Dimensional Analysis:

This process can be used to assist with a derivation or to check the result of a calculation. It is based on the principle that units can be treated as algebraic quantities. Here are some things you should keep in mind as you do problems:

quantities can be added or subtracted only if they have the same units
terms on both sides of an equation must have the same units

Factor Analysis:

Factor analysis is the process of using the units to determine whether to multiply or divide by a conversion. For example, to find the number of atoms in 45 pounds of magnesium, first set up a series of parentheses containing the units necessary to get the proper units in the result. (Here's an example.)  Some experience and knowledge is necessary also! In this case, you also need to know about Avogadro's number (the number of atoms per mole), what a mole is, and the units of the masses in the periodic table (Mg is 24.305 g/mole from Appendix A - page A.5).

Can you see how each of these terms has been chosen? The units divide out to give the proper units for the result. The units tell us how to put in the numbers:

After doing a calculation, you should look at the result! If this answer had been a small number, I'd be sure that I had made an error. If you're trying to find the mass of a car, and you obtain something on the order of the mass of the sun, you had better be aware that you have something wrong!

Here are two more examples.

Significant Digits:

Tthe rules for significant digits should be followed, especially if you plan on becoming a scientist. On the other hand, for most engineers (initial calculations anyway) 3-digit accuracy is sufficient.  Suppose you are working with data given to three significant digits.  You should:

carry through all intermediate calculations to five digits to avoid round off error
round off the result to three digits

The zero in "103 m/s" is significant, since the zero tells us that it is not a "1" or a "7", for example. The zero in the last example is significant for the same reason. The zeros before the decimal are not significant; they are there by custom to help people spot the decimal point and thus avoid errors. The zero after the decimal in 0.0562 m is also not significant, since it must be there to place the decimal. If you are in doubt, express it in scientific notation:

You should use scientific notation rather than lots of zeros.

Symbolic Notation:

proportional to
much greater than >>
approximately equal to
defined to be equal to
change in a quantity (change is always final value - initial value)
sum of items
absolute value
magnitude of a vector

sum of forces