1. Dimensional Analysis (a.k.a. unit conversions)
Dr. Chin Chu

2. In science, numbers with no units do not mean anything!
Humans always want to quantify things.
Instruments were invented to accomplish the task.
How to express the results from those measurements?
3.25
meters
numerical value
unit
In science, numbers with no units do not mean anything!

3. FACTOR-LABEL METHOD (a.k.a. DIMENSIONAL ANALYSIS)
Historically, many measuring systems were developed based on differing units.
How to convert between different units?
Example: How many yards are there in 3.25 meters?
FACTOR-LABEL METHOD
(a.k.a. DIMENSIONAL ANALYSIS)
A math protocol used to convert between different units.
Need:
Desired quantity.
Given quantity.
Conversion factors.

4. Conversion Factors: valid relationships or equities expressed as a fraction.
Example:
1 meter = yard
1 meter
yard
1 meter
yard
= 1
Conversion Factors
Note:
Basic conversion factors based on equivalency relationships always equal to 1.
They generally come in pairs with two units trading places.

5. STEPS INVOLVED IN FACTOR-LABEL METHOD:
Write down the given quantity /units.
Multiply given quantity/unit by 1’s.
Substitute 1’s with conversion factor(s) that eliminate unwanted units and achieve desired units.
Complete the math.

6. First write down the given quantity/unit
Factor label example 1
Q - How many yards are in 3.25 meters? (note: 1 m = yard)
3.25 meter
First write down the given quantity/unit

7. Factor label example 1 3.25 meter = 3.25 meter x1x1….
Next, multiply the given quantity/unit by 1’s.
Remember, anything multiply by 1 always stays the same!
Actually, the 1’s could go on forever!

8. Remember, those conversion factors always equal to 1.
Factor label example 1
3.25 meter = 3.25 meter x1x1….
1 meter
yard
= 1
Remember, those conversion factors always equal to 1.
So the 1’s in the above equation could be substituted by those conversion factors. Here, only one substitution is sufficient.
Which one?

9. Choose the one that will cancel out the meter.
Factor label example 1
3.25 meter = 3.25 meter x1
Choose the one that will cancel out the meter.
1 meter
yard
= 1
1 meter
yard
= 1

10. Bravo! Mission accomplished.
Factor label example 1
Q - How many yards are in 3.25 meters? (note: 1 m = yard)
1 meter
yard
3.25 meters =
3.25 meter x
Finish the math.
= 3.55 yard
Bravo! Mission accomplished.

11. Factor label example 1 = 3.55 yard 3.25 meter 1.0936 yard 3.25 meter =
THE MONKEY BAR:
3.25 meter
1 meter
yard
3.25 meter =
= 3.55 yard
If you are really good at this, an equivalent but simpler protocol could be used.

12. A bit more complicated than Example 1. But the approach is the same.
Factor label example 2
Q - How many seconds are there in a week? (note: 1 week = 7 days; 1 day = 24 hours; 1 hour = 60 minutes; 1 minute = 60 seconds)
1 week
1 week
7 days
1 day
24 hours
1 hour
60 min.
1 min.
60 sec.
1 week =
= 604,800 sec.
A bit more complicated than Example 1. But the approach is the same.

13. METRIC PREFIX SYSTEM
Often measurements return results of exceeding large or small values.
Examples:
meter
2,345,600,000,000 dollars
How to make scientist’s life a bit easier?
Here comes the metric prefix system: k m
Prefix
(conversion factor)
unit

14. Metric Prefixes
Note: Prefixes in red color need to be mastered without aid of the table.

15. What does the numerical values associated with metric prefixes mean?
Unit with prefix
Unit with prefix
Conversion Factors:
1 km = 103 m
1 km
103 m or
Factor
It provides information regarding the numerical relationship between unit with the prefix and unit without the prefix.
What does the numerical values associated with metric prefixes mean?

16. 4 Tm = ? nm
4 Tm
1 Tm
1012 m
10-9 m
1 nm
4 Tm =
= 4 x 1021 nm
Common strategy:
Convert the given quantity/unit to unit without prefix.
Convert the unit without prefix to the desired unit.
Factor-Label method could be readily applied to conversions between metric units.