Tutorial for Dimensional Analysis

Why dimensional analysis? Dimensional analysis (also called factor label) is a method of converting from one set of units to another. Let's say you wanted to convert 2.0 miles into millimeters. You'd use dimensional analysis to do this. Remember, all we are doing is converting the units -- we are not actually changing the real value (the distance involved) of the number. We are changing the numeral -- the representation of that distance.

What is the setup?

2.0 mi * 5280 feet * 12 inches * 2.54 cm * 1 m * 10³mm =
1 mi 1 foot 1 inch 10²cm 1 m

3.2 x 10⁶mm

First, we start with the "given" -- the 2.0 miles -- never start with an identity statement.

Notice that we always write in the DENOMINATOR the unit that we want to cancel from the previous NUMERATOR.

Notice that we are using a series of IDENTITY statements. We use these:
5280 feet = 1 mi
12 inches = 1 foot
2.54 cm = 1 inch
10²cm = 1 m
10³mm = 1 m

Be sure not to say something silly like: 12 feet = 1 inch or 10³m = 1 mm

Look at the sig figs. The given (2.0 miles) has 2 SF. The only other factor which has SF is the identity statement, 2.54 cm = 1 inch. The 2.54 cm has 3 SF. So, the answer has to have 2 SF (the factor with the fewest).

Remember, you want to make sure that all the units cancel out and that the last unit (the one that you want to have in your answer) should be in the numerator. If it isn't, you've messed up somewhere.

We will always give you whatever identity statements that you need (with the exception of the 6 metric prefixes that we told you to memorize -- see #1 below.). For example, 5280 feet = 1 mile and 2.54 cm = 1 inch, etc.

What things do you absolutely have to remember?

1. You must memorize the 6 metric prefixes that we gave you in class. They are:

2. You must memorize the 3 metric base units. They are:

3. You must be able to use of the metric prefixes with the appropriate base to make an identity statement using positive exponents. For example,

4. You must know the rule for significant figures involving multiplication and division. That is, your answer has the same number of SF as the factor with the fewest.

5. You must know how to round.

For example, when the number that you must round is EXACTLY HALF, if the number to the left is even, you round DOWN. If the number to the left is odd, you round UP. Here are a couple of examples:

When the number is MORE than HALF, you round up. When the number is LESS than HALF, you round down. Here are a couple of examples:

6. You must know how to enter a string of numbers into your calculator. See the instruction booklet (usually in microscopic print!) that comes with your calculator. However, a couple of common things include:

a. The EE or EXP button means x 10ⁿ . So, if you want to enter 4.5 x 10⁵ into your calculator, you would push these buttons:

b. If you have a string of numbers in the DENOMINATOR and you want to have all of them divided into the numerator, be sure to press the "DIVIDE" key on your calculator. So, if you want to perform this calculation:

4.5 x 10⁵ x 58.9
60 x 4 x 9

Here are the numbers you would push on your calculator:

7. You need to know which factors involve COUNTED numbers vs. MEASURED numbers in order to count the number of significant figures correctly. Here is a good rule of thumb:

If the identity statement is from English <--> English, COUNTED and therefore, NO SF.
If the identity statement is from Metric <-->Metric, COUNTED and therefore, NO SF.

If the identity statement is from English<-->Metric, MEASURED and therefore, SF. Which part of the identity statement is measured? Well, think about it: 1 pound = 454 grams. Which is the MEASURED part? The 454 grams, right? How many SF? 3

8. You must know when you MUST use scientific notation. You MUST use scientific notation if the number is greater than or equal to 1000 or if the number is less than or equal to .001. You MAY ALWAYS use scientific notation if you wish. Scientific notation is written as a number that is greater or equal to 1 multiplied by some power of ten. Here are some correct and incorrect ways of writing scientific notation: