You will need
1.5 Expanded Form and
• a calculator
Scientific Notation
GOAL
Express and compare numbers using expanded form and scientific notation.
Learn about the Math
Manuel’s grandfather is 75 years old and has a heart rate of about
70 beats per minute. Manuel uses his calculator to estimate the total
number of times his grandfather’s heart has beat over the 75 years.
Manuel’s grandfather has an African grey parrot.
The parrot is 45 years old and has a heart rate of
about 550 beats per minute. Manuel uses his
calculator to estimate the total number of times
the parrot’s heart has beat during its lifetime.
Manuel’s calculator displays a number in
scientific notation .
scientific notation
a way of writing a
number as a decimal
between 1 and 10,
multiplied by a power
of 10; for example,
70 120 is written as
7.012 ! 104
number between
1 and 10
7.012 ! 104
power of 10
expanded form
You can compare large numbers using scientific notation or
expanded form .
? Which heart has beat more times?
a way of writing a
number that shows
the value of each digit
as a power of 10; for
example, 1209 in
expanded form is
1 ! 103 " 2 ! 102 "
9!1
Communication Tip
The TI-15 calculator uses the carat symbol, ^, for scientific
notation. For example, 1.81 ! 1011 can be entered like this:
1.81 C 10 F 11
Some calculators use the letter
numbers in scientific notation.
in the display to express
Other calculators use a shorter form to express the
same scientific notation.
20
Chapter 1
NEL
Example 1: Comparing large numbers
Compare the number of heartbeats for Manuel’s grandfather and the parrot. Use scientific
notation and expanded form.
Manuel’s Solution
I had to determine which calculator answer, 2759400000 or 1.301 ! 10^10 , is greater.
1.301 ! 1010
1010 109 108 107 106 105 104 103 102 101
1.301 0 0
0 0
0 0 0 0 0
1
0
I decided to write both numbers in
expanded form. I represented
1.301 ! 1010 in a place value chart.
1010 109 108 107 106 105 104 103 102 101
1
3 0
1 0
0 0 0 0 0
1
0
I regrouped to determine the place
value of each digit.
1.301 ! 1010 $ 1 ! 1010 " 3 ! 109 " 1 ! 107
I showed each digit multiplied by its
place value. Then I showed the sum of
the products.
2 759 400 000
The number 2 759 400 000 is in standard
form. The front digit has the greatest
place value—billions or 109.
I showed each digit multiplied by its
place value. Then I showed the sum of
the products.
1010 109 108 107 106 105 104 103 102 101
2
7
5
9
4
0
0
0
0
1
0
2 759 400 000
$ 2 ! 109 " 7 ! 108 " 5 ! 107 " 9 ! 106 " 4 !105
1 ! 1010 " 3 ! 109 " 1 ! 107
> 2 ! 109 " 7 ! 108 " 5 ! 107 " 9 ! 106 " 4 ! 105
The parrot’s heart has beat more times than my
grandfather’s heart.
I compared the expanded forms of the
two numbers. One number has billions
and the other has 10 billions.
1 ! 1010 # 2 ! 109
I could have used scientific notation to compare
the two numbers.
2 759 400 000 $ 2.7594 ! 109
1.301 ! 1010 # 2.7594 ! 109
Reflecting
1. Why is ■.■ ! 109 always less than ■.■ ! 1010?
2. How do you know what power of 10 to use when you want to write
a number in scientific notation? Use an example to support your
explanation.
3. How do you know what powers of 10 to use when you want to write
the expanded form of a number? Use an example to support your
explanation.
NEL
Number Relationships
21
Work with the Math
Example 2: Comparing numbers using scientific notation
Which travelled farther?
• a ray of light travelling at a speed of about 300 000 km per second for one week
• the spacecraft Voyager 1 travelling at a speed of about 63 000 km per hour for 100 years
Solution
The ray of light travelled about
1.814 ! 1011 km.
Voyager 1 travelled about
5.519 ! 1010 km.
1.814 ! 1011 # 5.519 ! 1010
1011 #1010
The ray of light travelled farther.
A
Checking
B
4. State whether each number is in scientific
notation, standard form, or expanded form.
a) 1 300 000
b) 1.235 ! 10
c) 1 ! 103 " 2 ! 102 " 3 ! 10 " 5 ! 1
5. Copy and complete the chart.
Standard
form
Expanded
form
Scientific
notation
360
3 ! 102 " 6 ! 10
3.6 ! 102
3 ! 103 " 6 ! 102
3.6 ! 104
360 000
6. Which number is greatest?
4376
5 ! 103 " 4 ! 102
6.13 ! 103
22
Chapter 1
Practising
7. Express each number in expanded form.
a) 2345
b) 11 289
c) 105 284
d) 1 045 605
8. Express each number in standard form.
a) 1 ! 103 " 0 ! 102 " 4 ! 10 " 9 ! 1
b) 4 ! 104 " 3 ! 103 " 8 ! 102 "
7 ! 10 " 0 ! 1
c) 8 ! 106 " 1 ! 105 " 5 ! 1
9. Decide whether or not each number is in
scientific notation. Explain your reasoning.
a) 12.5 ! 107
b) 5.688 ! 1012
c) 0.43 ! 106
d) 4.5 ! 56
10. Express each number in scientific notation.
a) 1 300 000
b) 12 500
c) 882 500 000
d) 51 670 000
NEL
11. a) Calculate each power of 10.
103
106
109
1012
b) Compare the value of each power of
10 in part (a) with its exponent. What
pattern do you see?
c) Predict the value of 1015.
17. Answer each question. Show your work,
and write your answer in expanded form or
scientific notation.
a) About how many times has your heart
beat in your lifetime?
b) Most people blink about every 2 to 10 s.
About how many times have you blinked
in your lifetime?
c) The human eye can process about
36 000 bits of information every hour.
About how many bits of information
have you processed in your lifetime?
12. Express each number in standard form.
a) 1.235 ! 10
b) 8.01 ! 106
c) 5.688 ! 1012
d) 3.5 ! 1010
13. Which number is greatest? Explain your
reasoning.
987 098
6 ! 105 " 3 ! 10 " 1 ! 1
1.0 ! 106
14. Why does a calculator express some
numbers in scientific notation rather than in
standard form?
15. The media often use decimals to report
large numbers. Express each number using
scientific notation.
a) 1.4 million, the approximate number of
bytes of storage in a floppy diskette
(one million bytes $ one megabyte)
b) 1.4 billion, the approximate number
of bytes of computer storage needed
to store all the information in an
encyclopedia (one billion bytes $
one gigabyte)
c) 2.5 trillion, the approximate number
of bytes of storage capacity in the
computers at Statistics Canada
(one trillion bytes $ one terabyte)
16. a) Explain how you can change a number
in standard form to expanded form. Use
an example to clarify your explanation.
b) Explain how you can change a number
in scientific notation to standard form.
Use an example to clarify your
explanation.
NEL
18. Which prize is worth more? Explain your
reasoning.
• a prize on March 31, if the amount is
tripled each day, beginning with $3 on
March 1
• a prize after 50 years, if $1 million is
added each day
C
Extending
19. In 2004, William Gates III of Microsoft had
assets worth about $46.6 billion U.S.
a) What is this amount worth in Canadian
dollars? Each U.S. dollar was worth
about $1.29 Canadian.
b) Express the amount in Canadian dollars
using scientific notation.
Number Relationships
23