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CONTINUOUS VERSUS DISCRETE
BY A NATURAL NUMBER, we have meant a collection of indivisible
ones. And we have seen that we can always express in words the ratio of any two of them. A historical question has been whether it is possible to express the ratio of things that are not natural numbers, such a two lengths.
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Is one length necessarily a multiple of the other, a part of it, or parts of it? Will lengths have the same ratio as natural numbers?
To take up this question, we begin by explaining what we mean by continuous and discrete.
Now, half a chair is not also a chair; half a tree is not also a tree; and half an atom is surely not also an atom. A chair, a tree, and an atom are examples of a discrete unit. A discrete unit has no part. It is indivisible, in the sense that if it is divided, then what results will not be that unit, that thing, any more -- there is no such thing as half a person. A collection of discrete units therefore will have only certain parts. Ten people can be divided only in half, fifths, and tenths. You cannot take a third of them!
But consider the distance between A and B. That distance is not
made up of discrete units. It is not a number of anything. We say, instead, that it is continuous. As we go from A to B, there is no break.
(Continuous does not mean stretches on and on -- "continues"; it means not composed of discrete units. Thus if you study continuously for an hour, you do not take a break.)
Since the distance AB is continuous, not only could we take half of it, but we could take any part we please -- a tenth, a hundredth, or a billionth -- because AB is not composed of units! And most important, every part of AB, however small, will still be a length.
What is continuous has no limit to smallness. But if we keep dividing a natural number -- e.g. a bag of M&M's -- it will always have a limit; namely, one unit, one M&M.
This distinction between what is continuous and what is discrete makes for two aspects of number; namely, number as discrete units -- the natural numbers --and number itself as continuous. For we can imagine
1 to be like a length, and so it will have any part. Half, third, fourth, fifth, millionth.
This gives rise to the "fractions." We do not need fractions for counting. We need them for measuring. Fractions belong to what is continuous.
Problem 1.
a) Into which parts could 6 pencils be divided?
Halves, thirds, and sixths.
b) Into which parts could 6 meters be divided?
Any parts. 6 meters are continuous.
Problem 2. Which of these is continuous and which is discrete?
a) A stack of coins Discrete
b) The distance from here to the Moon.
Continuous. Distance is length, which is continuous. There is no limit to its smallness.
c) A bag of apples Discrete
d) Applesauce Continuous!
e) A dozen eggs Discrete
f) 60 minutes Continuous. Time is continuous.
g) Pearls on a necklace Discrete
h) The area of a circle As area, it is continuous; half an area is also an area. But as a form, it is discrete; half a circle is not also a circle.
i) The volume of a sphere As volume, it is continuous. As a form, it is discrete.
j) A gallon of water.
Continuous. We imagine that we could take any part.
But
k) Molecules of water.
Discrete. In other words, if we could keep dividing a quantity of water, then ultimately, in theory, we would come to one molecule. If we divided that, it would not be water any more!
l) The acceleration of a car as it goes from 0 to 60 mph Continuous. The speed is changing continuously.
m) The changing shape of a balloon as it's being inflated Continuous. The shape is changing continuously.
n) The evolution of biological forms; that is, from fish to man What do you think? Was it like a balloon being inflated? Or was each new form discrete?
o) Sentences Discrete. Half a sentence is surely not also a sentence.
p) Thoughts Discrete. (Half a thought?)
q) The names of numbers
Surely, the names of anything are discrete. Half a name makes no sense.
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