The Use of Units

Numbers are meaningless without the correct use of units. It makes no sense to say "the distance from my house to school is two", unless we follow that statement with "miles" or "kilometers" or whichever unit makes the statement true. We will distinguish between dimension, the abstract quality of a measurement without scale:

and unit, the quality of a number which specifies a previously agreed upon scale: While dimensional arguments are of primary importance in figuring out what kind of algebraic equation we need to solve a problem, units are necessary to actually solve the problem.

If we are simply adding quantities, such as

1 cup + 1 cup = 2 cups,
the units are trivial, but for a very important reason:
you cannot add, subtract or equate two quantities which do not have the same units.
For instance, it makes no sense to try to add 1 cup to 2 pints; it is first necessary to convert both quantities to a common set of units:
  1. 1 cup = 8 ounces
  2. 1 pint = 16 ounces
  3. 2 pints = 32 ounces
so
1 cup + 2 pints
is the same as
8 ounces + 32 ounces = 40 ounces.
There are implicit multiplications and divisions which occur when we convert one unit to another:
2 pints * (16 ounces per pint) = 32 ounces.
The quantity
16 ounces per pint
is called a conversion factor, and is written as a quotient:
16 ounces / pint
(the word "per" implies division).

In most of the computations in this chapter, we will need to multiply or divide more than one conversion factor in order to solve the problem at hand. It is both impractical and unnecessary to memorize equations for every single problem we might wish to solve; we can use the units of the conversion factors to create the appropriate equation when we need it (this is part of the mathematical sophistication which you have been gaining in this course). As an example, let us compute the number of teaspoons in a gallon. The conversion factors which we have to work with are:

We can express this problem as an equation:
? teaspoons = 1 gallon
The left hand side (LHS) of this equation has units of teaspoons; in order to change the units of the right hand side (RHS) to teaspoons, we must multiply the RHS by one or more conversion factors. Since the units of the RHS are gallons, we first multiply the RHS by a conversion factor which has gallons in the denominator:
1 gallon * 4 quarts / gallon = 4 quarts
The units in this equation work exactly like common factors in the numerator and denominator of a fraction: just as
15 / 6 = 5 * 3 / ( 3 * 2 ) = 5 / 2,
where we have used the fact that
3 / 3 = 1,
the units of gallons in the numerator and gallons in the denominator have "cancelled":
gallons / gallons = 1.
We proceed from here in exactly the same fashion: since we now have units of quarts on the RHS, we multiply by a conversion factor which has quarts in the denominator:
1 gallon * 4 quarts / gallon * 2 pints / quart = 8 pints,
where just as we used
gallons / gallons = 1,
in the first conversion, we now use
quarts / quarts = 1,
and the units of the RHS are now pints. The procedure from this point should be obvious:
  1. multiply by 2 cups / pint to get cups
  2. multiply by 8 ounces / cup to get ounces
  3. multiply by 2 tablespoons / ounce to get tablespoons
  4. multiply by 3 teaspoons / tablespoon to get teaspoons
At this point, we have achieved our goal of having both sides of the equation in the same units: both the LHS and the RHS are measured in teaspoons. The final computation is:
1 gallon * (4 quarts / gallon) * (2 pints / quart) * (2 cups / pint) * (8 ounces / cup) *
(2 tablespoons / ounce) * (3 teaspoons / tablespoon)
Notice how each multiplication by a conversion factor changes the units from those in the denominator to those in the numerator. Since we have established that both sides of the equation have the same units, the numerical computation can now be performed and we find that there are
1 * 4 * 2 * 2 * 8 * 2 * 3 = 768 teaspoons in a gallon.

We will have many occasions to do similar dimensional or unit analyses in the future, although you should keep in mind that we may sometimes be dividing by conversion factors as well as multiplying by them. For instance, if we have 6 teaspoons and need to know how many tablespoons that is:

? tablespoons = 6 teaspoons
we see that to get tablespoons on the RHS, we need to divide by our conversion factor of
3 teaspoons / tablespoon
(since 1 / ( a / b ) = b / a). The computation is then
6 teaspoons / (3 teaspoons / tablespoon) = 6 / 3 tablespoons = 2 tablespoons.

Now that we understand the basic procedure, let us investigate the geometry of hard disks in a computer.


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©2002, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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