Scientific Notation
Andrew Gloag
Melissa Kramer
Anne Gloag
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AUTHORS
Andrew Gloag
Melissa Kramer
Anne Gloag
EDITOR
Annamaria Farbizio
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Printed: October 13, 2013
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C ONCEPT
Concept 1. Scientific Notation
1
Scientific Notation
Here you’ll learn how to use scientific notation to rewrite very large or very small numbers as the product of a
decimal and 10 raised to a specific power.
Did you know that the average distance of the Earth from the Sun is about 92,000,000 miles? This is a big number!
Do you think that there is any way to write it more compactly? In this Concept, you’ll learn all about using scientific
notation so that you can express very large or very small numbers as the product of a decimal and 10 to a certain
power. This way, you’ll be able to express numbers such as 92,000,000 miles much more succinctly.
Guidance
Sometimes in mathematics numbers are huge. They are so huge that we use what is called scientific notation. It is
easier to work with such numbers when we shorten their decimal places and multiply them by 10 to a specific power.
In this Concept, you will learn how to express numbers using scientific notation.
Definition: A number is expressed in scientific notation when it is in the form
N × 10n
where 1 ≤ N < 10 and n is an integer.
For example, 2.35 × 1037 is a number expressed in scientific notation. Notice there is only one number in front of
the decimal place.
Since the scientific notation uses powers of ten, we want to be comfortable expressing different powers of ten.
Powers of 10:
100, 000 = 105
10, 000 = 104
1, 000 = 103
100 = 102
10 = 101
Using Scientific Notation for Large Numbers
If we divide 643,297 by 100,000 we get 6.43297. If we multiply 6.43297 by 100,000, we get back to our original
number, 643,297. But we have just seen that 100,000 is the same as 105 , so if we multiply 6.43297 by 105 , we
should also get our original number, 643,297, as well. In other words 6.43297 × 105 = 643, 297. Because there are
five zeros, the decimal moves over five places.
Example A
Look at the following examples:
1
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2.08 × 104 = 20, 800
2.08 × 103 = 2, 080
2.08 × 102 = 208
2.08 × 101 = 20.8
2.08 × 100 = 2.08
The power tells how many decimal places to move; positive powers mean the decimal moves to the right. A positive
4 means the decimal moves four positions to the right.
Example B
Write in scientific notation.
653,937,000
Solution:
653, 937, 000 = 6.53937000 × 100, 000, 000 = 6.53937 × 108
Oftentimes, we do not keep more than a few decimal places when using scientific notation, and we round the number
to the nearest whole number, tenth, or hundredth depending on what the directions say. Rounding Example A could
look like 6.5 × 108 .
Using Scientific Notation for Small Numbers
We’ve seen that scientific notation is useful when dealing with large numbers. It is also good to use when dealing
with extremely small numbers.
Example C
Look at the following examples:
2.08 × 10−1 = 0.208
2.08 × 10−2 = 0.0208
2.08 × 10−3 = 0.00208
2.08 × 10−4 = 0.000208
Video Review
MEDIA
Click image to the left for more content.
2
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Concept 1. Scientific Notation
MEDIA
Click image to the left for more content.
Guided Practice
The time taken for a light beam to cross a football pitch is 0.0000004 seconds. Write in scientific notation.
Solution:
1
0.0000004 = 4 × 0.0000001 = 4 × 10,000,000
= 4 × 1017 = 4 × 10−7
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note
that there is not always a match between the number of the practice exercise in the video and the number of the
practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba
sic Algebra: Scientific Notation (14:26)
MEDIA
Click image to the left for more content.
Write the numerical value of the following expressions.
1.
2.
3.
4.
5.
3.102 × 102
7.4 × 104
1.75 × 10−3
2.9 × 10−5
9.99 × 10−9
Write the following numbers in scientific notation.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
120,000
1,765,244
63
9,654
653,937,000
1,000,000,006
12
0.00281
0.000000027
0.003
0.000056
0.00005007
0.00000000000954
3
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Quick Quiz
−4 3 −3
−3 −2
) ·x y
1. Simplify: (2x y−2x
.
0 y2
2. The formula A = 1, 500(1.0025)t gives the total amount of money in a bank account with a balance of
$1,500.00, earning 0.25% interest, compounded annually. How much money would be in the account five
years in the past?
−3
3. True or false? 45
= − 125
64
4