.25
1/16
2/16
What is special about 98.6ºF?
3/16
It is normal human body temperature.
)
= 0 feet
What is special about the time t = 11.25?
It is the exact time the object hits the ground.
4/16
3. Mark the number line with the following values: 0.5, 0.05, 0.49, 0.51.
5/16
6/16
7/16
8/16
4. Fill in the missing values in the chart.
5. Use the chart and the inch ruler to answer the questions.
Fraction in
Lowest Terms
Sixteenths
1/16
1/16
0.0625
1/8
2/16
0.1250
3/16
3/16
0.1875
0.1875 < 0.2 < 0.25
1/4
4/16
0.2500
b. Between which two fractions on the ruler would 0.2 inches
be located?
5/16
5/16
0.3125
Between 3/16 and 1/4.
3/8
6/16
0.3750
c. Between which two fractions on the ruler would 0.3 inches
be located?
7/16
7/16
0.4375
0.25 < 0.3 < 0.3125
1/2
8/16
0.5000
Between 1/4 and 5/16
© 2010 Cheryl Wilcox
Decimal
Equivalent
a. Between which two decimals in the chart in #4 would 0.2
be located?
Free Pre-Algebra
Lesson 36 ! page 9
2010 Giants Season Statistics
Player
At Bats
Hits
Huff
569
165
Fontenot
240
68
7. What is the ratio of Huff’s At Bats to Fontenot’s?
569 / 240 = 2.37
Write a sentence comparing the number of At Bats.
Huff had more than twice as
many at bats as Fontenot.
6. Write a sentence to compare the data.
Huff had more at bats and more hits than
teammate Fontenot.
8. If you divide Hits by At Bats, then round to three decimal
places, you have calculated the rate called the batting
average. Compute the batting average for each player.
Huff: 165 / 569 = 0.290
Fontenot: 68 / 240 = 0.283
Write a sentence comparing the player’s batting averages.
Huff has the higher batting average at 0.290,
with Fontenot close at 0.283.
9. Write the numbers in scientific notation.
a. 888,800,000,000
8.888 x 1011
b. 5,009,000,000,000,000,000
c. 11,780,000,000
5.009 x 1018
1.178 x 1010
10. Write the numbers in standard notation.
11. a. Re-write with a place value name.
a. 9.05 x 108
Human stone age cultures arose about 2,500,000 years ago.
905,000,000
2.5 million
b. 7.992 x 1012
c. 5.0 x 107
7,992,000,000,000
50,000,000
b. Translate the written name to standard notation.
Evidence from radiometric dating indicates that the Earth is
about 4.570 billion years old.
4,570,000,000
12. Write the numbers from #11 in scientific notation:
13. Simplify, then solve the equation.
a. Human stone age culture: 2,500,000 = 2.5 x 106
0.04 10,000 ! x + 0.07x = 475
b. Age of Earth: 4.570 x
109
c. What is the difference in order of magnitude of the two
numbers? The exponents are 9 and 6, so the
difference is 9 – 6 = 3.
d. How many times greater is the age of the earth than the
time since stone age culture began? 103 = 1000 times
© 2010 Cheryl Wilcox
(
)
400 ! 0.04x + 0.07x = 475
400 + (!0.04x + 0.07x ) = 475
0.03x + 400 = 475
0.03x + 400 ! 400 = 475 ! 400
0.03x = 75
0.03x / 0.03 = 75 / 0.03
x = 2500
Free Pre-Algebra
Lesson 36 ! page 10
Lesson 36: Scientific Notation
Homework 36B
Name _________________________________________
1. Use the formula C = (F – 32) / 1.8 to find the Celsius
temperature equivalent to 212ºF.
2. Find the height of an object shot into the air at 252 ft/sec
when t = 15.75 seconds using the equation
h = !16t 2 + 252t
What is special about the time t = 15.75?
What is special about 212ºF?
3. Mark the number line with the following values: 0.8, 0.08, 0.78, 0.88.
4. Fill in the missing values in the chart.
Fraction in
Lowest Terms
Sixteenths
5. Use the chart and the inch ruler to answer the questions.
Decimal
Equivalent
9/16
a. Between which two decimals in the chart in #4 would 0.8
be located?
10/16
11/16
b. Between which two fractions on the ruler would 0.2 inches
be located?
12/16
13/16
c. Between which two fractions on the ruler would 0.6 inches
be located?
14/16
15/16
16/16
Fraction in
Lowest Terms
Sixteenths
Decimal
Equivalent
9/16
9/16
0.5625
5/8
10/16
0.6250
© 2010 Cheryl Wilcox
11/16
11/16
0.6875
3/4
12/16
0.7500
Free Pre-Algebra
Lesson 36 ! page 11
2010 Major League Statistics
Player
At Bats Hits
Suzuki (SEA)
680
214
Tulowitzki (COL) 470
148
7. Find the difference of Suzuki’s At Bats to Tulowitski’s.
Write a sentence comparing the number of At Bats.
6. Write a sentence to compare the data.
8. If you divide Hits by At Bats, then round to three decimal
places, you have calculated the rate called the batting
average. Compute the batting average for each player.
9. Write the numbers in scientific notation.
a. 204,800,000
b. 50,000,000,000,000,000,000
Write a sentence comparing the player’s batting averages.
c. 1,780,000,000,000
10. Write the numbers in standard notation.
11. a. Re-write with a place value name.
a. 3.5 x 1018
The total surface area of earth’s oceans is about
139,800,000 square miles.
b. 2.02 x 106
b. Translate the written name to standard notation.
c. 4.111 x 1015
The surface of the great Salt Lake is 1.8 thousand square
miles.
12. Write the numbers from #11 in scientific notation:
13. Simplify, then solve the equation.
a. Ocean Surface:
0.05 20,000 ! x + 0.06x = 1070
b. Salt Lake Surface:
c. What is the difference in order of magnitude of the two
numbers?
d. How many times greater is the surface of the oceans than
the surface of Salt Lake?
© 2010 Cheryl Wilcox
(
)