Last Update: 29 June 2013
Note: or material is highlighted |
Rather, it's a description of a set of branches of math that all have in common the feature that they are "discrete" rather than "continuous".
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Here's what this means:
I.e., continuous sets have no gaps.
Set | Name | Members | Has gaps? | Countable? | Dense? | Continuous? | Discrete? |
---|---|---|---|---|---|---|---|
W | positive naturals | 1,2,3… | yes | yes | no | no | yes |
N | naturals | 0,1,2,3… | yes | yes | no | no | yes |
Z | integers | …–3,–2,–1,0,1,2,3… | yes | yes | no | no | yes |
Q | rationals | fractions/repeating decimals | yes | yes | yes | no | ??? |
R | reals | all decimals, including non-repeating | no | no | yes | yes | no |
From this, we can see that W, N, and Z are clearly "discrete" and that R is clearly not discrete.
But Q stands somewhere in the middle:
The following are especially good:
"Continuous vs. Discrete Mathematics
The world of mathematics can be divided roughly into two realms: the continuous and the discrete. The difference is nicely illustrated by wristwatches. Continuous mathematics corresponds to analog watches - the kind with separate hour, minute, and second hands. The hands move smoothly over time. From an analog watch perspective, between 12:02 P.m. and 12:03 P.m. there are infinitely many possible different times as the second hand sweeps around the watch face. Continuous mathematics studies concepts that are infinite in scope, where one object can blend smoothly into the next. The real-number system lies at the core of continuous mathematics and - just like the watch - between any two real numbers, there is an infinity of real numbers. Continuous mathematics provides excellent models and tools for analysing real-world phenomena that change smoothly over time, including the motion of planets around the sun or the flow of blood through the body.
Discrete mathematics, on the other hand, is comparable to a digital watch. On a digital watch, there are only finitely many possible different times between 12:02 P.m. and 12:03 P.m. A digital watch does not acknowledge split seconds! There is no time between 12:02:03 and 12:02:04. The watch leaps from one time to the next. A digital watch can show only finitely many different times, and the transition from one time to the next is sharp and unambiguous. Just as the real-number system plays a central role in continuous mathematics, integers are the primary tool of discrete mathematics. Discrete mathematics provides excellent models and tools for analysing real-world phenomena that change abruptly and that lie clearly in one state or another. Discrete mathematics is the tool of choice in a host of applications, from computers to telephone call routing and from personnel assignments to genetics.
Edward R. Scheinerman, Mathematics, A Discrete Introduction (Brooks/Cole, Pacific Grove, CA, 2000): xvii–xviii."