CHAPTER 3
Chapter 3 - The Mathematical function and its Graph
Section 3.3 - Dimensions
The mathematical function is a relationship that defines a situation in terms of
a set of interacting conditions. Neither the situation nor the condition can exist
independent of each other. The function states how a set of fixed conditions come together
to define a situation. In mathematics, conditions are denoted by the term, dimension.
Before one can understand the concept of the function, we need to explain what a dimension
is.
In science there are four fundamental dimensions used to
describe any physical situation in the natural world. They are mass,
length, temperature and time. While
mass, length and temperature refer to static situations, time is a dynamic
dimension used to describe changes that define
an action.
Think of time as a passage of events. For example a
summer spent doing nothing but eating, sleeping and watching TV would seem to
have gone by in a few days. On the other hand a summer spent traveling around
the world would seem to have lasted forever. If we assume each vacation to have
been 80 days, ones memory of the uneventful vacation consists only of a few
events that could have been done in a day whereas the vacation around the world
consisted of a plethora of events, such that relative to the boring summer,
they took longer to finish.
For time to pass something must change with respect to
itself. The units of time, the second, is nothing more than a reference for any
change. If one second refers to a pendulum’s swing through one arc, then any changing dimension can be measured
relative to the standard of one second. For example a moving car’s distance
(length) from a reference point is constantly changing. The changing distance
can only be measured with respect to time.
It is important to understand that time is the dimension for actions, while length, mass
and temperature refer to static conditions. If length, mass or temperature were changing
with respect to itself then the change would
have to be analyzed with respect to time. To summarize, for time to pass some action must
occur and for an action to occur some dimension must be changing with respect to
itself.
Length, mass, temperature, and time refer to the simplest
absolute dimensions the physical world can be reduced to. Other dimensions derived
from these fundamental
dimensions include, force (dependent on mass), stress (dependent on force),
elasticity (dependent on force and stress), velocity (dependent on length and
time), kinetic energy (dependent on mass and velocity), etc.
Many derived dimensions are dependent on other derived
dimensions such as kinetic energy is dependent on velocity which is dependent
on time. But what about people, dollars, etc. Clearly these are also measurable
conditions; however, they can not be easily expressed in terms of the
fundamental dimensions of mass, length, time, and temperature. Since they are unique dimensions we need to come up
with a consistent definition for a dimension that applies to all fundamental,
derived, and unique dimensions.
A dimension is simply a quantifiable condition that describes
a situation. In science we always study situations where external factors are
removed from the analysis. It is the objects that remain along with their
properties that become our focus of study. A dimension is a measurement of a
property that is relevant to the situation being analyzed. By themselves,
dimensions are meaningless. They must refer to certain conditions specific to
the situation being studied. One cannot refer to just mass or length. One has
to specify mass and length of what part of the system? The units of the various
dimensions, meter, seconds, Celsius, kg, serves as standards to measure the
dimension relative to.
Understanding which dimensions to include in your
scientific analysis all depends on the type of situation being studied. The
more familiar you are with the fundamental and derived dimensions, the easier
it will be for you to understand which dimensions are significant and how they
define the situation.
