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Rational and irrational numbers

ADJECTIVES MODIFY nouns. A short person, a tall person, a real person. We do not expect that an adjective will completely change the meaning of the noun; we expect the stability of language.  Yet when mathematics speaks of a real number these days, that adjective completely changes the expected meaning of the word number.

Let us begin with the natural numbers.  Each one has a defined name, a symbol, and a position in the sequence. ("5" comes before "6" and after "4.") Those same properties must belong to the numbers we need for measuring, the rationals and irrationals. Calculus is a theory of measuring.

It is the name, symbol, and sequential position of
what we call a number
that allow us to count, measure, and calculate.

That is the customary and expected meaning -- use -- of the word number. That is certainly how we use that word in arithmetic. And arithmetic is never absent from calculus.  A limit is a number that we calculate and name.

Rational and irrational numbers

Every rational number has a defined name, a symbol, and a knowable

  position in the sequence. (  3
 is more than  2
 and less than  5
.)  Rational

numbers have the customary meaning of the word number.

Complex or imaginary numbers do not have that meaning, because we cannot place them with respect to order. They are algebraic entities. They obey the laws of computation, and they can be solutions to an equation.

Irrational numbers can also have that customary meaning.  Apart from unique irrationals such as π and e, the names and definitions of the irrationals come from the categories of functions:  roots, sines, arcsines, logarithms, and so on.

But we cannot determine the order of irrational numbers from their

  names. Is Irrationals more than or less tan  5π
?  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?

For example,

1.414213562373095 < Square root of 2 < 1.414213562373096.

There is a procedure that enables us to calculate as many digits as we please of Square root of 2.  Therefore we can place it with respect to order. That guarantees that it is a number. For if it did not have that customary meaning, it would be of no use to calculus, geometry, or science.

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.

It should not be unwarranted to say that if a theory of irrationals does not satisfy those conditions, then it lacks a firm foundation.

(See Kronecker's Algorithmic Mathematics.)

Every irrational number, then -- by which we mean every one that exists -- will thus have the customary meaning of the word number.

An arithmetical continuum?

The concept of a continuum comes from geometry.  A line is the classic example. It is a continuum of length. The job of arithmetic when confronted with what is continuous is to come up with the name of a number to be its measure, relative to a unit of measure.

Real numbers?

In coördinate geometry, we measure length as the distance from 0 along the x-axis.  That distance will be the x-coördinate of the point P.

Now since length is continuous, it was thought that the values of x should reflect that by being a continuum of numbers. That is, to every point on the x-axis -- every distance from 0 -- there should correspond a number.

Can the concept of a continuum be applied to numbers? Or is that a figment of the imagination? For if the word number is to retain its customary meaning, implying both a name and a symbol, then such a continuum is impossible. We cannot name every element in a continuum. Names are discrete.  A continuum of names -- the differences between them being arbitrarily small -- is an absurdity.

There is no arithmetical continuum.

(That simple argument is called the semantic rejection.)

Time, distance, motion 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.  A continuum of the numbers we need for measuring and that corresponds to an actual continuum -- time, distance, motion -- does not exist.

Whether it is even necessary is another question.

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.

The real numbers

The term real number 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 (Square root of 2).  It is perfectly clear, then, when by a real number we mean any rational number or any irrational number that exists. The word number has its customary meaning. And they will not form a continuum.

Yet it is now claimed that the real numbers are a continuum; that they are like the points on a line, and corresponding to every point there is a real number. To claim that, the word number had to be given a completely different meaning -- having nothing to do with measuring. What distinguishes that meaning is that, in addition to the customary rational and irrationals, there are now "numbers" with no names. In fact, the reals will be teeming with nameless "numbers." Otherwise, they could not fill out a continuum.

Such "numbers" clearly were not intended to be useful. Something without name or symbol cannot obey laws of computation. And they cannot be solutions to an equation. They are the "numbers" that truly deserve to be called imaginary.

Infinite decimals

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 is called an infinite decimal.  It represents 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, and especially an irrational, could be symbolized by an infinite decimal.

To actually construct a decimal expansion, of course, there must be an algorithm, a rule. But if there are to be rules for computing a continuum of numbers, then there must be a continuum of rules -- the differences betweeen which can be made as small as we please. Again that is absurd. Rules are discrete.

If it is not possible then to compute each next digit of what might appear as

.24059165378. . . ,

then it does not signify a limit. What is more, we cannot place it with respect to order.  It is not the symbol of a number.

In fact, 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.

THIS WHOLE PROBLEM of a continuum of numbers began with the assumption, the concept, that a line -- the x-axis -- is composed of points. But does calculus really require that? Or is "point" simply the word we use to indicate to a specific place, such as the boundary of an interval or where two lines meet? Points are individuals, and we indicate them one at a time. That is what we do. And that is all we need to mean when we say that that point exists.

(We may say there are an infinite number of points on a line, which is a brief way of saying that there is no limit to the number of points we could indicate.)

Again, the obsession with an infinity of points and a continuum of numbers seemed to be demanded by coördinate geometry. To "every" point on the x-axis there should correspond a number that is its coördinate. But nothing in our experience requires that. When we let a variable approach a limit or do a calculation, we name a number. That is all anyone has ever done or will ever need to. A limit is a number that we name.

By expressions such as "all" values -- or "all" anything -- it is sufficient to mean all that we name or produce.  We have the clearest example of that in Euclid's Elements, in which statements with the word "all" or "every" -- such as "All right angles are equal" -- refer to all that exist; that is, all we have actually drawn.

When a number exists

We have the decimal system of positional numeration. Together with the names of functions, that is what enables us to name any number. That is our instrument of construction.

A number -- that individual -- will exist, we assert, at the moment it is named. It will exist in the most actual sense of the word.  Naming is how we produce a number, how we bring a verbal or written symbol for that idea into this world. For if we cannot bring an idea into this world -- if we cannot experience it -- then it is nothing but an idea, which is to say, a fantasy.

("Do you mean to say that the number 100 does not exist until I name it?" That is correct, and you have just named it.)

Opposing that is the idea of an actual infinity -- an infinite collection of names and numerals: "The natural numbers," "The real numbers" -- every one of which is imagined to exist now. Although that may exist as an idea, an image in our mind, we cannot experience an infinity of anything. "The natural numbers," "The real numbers" exist only as names. They are kinds of numbers. To say that there are an infinite number of them does not lead to any consequences, and therefore is not necessary. Calculus has need of numbers only as we name them and bring them into this world.

It is pointless to do with more what can be done with less.

Occam's razor

At first, the infinite was a philosophical and religious concept. As for the use of that word in calculus, it is completely different from its definition in the theory of infinite sets. Here is how the "Prince of Mathematicians," Carl Friedrich Gauss, put it: "I protest against the use of an infinite quantity as something completed; this is never allowed in mathematics. The infinite is only a manner of speaking, the true meaning being a limit to which certain ratios can come as near as desired, while others are permitted to increase without bound."

To summarize:  For numbers to be useful to calculus and science, they must have names; the word number must have its customary meaning. As for a "real number," the original definition is perfectly clear and sufficient. It is what we call any rational or irrational number. Any definition that defines them so that they form a continuum, completely departs from that meaning, and has nothing to do with measuring -- nor was it ever intended to. That theory of real numbers belongs to 19th century modernism, a movement which sought "freedom" from the values of the past  and from what was accessible only to the many. Those real numbers are an abstract creation; a kind of logical sport; and the most prominent current example of fantasy mathematics.

End of the lesson

Appendix 2:  Is a line really composed of points?

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Copyright © 2015 Lawrence Spector