Download Scientific Notation and Dimensional Analysis2012

Scientific Notation &
Dimensional Analysis
Chapter 2.2
What is scientific notation?
• A short-hand way to write a very small or large number
without all the zeros.
• 0.00000000000000678=6.78x10-15
• 7958900000000=7.9589x1012
• Numbers expressed as a multiple of two factors.
• A number between 1 to 10 and ten raised to a power.
• The exponent or power tells you how many times the first factor
must be multiplied by 10.
• Numbers greater than 1→ exponent=positive (+)
• Numbers less than 1 → exponent=negative (-)
Converting data into
scientific notation
• STEP 1→ Move the decimal place so only
one number is between 1 and 9.
• STEP 2→ Remove the extra zeros at the
end or beginning or the number.
• STEP 3→ Multiply the result by 10n where n
is the number of decimal places
moved.
• If the number was less than 1, then n will be negative.
(The decimal place moved to the right).
• If the number was greater than 1, then n will be positive.
(The decimal place moved to the left).
EXAMPLE:
The Distance From the Sun to the Earth
93,000,000
Step 1
• Move decimal
• Leave only one number in front
of decimal
Step 2
• Write number without zeros
Step 3
• Count how many places you
moved decimal
• Make that your power of ten
The power of ten is 7
because the decimal moved
7 places.
• 93,000,000 →Standard Form
• 9.3 x 107 → Scientific Notation
More Examples
•Write in decimal form w/o zeros.
•Determine the power of ten.
1
98,500,000
9.85 x 10?
9.85 x 107
2
64,000,000,000
6.4 x 10?
6.4 x 1010
3
0.000000027
2.7 x 10?
2.7 x 10-8
4
0.00000000429
4.29 x 10?
4.29 x 10-9
Practice Problems
Write the number in scientific notation.
•
•
•
•
•
734,000,000
870,000,000,000
90,000,000,000
0.00000765
0.00000000034
Answers
1)
2)
3)
4)
5)
734,000,000 = 7.34 x 108
870,000,000,000 = 8.7 x 1011
90,000,000,000 = 9.0 x 1010
0.00000465 = 4.65 x 10-6
0.00000000034 = 3.4 x 10-10
Converting data out of
scientific notation
• Move the decimal place the number of
decimal places indicated by the
exponent.
• If exponent is negative, the decimal place will
move to the left and the result will be less than 1.
• If exponent is positive, the decimal place will move
to the right and the result will be greater than 1.
Practice Problems
Write the number in standard form.
• 9.5 x 106
• 5.47 x 10-4
• 6.775 x 1010
Answers
• 9.5 x 106 = 9,500,000
• 5.47 x 10-4 = 0.000547
• 6.775 x 1010 = 67,750,000,000
Using a calculator for
scientific notation
• The EE button on your calculator will
allow you to enter a number in scientific
notation.
EXAMPLE→3.95 X 10-7
•
•
•
•
•
Enter 3.95
Press EE
Enter (-)
Enter 7
Proceed with manipulation as usual.
Practice Problems
Add 2.7 x 107, 5.35 x 106 , & 7.49 x 108
FINAL ANSWER: 7.8 x 108
Divide 5.65 x 107 by 3.9 x 10-3
FINAL ANSWER: 1.4 x 1010
Dimensional Analysis
How to convert from one set of
units to another
AKA Factor Label Method
What is a conversion factor?
• Two numbers that are equivalent but written in different
units.
• Written as a fraction and the same no matter which
number is written on top.
Example:
365 days = 1 year
365 days
1 year
O
R
1 year
365 days
Steps of Dimensional Analysis
•
•
•
•
Read the problem.
Determine the given.
Determine the unknown.
Think about the conversions factors that
you know to get from the given to the
unknown.
Example: How many yards are in 56 inches?
56 inches
Write the given with it’s unit in the first
numerator of your “railroad” track.
• What conversion factors do you know
that will get you closer to the unit you
want in the final answer?
• I have 56
• I want
inches
yards
Have
Want
I Know
(conversion factors)
I know
12 in = 1 foot
and
3 feet = 1 yard
• Pick a conversion factor that will get you
closer to the unit you want in the final
answer. This becomes the unit at the top
of the new fraction.
• I know 12 inches = 1 foot
56 inches
1 foot
12 inches
• Is the unit on TOP the one you
WANT to end up with?
YES
then do the math
NO
find another conversion factor
that gets you closer
• My “given” unit cancelled and I’m left with
feet.
• I WANT yards
• I HAVE feet
56 inches
1 foot
1 yard
12 inches
3 feet
I KNOW 3 feet = 1 yard
so . . .
• The last step is to check your
cancelled units and do the math!
56 inches
1 foot
1 yard
12 inches
3 feet
• 56 x 1 x 1 ÷ 12 ÷ 3 = 1.6 yards
• Congratulations! Now you can
solve word problems by
dimensional analysis!
Rules of Dimensional Analysis
1. Write ALL numbers as fractions.
2. Include units with all numbers (No Naked Numbers!).
3. Figure out what conversion factors are needed to get to the
wanted units.
4. Arrange conversion factors so the units cancel.
5. Set up the entire problem first, then do the math.
6. Multiply the numbers on top (avoid calculation error-don’t stop in
middle of calculation).
7. Divide result of top by result of bottom.
Show WORK and DON’T skip steps.
If you are 16 years old, what is your age in seconds?
•Given: 16 years
•Wanted: Age in seconds
•Conversion factors: days/yr, hours/day, minutes/hour, & seconds/minutes
16 years
365 days
24 hours
60 minutes
60 seconds
1 year
1 day
1 hour
1 minute
= 504576000 seconds
What is that in scientific notation and sig figs?
= 5.0 x 108 seconds
You want download your music
collection to iTunes. You have 225
CDs. If each CD has 12 songs and it
takes 85 seconds to download 1 song,
how many hours will it take to
download your entire collection?
•
•
•
Given: 225 CDs
Wanted: Hours to Download
Conversion factors: songs/CD, sec/song, sec/min, min/hour
225 CDs
12 songs
85 seconds
1 minute
1 hour
1 CD
1 song
60 seconds
60 minutes
=63.75 hours
The density of copper is 8.96 g/mL.
What is its density in kg/m3?
8.96 g
1 kg
1 mL
1,000,000 cm3
1 mL
1000 g
1 cm3
1 m3
=8960 kg/m3 or 8.96 x 103 kg/m3