As we go to more complex physics, we can add additional dimensions.
An important thing to realize is that at our current state of knowledge it appears that the choice of dimension is arbitrary. We typically measure distance and time, but we now know that there is a fundamental speed associated with the universe: the speed of light, c. This speed is an invariant (in empty space) -- the same for all observers. We could therefore choose c = 1 (with no units) and take our measure of distance to be the amount of time it takes light to travel that distance. We could thus measure all of our distances using time units. (1 meter corresponds to about 1 nanosecond.)
Because there are a number of fundamental universal constants (the speed of light, c, Planck's constant, h, Newton's gravitational constant, G, and the charge on the electron, e) all of the quantitites we typically choose to compare to arbitrary standards could be "counted" in terms of fundamental units. But at present, it's impossible to count the acutal number of excess electrons on a flannel shirt just coming out of the dryer. It's even worse to try to imagine "counting" a distance in terms of the number of Planck lengths (~10^{-33} m) that fit into it. (Try the problem on constructing these fundamental units of measurement.)
As a result, our use of dimensions is a convenience. We choose dimensions that match what we can measure at a given time in a given situation. This helps us both to find mistakes in our calculations and to generate new hypotheses for plausible new physics.
Note that the term "dimension" refers to the general fact that we have an arbitrary choice to make when we define a particular kind of measurement. Thus, a "length" is a dimension. The specific choice we make to measure length with -- feet, furlongs, meters, or light years -- is referred to as a unit.
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Last revision 11. September, 2005.