Geography Reference
In-Depth Information
Continuous: A billiard ball on a table has, at a particular moment in time, a certain position. When it
moves, its velocity, acceleration, jerk, direction, and so on, are vital to the outcome of its ending position,
and the ending positions of other balls. The smallest difference in position, velocity, and spin can make a
major difference in whether a ball drops into a pocket or not.
Continuous phenomena are characterized by the following:
1. The existence of an infinite number of states over independent variables, for instance, time.
2. When there is a finite but extremely small (infinitesimal) difference between values of an indepen-
dent variable, there is at most an infinitesimal difference in one or more dependent variables.
(A served tennis ball will change position very slightly in a fraction of a second.)
3. No matter how carefully measurements are undertaken, the state of the system can never be
determined exactly.
Discrete phenomena are characterized by the following:
A finite number of states (There are only so many combinations of pieces in positions on a chess board.)
❏
The smallest possible difference in an independent variable may result in a significant finite differ-
ence between states. (If “move number” increases by one, the positions of pieces on the chessboard
will be in a distinctly different state.)
❏
The exact state of the system can be determined.
❏
To further illustrate the difference between continuous and discrete phenomena, let's look at graphs of
each. Figure 6-1 might illustrate the amount of water in a stoppered sink as it is being filled from a faucet,
using time as the independent variable. Note the connectedness of the line.
Figure 6-2 shows the number of students “in” a classroom as they enter in the minutes before the class
starts. Since students come in packages of one, you see jumps in the graph.
With discrete phenomena, exactness is possible. With continuous phenomena, it is not. A basket may
contain exactly eight eggs. A bathtub cannot contain exactly eight liters of water.
Table 6-1 shows some examples of continuous and discrete phenomena. We might argue over some of
the categorizations. Further, discrete phenomena may have continuous parts and vice versa. But you will
probably get the idea from the list.
With our languages, we respect the difference between continuous and discrete. “How much” applies
to continuous things. “How many” applies to things of a discrete nature. We wouldn't speak about an
amount of votes—we would speak of a number of votes.
If we can use computers with enough precision to represent the world, does it matter that they are
discrete machines and that that world is a mixture of continuous and discrete? (Can you tell if recorded
music comes from a CD or an LP? Actually, some people can.) Is there an impact in using discrete
machines to represent continuous phenomena? Usually only a little—but sometimes a lot. The idea of this
section is to be sure that you understand that the potential exists for inaccuracies and errors when we use
completely discrete machines to represent continuous phenomena. Punch line: computers are completely
discrete machines!
Search WWH ::
Custom Search