ARE THE REAL NUMBERS
|position in the sequence. (||3
|is more than||2
|and less than||5
Rational numbers have the customary meaning.
An irrational numbers can also have that meaning. But we cannot determine the order of irrational numbers from their names. Is more than or less tan? The only way to decide is to compare their rational approximations. And that will depend on the existence of a method, an algorithm, to actually produce one.
Specifically, we must be able to decide whether an irrational number is less than or greater than any rational number we specify. For it is the rational numbers whose order we know. Is the irrational number less than or greater than 2.71828103594612074?
We say, then, that the sentence "This is an irrational number" means:
|1)||This irrational number has a name; and|
|2)||we can decide whether it is less than or greater than any rational number we specify.|
Every irrational number, thenby which we mean every one that existswill thus have the customary meaning of the word number.
A continuum of numbers?
The concept of a continuum comes from geometry. A line is the classic example; it is a continuum of length; there is no smallest length; equivalently, there is no limit to the smallness of the difference between two lengths.
To create the theory of real numbers, the word continuum had to be given a completely different meaning. A continuum was defined as a set of what were called "points." Those abstract points were then graphically identified with the geometrical points on the x-axis. It was then stated as an axiom, "Corresponding to every point on a line, there is a real number." Since those points constituted a continuum—there is no limit to the smallness of the distance between two points—there should be no limit to the smallness of the difference between two real numbers.
But differences between names are not arbitrarily small. Although the reals are called "numbers," they clearly are not numbers in the conventional sense. What are they? They are points. Points masquerading as numbers.
There is a method, an algorithm, that allows us to construct as many decimal digits as we please for the irrational number π:
π = 3.141592653589793. . .
The symbol on the right with three dots (ellipsis) is called an infinite decimal. It signifies this sequence of rational numbers:
3.1, 3.14, 3.141, 3.1415, . . .
π is the limit of that sequence.
Abstracting from that, it was asserted that every real number could be symbolized by an infinite decimal.
.24059165378. . .
A decimal is a way of representing a number. And numbers have names. The name of this decimal --
-- is "Two thousand four hundred five ten-thousandths."
An infinite decimal however has no name. It is not that we will never
.24059165378. . .
finish naming it. We cannot even begin.
Finally, the English mathematician and father of computer science Alan Turing proved the following:
To compute the decimal expansion of a real number, it is possible to create an algorithm for only a countable number of them.
If it is not possible then to compute each next digit of what might appear as
.24059165378. . .
—then it does not signify a limit. We cannot place it with respect to order. It is not the symbol of a number.
What is it a symbol for? One of those points that compose a continuum. Although each infinite decimal is called a real number, each one is a symbol for—a mark that represents—a point. Hence the axiom that relates them—"Corresponding to every point on a line there is a real number, that is, there is a point on a line"—is a tautology; a true statement devoid of substance.
The obsession with a continuum of numbers seemed to be demanded by coördinate geometry. The x-axis was conceived to be composed of points, and to "every" point there must correspond a number. But nothing in the actual practice of calculus requires that. When we say that a variable approaches a limit, we mean as a sequence of rational numbers. When we evaluate a function, we name numbers, one at a time.
Thus if we say that the domain of f(x) = x2 is
−∞ < x < ∞,
we mean that x may take any rational or irrational value we name.
"Let x = 5." "Let x = −1." "Let x = ."
Upon naming the number, we may then say that it exists. It will exist at the moment we say, write, read, or hear its name. A specific number will exist when we are conscious of it.
"Do you mean to say that the number 100 does not exist until I say its name?" That is correct. And you have just done so.
Now, what is it that enables us to name a number? It is the decimal system of positional numeration. Together with the names of functions, that is our instrument of construction.
"Fifth root of 238,096,608,009,650,000,412."
Calculus gets along perfectly well without the idea of a continuum of numbers. Calculus requires only numbers with names.
When faced with what is continuous, the job of arithmetic is to come up with the name of a number to be its measure. Of those numbers—those names—there is no continuum. It will not be possible to name the length of every line. Equivalently, it will not be possible to assign a coördinate to every distance from 0. Time, distance, motion: they are continuous. Numbers are not. That is the tension between geometry and arithmetic, a tension realized by Pythagoras with his discovery of what we call the irrational—and he called "without a name" (alogos). That tension was brought to a head with the introduction of coördinate geometry, which has been the dominant methodology since the 17th century, and which of course we take for granted. Geometry is concerned with continuous objects, while the domain of arithmetic is numbers and their discrete names.
It should be no wonder, then, that neither a teacher nor a text can give an example of a variable approaching a limit continuously, but only as a sequence of discrete rational numbers. Why not? Because no such thing exists.
As for the term real number, it was coined by René Descartes in 1637. It was to distinguish it from an imaginary or complex number. Now, we can define a rational number, and they exist. An irrational number can be defined (not rational), and they will exist (). That original definition is perfectly clear and sufficient. A real number is what we call any rational or irrational number. And they will not—they need not—form a continuum.
Copyright © 2021 Lawrence Spector