Using the SI System, Scientific Notation and the Factor-Label
Method of Conversions
(AKA Dimensional Analysis)
 The International System of
units
 Uses prefixes and base
units.
 The prefixes are here 
 And the base units you
need to know are:




Meter (length)
Kilogram (mass)
Joule (energy)
Second (time)
 Use the stairs to
convert 1.5 hm to
mm.
 How about 1.5 cJ
to kJ?
 In science, numbers are often VERY LARGE, or very
To deal with that, you should consider using
scientific notation.
 Students new to sci notation tend to think it’s difficult,
but that’s because they’ve memorized the rules the
wrong way, so they only get the right answer half the
time.
 If you aren’t excellent at scientific notation, take a
second and forget what you’ve learned about it. Trust
me, my way’s easier than what you learned.
small.
 Step 1: find the decimal. If there’s not one, put one at the right




hand end of the number.
Step 2: Move the decimal so that there is only ONE non-zero
number to the left of the decimal.
Step 3: Re-write the number with the decimal in the new spot. If
there are zeros to the left of your decimal now (leading zeros),
leave them off. If the digit farthest to the right is a zero, leave it
off, along with any other “trailing zeros” (at the end).
Step 4: Add “x10” to the right of what you’ve written now.
Step5: Look at the original number. Is it greater than 1 (would
you be happy if someone gave you that many dollars)? If so,
write a + sign after the 10. If not, write a – sign.
Step 6: In step 2, how many places did you move the decimal?
Write that number after the + or – sign.
Normal number:
0.000035
0.000035000
35,000,000,000
35,000,648,000
35.0035
In scientific notation:
Normal number:
In scientific notation:
6 x 104
6 x 10-4
5.02 x 10-3
5.02 x 103
 Multiply:
 Multiply the big numbers, add the exponents
 Divide:
 Divide the big numbers, subtract the exponents
 Add/subtract:
 If exponents match, add/subtract the big numbers, keep
the exponents.
 If exponents don’t match, change them so they do, or
write the numbers out in normal notation
 The stair step method – Works great for conversions
within the SI system, but ONLY if you know how
(remember 3 steps between k, M, G, T and m, μ, n, p)
and ONLY applies within SI/metric systems.
 Guessing and/or logic – Works well if the problem isn’t
too complex and you’re good at it.
 Factor-label method – Works right every time, if you
know the steps (and follow them).
 Warning: This starts out kind of
confusing. Be patient.
 Comments from many past students:
 I didn’t get it at first, but now it’s SO easy!
 That factor-labeling thing really came in handy!
 I wish I had paid attention when you first explained
factor-labeling!
 Because:
 You will use it if you ever take another chemistry, physics
or engineering class ever in your life.
 It’s useful outside of science too.
 It makes you look smart
 I said so! 
1. Start with the number in the
problem with its units.
Make it into a fraction
(put it over 1).
2. Multiply by a new fraction so that:
a. The unit that you don’t want (in the
first fraction) cancels out AND
b. The top of the second fraction is
equal to the bottom of the second
fraction.
3. Cancel out the units and check:
a. If you have the units you want, go to
step 4.
b. If you don’t have the units you want,
repeat steps 2 and 3.
4. Solve:
a. Multiply across the tops of the fractions
and write it down.
b. Multiply across the bottoms of the
fractions and write it down.
c. Divide: top divided by bottom.
5.
Double check:
 Check for significant figures (the APES
exam is lenient with this, as long as you
show your work. Keep three non-zero
digits and round the rest.)
 Don’t leave the number naked! (Add
the correct unit!)
 Check to see if the answer MAKES
SENSE.
 No really, that’s all!
 Since easy problems can be solved other ways, let’s try
a complex problem.
 Please trust that I will give you the occasional problem
that is too complex for you to use logic on.
 Use Factor-Labeling.
 How many centimeters are in 156 miles?
 Known conversion factors (use these even if
you have access to others, for today):
 .394 in = 1 cm
 12 in = 1 ft
 1 mi = 5280 ft
 How many centimeters are in 156 miles?
156 mi
1
156 mi x 5280 ft
1
1 mi
156 mi x 5280 ft
1
1 mi
We do not have the units we want (centimeters).
Repeat steps 2 and 3.
156 mi x
1
5280 ft
1
mi
12 in
1 cm
1 ft
.394 in
This is the unit we want, so we
go on to the last step…
156
1
5280
1
12
1 cm
1
.394
156 x 5280 x 12 =
.394
25,086,700.5 cm