Scientific Notation

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Transcript Scientific Notation

Scientific Notation
The Ins, Outs and How -to
Objectives
• To express numbers in standard decimal
notation and scientific notation
• To convert numbers between scientific
notation and standard notation
• To explain the benefits of using scientific
notation
• To convert between scientific units
• To develop undersatnding and basic examples
of the Powers of Ten
Benefits
• Minimizes mistakes when using really large or
small numbers in calculations
• Saves time when writing really large or small
numbers
Example of Benefit
Let’s suppose we had the technology to travel to
the starts. You were trying to calculate how
much fuel would be needed to get from here to
Proxima Centauri, the nearest start other than
our Sun.
A mere 39,900,000,000,000 km away…
Now if you had to enter that number into a
calculator, what’s the likelihood you would miss
or add a zero?
Express the following in scientific notation:
39,900,000,000,000 km
1. Leave some space and
write x 10
2. Locate the decimal
point, move it so that
you only have 1 digit
to the left of it.*
3. Count the number of
spaces you moved the
decimal point and
write it as an exponent
with the 10**
1. _______ x10
2. 39,900,000,000,000
1. _______ x1013
Express the following in scientific notation:
39,900,000,000,000 km
(cont)
4. Write the number that
remains in the space you
left.
5. If necessary round off
so that there are only 2
digits to the right of the
decimal point.
6. Don’t forget to rewrite
the units!!
4. 3.99 x1013
5. (not necessary this
time)
6. 3.99 x1013 km
Just as an FYI
Later in this presentation we will be converting between units
but for now…
Did you know that 39,900,000,000,000 km is the same as
271,000 AU (astronomical units). As you can see even that
number has several zeros and would be better expresseed in
scientific notation.
For that reason, when we talk about the distances to the
stars, we no longer use the AU; commonly, the light year is
used.
A light year is the distance light travels in one year - it is equal
to 9.46 x 1012 km.
Making the distance to Proxima Centauri a mere 4.22 light
years.
Practice Time
Convert the following into
Scientific Notation
a. 247,800,000,000 L
c. 0.0000000346 kg
b. 987,000,000 mg
d. 0.0001456 ms
Answer Check
a. 247,800,000,000 L =
2.48 x 1011 L
c. 0.0000000346 kg =
3.46 x10-8 kg
b. 987,000,000 mg =
9.87 x 10 8 mg
d. 0.000000000001456 ms
= 1.46 x10-12ms
Opener
• What benefit does writing numbers in
scientific notation have?
• Convert the following
1.
2.
3.
4.
346,000,000 s
0.00798 m
-753,000 g
-.0002156 N
Express the following in Standard Notation 6.02 x1023
atoms (aka Avagadro’s Number)
1. Copy the decimal
number
2. Move the decimal
point the number of
spaces indicated by
the exponent*
3. Fill in empty places
with zeros.
1. 6.02
2. 6.0200000000000000
0000000
Express the following in Standard Notation 6.02 x1023 atoms
(aka Avagadro’s Number)
(cont.)
3. Now remove the
decimal point and add the
commas (if necessary)
3. 602,000,000,000,000,
000,000,000
4. Don’t forget to rewrite
the units
4. 602,000,000,000,000,
000,000,000 atoms
Practice Time
a. 5.87 x 10 4 km
c. 5.39 x 10-2 cm
a. 9.63 x 10- 7 kg
d. 2.94 x 106 mL
Answer Check
a. 5.87 x 10 4 km =
58,700 km
a. 9.63 x 10- 7 kg =
0.000000963 kg
c. 5.39 x 10-2 cm =
0.0539 cm
d. 2.94 x 106 mL =
2,940,000 mL
Converting to and from different
Scientific Units
Now, that you have developed some proficiency
with scientific notation the next step is to be
able to convert from meters to kilometers or
vice versa.
Why would you want to do that?
Just like when writing you choose words based
on what best conveys your meaning. So it is
similar with science units.
For example:
Which do you think is a better way of expressing
this value:
1 km or 100,000 cm?
True they are both the same BUT the first is a
better expression of the measurement.
So how do you do this…
• One way is to memorize a chart and move
your decimal accordingly. If you have learned
that way great stick with it
• Dimensional Analysis is a process where using
a few basic relationships I can consistently get
a correct answer.
What is a “Base Unit?”
• A Base Unit is a unit of measuring that does
not have a prefix.
• Examples:
– Meter (m)
– Gram (g)
– Liter (L)
– Second (s)
• Not: km, mL, mg, etc
Conversion Factors
• SI Prefixes:
pico = p = 10−12
nano = n* = 10−9
micro = μ* =
10−6
milli = m* = 10−3
centi = c* = 10−2
deci = d* = 10−1
kilo = k* = 103
mega = M* = 106
giga = G = 109
• (* = You need to know these commonly used prefixes and
their abbreviations)
• When setting up a conversion factor to or from a base unit,
use 1 for the prefixed unit and the power of 10 for the
base unit.
• Examples: 1 mg = 10-3 g and 1 kg = 103 g.
You will always be given the SI Prefixes & number values
• Note: a conversion factor is always equal to the number 1
• Complete prefix Practice now to 550 mm, rest for HMWK
Dimensional Analysis
1. Write the original number
in scientific notation
2. Draw a railroad track.
Write the original number in
scientific notation in the first
position (track) with the
units.
3. Place the unit you are
canceling in the bottom and
the “new unit” on the top.
One of these should be the
base unit.
Convert 4.7 kg to grams.
• Solution :
4.7 kg
g
kg
4. Place the number 1
with the unit that has the
prefix.
• 5. Now you will use the
chart to help you figure
out the number to put
with the “new unit”
– Find the prefix on the
chart
– Write its number value
with the base unit (the
blank space)
4.7 kg
4.7 kg
g
1 kg
103 g
1 kg
6. Multiply the tops x tops
7. If necessary multiply
bottoms x bottoms
8. Divide top by the
bottom
9. Write your final answer
with units.
Note to self.... Know your
calculator...
• “ee” on calculators
means x10
• use parentheses
4.7 kg
103 g
1kg
4.7 x 103 g.
Your Turn
Convert 0.125 meters to
mm.
milli = m* = 10−3 becomes
1 mm = 10-3 m
Convert 1,086 m to km
kilo = k* = 103
Answer Check
Convert 0.125 meters to
mm.
0.125 m
1 mm
10-3 m
= 125 mm
= 1.25 x102 mm
Convert 1,086 m to km
1,086 m
= 1.086 km
= 1.09 km
1 km
103m
And now for the Advanced
• These will be conversions between units when
they both have prefixes!
• Have no fear you already know how. It’s just
going to take 2 steps to convert.
• In other words, first convert to the base unit
(meters, grams, seconds, or liters) then to the
desired unit.
Let’s Try One Together
Convert 76 cm to mm.
Let's try to convert this problem into
two problems that we already know
how to solve. First, look at the
conversion factors we know involving
cm and mm.
•
1 cm = 1 x 10-2 m
1 mm = 1 x 10-3 m
Both factors involve meters, which
means we can convert cm to m, and
then m to mm.
In other words, this conversion will
take two steps.
Solution:
First, convert 76 cm to m.
76 cm
10-2 m
1 cm
= 0.76 m
Next, convert 0.76 m to mm
• 0.76 m
1 mm
10-3 m
use your calculator
= 760 mm
Alternative Solution
Solution:
We could also perform this conversion in one equation by
making one long train track. Those of you who eventually take
Chemistry or Physics will work a lot of problems by stringing out
conversion factors in this manner. Just make sure that the units
cancel out appropriately.
76 cm
10-2 m
1 cm
1 mm
10-3 m
= 760 mm
A little Math Note
• How do you multiply numbers in scientific
notation?
– Add the exponents!
– (1 x 104) x (1 x 103) = 1 x 104+3 = 1x 107
• How do you divide numbers in scientific
notation?
– Subtract the exponents!
(1 x 104)
(1 x 103) = 1 x 104-3 = 1x101
Closer
• What is scientific notation?
• When will you use or have a negative
exponent?
• Do negative numbers have an exponent?
Explain
• What some benefits of scientific notation?
Homework- due next class period
Complete Prefix practice
Homework 2*
Basic Scientific Notation = 2
Conversions to or from a base unit = 3
Conversions when both units have prefixes =4