The Evolution of the

R E A L  N U M B E R S

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A NATURAL NUMBER is a collection of indivisible ones.

Natural numbers

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 as two lengths.



Is one length necessarily a multiple of the other, a part of it, or parts of it?  Will lengths have the same ratio to one another as natural numbers?

To take up this question, we must explain what we mean by continuous versus 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 is indivisible, in the sense that if it is divided, then what results will not be that unit -- it will not have that same name -- any more. Half a person is not also a person.

We count things that are discrete.  One person, two, three, four, and so on.  What is more, a collection of discrete units will have only certain parts (Lesson 1).  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

The distance between A and B

made up of indivisible units. There is nothing to count. It is not a number of anything.  We say instead that it is a continuous whole.  That means that as we go from A to B, the line "continues" without a break.

Since the length AB is continuous, not only could we take half of it, we could take any part we please -- a tenth, a hundredth, or a billionth. And most important, any 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 box of chocolates -- it will always have a limit, namely one chocolate.

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 as the measure of things that are

A number line

continuous.  This gives rise to the "fractions." We do not need fractions for counting.  We need them for measuring; for assigning a number as the size of something that 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, which is a length, 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. We can imagine half of that distance, or a third, or a fourth, and so on.

c)  A bag of apples.   Discrete

d)  Applesauce.   Continuous!

e)  A dozen eggs.   Discrete

f)  60 minutes.

Continuous. Our idea of time, like our idea of distance, is that there is no smallest unit.

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, a circle is discrete; half a circle is not also a circle.

i)  The volume of a sphere.

As volume, it is continuous. As a form, a sphere is discrete.

j)  A gallon of water.

Continuous. We imagine that we could take any part.


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. There is no limit to the smallness of the differences between shapes.

n)  The evolution of biological forms; that is, from fish to man
n)   (according to the theory).

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.

End of Section

Next Topic:  Fractions

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