1. Math Review
Units Review
Science 111
CSUB
Jeff Lewis ©2008

2. Units and the Metric System
• This is material covered in Appendix A of
the textbook.
• The (probably) more familiar units are the
USCS: United States Customary System.
• The metric system is also known as the
International System or SI system.

3. Why Why Why?
• Why use the metric system when you are
more familiar with the U.S. system?
1. The metric system is the better system.
2. The metric system is the standard
throughout the rest of the world.
3. Anyone doing or learning science needs to
learn the metric system.

4. Standard (or Base) Units
• USCS
Length foot (ft)
Force pound (lb)
Time second (sec)
Note: the pound is a unit
of force, not mass.
• SI
Length meter (m)
Mass kilogram (kg)
Time second (s)
Same time unit in both
systems.

5. Better? Why?
• Why is the metric system a “better” system
than the USCS?
• Not because the meter is better than the
foot.
• What makes the metric system better are the
metric prefixes used to construct larger and
smaller size units.

6. Prefix Advantages
• The same prefixes are used with all units to
make the new, larger or smaller, units.
• And the new units are always simple
multiples of 10 (or 100 or 1000 etc.) larger
or smaller.

7. Micro prefix
• micro- makes a new unit 1,000,000 times
smaller.
Microgram, microsecond, micrometer,
microliter, microvolt, microphone (whoops,
ignore that last one).
1 microgram = .000001 grams = 1 µg

8. Milli prefix
• milli- makes a new unit 1/1000 times the
size of the original unit.
Milligram, millisecond, millimeter, milliliter,
millidegree, milliamp, millipede (oops).
1 millisecond = .001 seconds = 1 ms

9. Centi prefix
• centi- makes a new unit 1/100 (.01) the
size when in front of anything.
Centimeter, centigram, centisecond,
centidegree, centijoule.
1 centimeter = .01 meters = 1 cm

10. Kilo prefix
• kilo- make a new unit 1000 times larger.
Kilogram, kilometer, kilosecond, kilopound,
kilowatt, kilovolt.
1 kilogram = 1000 grams = 1 kg

11. Mega prefix
• mega- makes a new unit 1,000,000 times
larger.
Megameter, megasecond, megagram,
megahertz, megaton, megaparsec.
1 megaton = 1,000,000 tons = 1 Mton

12. Standard Prefixes
• Whatever unit you want larger or smaller
versions of, you use the same prefixes.
• Not like the USCS where every unit works
differently (like inch-foot-yard-mile or
ounce-pint-quart-gallon).

13. Learn the Metric System!
• Learn the prefixes we’ve talked about and
how much each stands for.
• You’ll need to know the metric system for
homework, labs, and exams.

14. MKS vs CGS
• In the mks version of
the metric system, the
meter, kilogram, and
second are considered
the “fundamental”
units.
• In the cgs version, the
“fundamental” units
are centimeters,
grams, and seconds.
• Does it matter which
is fundamental? No!

15. We will use mks system
• The systems do differ when you talk about
derived units (units that are combinations of
the base units).
• mks: force – newtons, energy – joules
• cgs: force – dynes, energy – ergs
• We’ll use mks, newtons and joules.

16. Conversion of Units
• A very important mathematical technique is
being able to convert units.
• Meaning, being able to take a value
expressed in one unit and figure out its
equivalence expressed in a different unit.

17. Sample Conversion Problems
• For example, you might want to know
– What is 5 kilometers in miles?
– What is 100 meters in yards?
– How many ounces are in one liter?
– What is 90 kg in pounds?
– How many minutes is 505 seconds?
– How many meters is 345 centimeters?

18. Basic Method
• Look up the equivalence that relates the two
units you are trying to convert between.
• Such as, 1 ft = 12 in or 1 km = 1000 m
• This equivalence is then used to construct a
“conversion factor”, a fraction with one of
the values on top and the other on the
bottom.

19. Conversion Factors
( )1 ft
12 in ( )12 in
1 ft ( )1000 m
1 km
or or
What makes these conversion factors
special is that each is equal to one (because
numerator and denominator are the same!).
Mathematically, this means we can multiply
them anywhere, anytime, without changing
the value.

20. Sample Problem with Solution
Problem: How many inches are in 15 feet?
That is, we are trying to convert 15 ft into the
equivalent number of inches.
Solution: Create an equality, 15 ft = 15 ft
Then multiply by a conversion factor that will cancel
the ‘ft’ and give ‘in’ instead.

21. 15 ft = 15 ft = 180 in
Note how I chose the conversion factor with ft
on the bottom so that the ft would cancel.
My original, trivial, 15 ft = 15 ft equality is still
valid even though I multiplied on the right side
only because I multiplied by a factor of one.
( )12 in
1 ft

22. New Problem with a Twist
Problem: What is an area of 100 square feet (10 ft by
10 ft or 100 standard floor tiles) in units of square
inches?
Solution: Start with 100 ft2
= 100 ft2
(ft2
= ft x ft, an area unit).
The twist? To do this conversion we have to multiply
by the conversion factor twice.

23. 100 ft2
= 100 ft2
= 14,400 in2
The ft2
unit is really two factors of ft, so I had to
convert both of them.
In converting volumes, there would be three
length units to convert.
( )12 in
1 ft ( )12 in
1 ft

24. Problem with a Different Twist
Problem: How many seconds are there in one year?
That is, we are converting the duration of 1 yr
into the equivalent number of seconds.
Solution: Start with 1 yr = 1 yr
The twist? I don’t know the equivalence factor
between years and seconds (that is what we are trying
to figure out).
Instead, I can do this with a chain of conversions.

25. 1 yr = 1 yr
= 31536000 s = 31,536,000 s
= 3.15 x 107
s
I didn’t know the direct conversion from
years to seconds but I knew the intermediate
conversions.
Note that I figured out what to put on top
and bottom based on how units will cancel.
( )365 day
1 yr ( )24 hr
1 day ( )60 min
1 hr ( )60 s
1 min

26. Unit Conversion Summary
• Converting units is a very common
problem, especially in labs.
• I urge you to carefully follow the method
I’ve outlined here.
• Students who have trouble usually don’t
write the steps down and instead try just to
do it in their head.

27. Practice Problems (do now!)
1. What is 5 kilometers in miles?
(1 mi = 1.609 km)
2. What is 60 mi/hr in km/hr?
3. What is 1 m/s in mi/hr? (1 km = 1000 m)
4. What is 500,000 ft3
in m3
? (1 m = 3.28 ft)
5. What is 32.2 ft/s2
in cm/s2
?
(100 cm = 3.28 ft)

28. Answers:
1. 3.11 mi
2. 96.54 km/hr
3. 2.24 mi/hr
4. 14,169 m3
5. 982 cm/s2

29. Scientific Notation
• Powers-of-ten notation:
– 105
means 10 x 10 x 10 x 10 x 10 = 100,000
– 1024
= 1 followed by 24 zeroes
– 10-1
= 1/10 = 0.1
– 10-4
= 1/10 x 1/10 x 1/10 x 1/10 = 1/10,000 =
0.0001
– 3.21 x 103
= 3.21 x 1000 = 3,210
– 3.21 x 10-5
= 0.0000321

30. Scientific Notation Advantages
• Scientific (or powers-of-ten) notation is a
simple way to write out very large or very
small numbers.
• While we won’t be doing much math this
quarter, you will be expected to recognize
and understand values written in scientific
notation when you see them.
• And if you need to do a calculation…

31. Sci Not on your Calculator
• All scientific calculators come with a
shortcut button for inputting numbers
written in scientific notation.
• Look on your calculator for a button labeled
“E”, “EE”, or “Exp”.
– I’ll assume it’s called “EE” in the following.
• To enter the value 4.2 x 1015
into your
calculator, you push “4 . 2 EE 1 5”

32. Sample Problem
• Calculate 4.2 x 1015
/ 2.1 x 10-5
• Solution: Push “4 . 2 EE 1 5 / 2 . 1 EE +/- 5”
• Answer: 2 x 1020
• Notes: If your calculator says “2 20
”, you
need to realize that that means 2 x 1020
.
• Without the EE button, you’d have to push
“4 . 2 x 1 0 ^ 1 5 / ( 2 . 1 x 1 0 ^ +/- 5 )” and
you would get the wrong answer without the
parentheses.

33. Practice Problems
6. Simplify (2 x 1010
) x (3 x 1020
)
7. Simplify (25 x 1010
) / (5 x 1012
)
8. Simplify (6.02 x 1023
) (105
)
18
• Hint: 105
is “1 EE 5” or “1 0 ^ 5”, not “1 0 EE
5”
9. Simplify 4 (1.496 x 10π 8
)2
10. Simplify 1.05 x 10-22
(10-14
) (4.32 x 10-9
)

34. Answers
6. 6 x 1030
7. .05 ( = 5 x 10-2
)
8. 3.34 x 1027
9. 2.81 x 1017
10. 2.43
One way to do #10: “1 . 0 5 EE - 2 2 / 1
EE - 1 4 / 4 . 3 2 EE - 9 =”

35. Combined Units
• We learned about base units (kg, m, s)
before, but some types of quantities have
units that are combinations of these.
• Speed or velocity units: distance/time,
units like mi/hr, m/s, or km/min.
• Acceleration units: distance/time/time,
units like m/s2
, mi/hr/sec, or ft/s2
.

36. More Combined Units
• The metric unit of force, the newton (N), is
a combined unit: N = kg m / s2
• The USCS units of mass, the slug, is a
combined unit: slug = lb s2
/ ft
• The metric unit of energy, the joule (J), is a
combined unit: J = kg m2
/ s2
– This can also be written as J = N m

37. Still More Combined Units
• The USCS unit of energy is the (lb ft), a
combined unit without a special name.
• The metric unit of momentum is the (kg
m/s), a combined unit without a special
name.
• Area units are distance x distance, like m2
.
• Volume units are distance cubed, like m3
.

38. Temperature Units
• There are three commonly used temperature
scales: Fahrenheit (°F), Celsius (°C), and
Kelvin (K), these will be discussed in more
detail in chapter 7.
• The conversion formulae:
– C = (5/9) (F - 32)
– F = (9/5) C + 32
– C = K -273 K = C + 273

39. Practice Problems
11. Simplify (N m3
) / (kg m/s)
[N = kg m / s2
]
12. Convert 78°F into °C
[C = (5/9) (F - 32)]
13. Convert the answer in the previous
problem into K
[K = C + 273]

40. Answers
11. (N m3
) / (kg m/s) = (kg m4
/ s2
) / (kg m/s) =
m3
/s
12. C = (5/9) (78 - 32) = (5/9) (46) = 25.6
13. K = 25.6 + 273 = 298.6
There’s more?? Will this day never end?
Okay, maybe we should take a short break before
reviewing basic algebra and doing review for the
first exam.

41. Algebra Review
• Algebra is the manipulation of an equation
to solve for an unknown.
• The basic rule is that the equality remains
valid so long as you do the same thing to
both sides of the equation.
• Example: 4 x = 12, solve for x.
• Solution: Divide both sides by 4 and you
get x = 12/4 = 3

42. • Example: (3/x) = 15, solve for x.
• Solution: Multiply both sides by x, giving
15 x = 3, then divide both sides by 15,
x = 3/15 = 0.2
– Alternate solution 1. Think of the 15 as being
(15/1) and cross-multiply.
– Alternate solution 2. Again think of 15 as
(15/1) and ‘flip’ both sides [giving (x/3) =
(1/15), then multiply both sides by 3].

43. • Example: y4
= 1.6 x 108
, solve for y.
• Solution: Take the “fourth-root” of both
sides of the equation. This can be done by
using either a “x
y” button on your√
calculator or by raising both sides of the
equation by the (1/4) power.
y = (1.6 x 108
)1/4
= 112.5

44. Practice Problems
14. (1/x) = 16, solve for x.
15. x3
= 216, solve for x.
16. 20 x2
= 4000, solve for x.
Yeah, I’m getting tired too. So that’s enough
algebra. Now lets talk about the exam.