Lesson
6 Writing Very Small Numbers
Problem Solving:
Probability—One or the Other
Writing Very Small Numbers
How does scientific notation help us write very
small numbers?
In previous lessons of this unit, we used scientific notation to rewrite
large numbers. These numbers can be difficult to read and confusing
to understand if they are written in standard notation. For example,
scientists find it more difficult to read 4,300,000,000,000 than
4.3 × 1012.
Does this kind of thinking apply to small numbers? Yes, it does.
Here are some everyday examples of small numbers. Each of the
examples is compared to the length of one meter. Remember that a
meter is a little longer than a baseball bat.
Living Examples of Small Numbers
The width of a human hair: 0.0070 centimeter
A dust mite: 0.00080 centimeter
The deadly Ebola virus: 0.00020 centimeter
cm
1
2
3
4
These living things are so small we cannot simply measure them with
a ruler. We see that the decimal numbers for these examples are hard
to read.
506 Unit 7 • Lesson 6
Lesson 6
Mathematicians had to choose a method for writing small numbers.
The table shows how they thought about representing small decimal
numbers. They decided to use negative exponents.
Hundreds
100
102
Tens
10
101
Ones
1
100
•
Tenths
0.1
10−1
Remember, the red dot
represents the decimal
point.
Hundredths
0.01
10−2
Thousandths
0.001
10−3
Decimal Numbers
The method for writing small numbers using scientific notation is the
opposite for writing big numbers.
Steps for Writing Exponents for Very Small Numbers
Step 1
Start with the decimal point on the left.
0.00021
Step 2
Move the decimal point to a
number between 1 and 10.
0.00021 = 0.0 0 0 2 1
Step 3
Count the number of places that
the decimal point moved.
The number of places we moved the
decimal point is reflected in the exponent
for the power of 10.
0.00021 = 0.0 0 0 2 1 = 2.1 x 10-4
However, there is one main difference. We use a negative exponent to
show that we moved the decimal point in the opposite direction.
Remember, scientific notation always uses a base of 10.
2.1 × 10−4
-4th power of 10
Unit 7 • Lesson 6 507
Lesson 6
Be careful not to confuse negative exponents with negative numbers.
We will learn more about negative numbers in the next unit. For now,
just remember that numbers with negative exponents are very small
numbers between 0 and 1.
The table shows the relationship between a power of 10 with a negative
exponent and its fractional and decimal number equivalent for tenths
through 100,000ths. This pattern continues on forever as decimal
numbers get smaller and smaller.
Example 1
Show the relationship between powers of 10 and small numbers.
Power of 10
Fraction
10−1
1
10
1
100
1
1,000
1
10,000
1
100,000
10−2
10−3
10−4
10−5
Decimal Number
0.1
0.01
0.001
0.0001
0.00001
We use negative exponents when we work with very small numbers in
scientific notation.
Apply Skills
Turn to Interactive Text,
page 263.
508 Unit 7 • Lesson 6
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
Lesson 6
Problem Solving: Probability—One or the Other
What is the probability when it is one thing
or the other?
When we work with probabilities, we should ask ourselves, “Did I set
up the problem correctly? Am I thinking about the right numbers?”
These are important questions when we are given problems in which
we have two or more possible things that could happen. Here is a
simple example.
The spinner below is divided into 4 equal parts. Each part is 1
4 or 0.25 of
the circle. What are the chances of the spinner landing on red or blue?
Example 1
Find the probability of the spinner landing on either red or blue.
The spinner can land on red or blue, but not both. The chance for landing
on red is 0.25. The chance for landing on blue is also 0.25. That means
the chance is 0.25 + 0.25, or 0.50, for landing on red or blue. The
spinner can’t land on both, but we still need to consider the increased
probability of landing on one or the other.
0.25 Green
0.25 Yellow
0.25 Red
0.25 Blue
• Probability of landing on red: 0.25
• Probability of landing on blue: 0.25
The probability of landing on red or blue: 0.25 + 0.25, or 0.50
We can expand this discussion by looking at more than one “or”
statement in a probability.
Unit 7 • Lesson 6 509
Lesson 6
Now let’s find the probability of a statement with more than one “or”
statement.
Example 2
Find the probability of landing on
red or blue or yellow.
0.25 Green
Probability of landing on red: 0.25
Probability of landing on blue: 0.25
Probability of landing on yellow: 0.25
0.25 Red
0.25 Yellow
0.25 Blue
The probability of landing on red, blue, or yellow:
0.25 + 0.25 + 0.25 or 0.75
What if we wanted to know the probability of landing on red or blue or
yellow or green? Now we have all the possible outcomes of a spin. This
means there is a 100 percent chance of landing on one of the colors.
Example 3
Find the probability of landing on
red or blue or yellow or green.
0.25 Green
Probability of landing on red: 0.25
Probability of landing on blue: 0.25
Probability of landing on yellow: 0.25
Probability of landing on green: 0.25
0.25 Red
The probability is 0.25 + 0.25 + 0.25 + 0.25 = 1.00
Again, the decimal number 1.00 is the same as 100 percent.
Problem-Solving Activity
Turn to Interactive Text,
page 264.
510 Unit 7 • Lesson 6
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
0.25 Yellow
0.25 Blue
Lesson 6
Homework
Activity 1
Choose the scientific notation that is correct for each number. Write a, b, or c
on your paper.
1. 0.01 ​
(a) 1.0 · 102
(b) 1.0 · 10−2
(c) 10.0 · 10−2
2. 430 ​
(a) 4.3 · 10−2
(b) 4.3 · 10−3
(c) 4.3 · 102
3. 0.0025 ​
(a) 2.5 · 103
(b) 25.0 · 10−3
(c) 2.5 · 10−3
4. 0.000049 ​
(a) 4.9 · 10−5
(b) 4.9 · 105
(c) 4.9 · 10−4
5. 3,800 ​
(a) 3.8 · 105
(b) 3.8 · 10−3
(c) 3.8 · 103
Activity 2
Rewrite the numbers in scientific notation.
Model0.0023 = 2.3 · 10−3
1. 0.023 ​
2. 0.45 ​
3. 0.00056 ​
4. 0.1 ​
Activity 3
Tell the percentage probability of each outcome using this spinner:
1. What is the probability the spinner will
land on green or yellow? ​
2. What is the probability the spinner will
land on blue or red? ​
0.25
Purple
0.25 Red
3. What is the probability the spinner will
land on green, blue, or red? ​
4. What is the probability the spinner will
land on purple, green, or blue? ​
0.15 Blue
0.25 Yellow
0.10
Green
5. What is the probability the spinner will
land on purple, red, blue, green or yellow? ​
Copyright 2010 by Cambium Learning Sopris West®. All rights reserved. Permission is granted to reproduce this page for student use.
Unit 7 • Lesson 6 511
Lesson 6
Homework
Activity 4 • Distributed Practice
Solve.
1. 3.2 + 1.5 + 2.7 + 8.9 + 1.0 + 5.2 + 4.4 + 6.8
16
45
1
1 15
8 + 7 56
4
8
2. 9 · 10 3.
4. 12.9 · 0.1 5. 167.01 − 89.93 6. Convert 27% to a fraction. 18
27
100
7. Convert 36 to a decimal number. 8. Convert 1.07 to a percent. 512 Unit 7 • Lesson 6
Copyright 2010 by Cambium Learning Sopris West®. All rights reserved. Permission is granted to reproduce this page for student use.