CHAPTER VII: RATIONAL, REAL, AND COMPLEX NUMBERS

[page 63]

We have now seen how to define cardinal numbers, and also relation-numbers, of which what are commonly called ordinal numbers are a particular species. It will be found that each of these kinds of number may be infinite just as well as finite. But neither is capable, as it stands, of the more familiar extensions of the idea of number, namely, the extensions to negative, fractional, irrational, and complex numbers. In the present chapter we shall briefly supply logical definitions of these various extensions.

One of the mistakes that have delayed the discovery of correct definitions in this region is the common idea that each extension of number included the previous sorts as special cases. It was thought that, in dealing with positive and negative integers, the positive integers might be identified with the original signless integers. Again it was thought that a fraction whose denominator is 1 may be identified with the natural number which is its numerator. And the irrational numbers, such as the square root of 2, were supposed to find their place among rational fractions, as being greater than some of them and less than the others, so that rational and irrational numbers could be taken together as one class, called “real numbers.” And when the idea of number was further extended so as to include “complex” numbers, i.e. numbers involving the square root of −1, it was thought that real numbers could be regarded as those among complex numbers in which the imaginary part (i.e. the part [page 64] which was a multiple of the square root of −1) was zero. All these suppositions were erroneous, and must be discarded, as we shall find, if correct definitions are to be given.

Let us begin with positive and negative integers. It is obvious on a moment’s consideration that +1 and −1 must both be relations, and in fact must be each other’s converses. The obvious and sufficient definition is that +1 is the relation of n+1 to n, and −1 is the relation of n to n+1. Generally, if m is any inductive number, +m will be the relation of n+m to n (for any n), and −m will be the relation of n to n+m. According to this definition, +m is a relation which is one-one so long as n is a cardinal number (finite or infinite) and m is an inductive cardinal number. But +m is under no circumstances capable of being identified with m, which is not a relation, but a class of classes. Indeed, +m is every bit as distinct from m as −m is.

Fractions are more interesting than positive or negative integers. We need fractions for many purposes, but perhaps most obviously for purposes of measurement. My friend and collaborator Dr A. N. Whitehead has developed a theory of fractions specially adapted for their application to measurement, which is set forth in Principia Mathematica.1 But if all that is needed is to define objects having the required purely mathematical properties, this purpose can be achieved by a simpler method, which we shall here adopt. We shall define the fraction m/n as being that relation which holds between two inductive numbers x, y when xn=ym. This definition enables us to prove that m/n is a one-one relation, provided neither m nor n is zero. And of course n/m is the converse relation to m/n.

From the above definition it is clear that the fraction m/1 is that relation between two integers x and y which consists in the fact that x=my. This relation, like the relation +m, is by no means capable of being identified with the inductive cardinal number m, because a relation and a class of classes are objects [page 65] of utterly different kinds.2 It will be seen that 0/n is always the same relation, whatever inductive number n may be; it is, in short, the relation of 0 to any other inductive cardinal. We may call this the zero of rational numbers; it is not, of course, identical with the cardinal number 0. Conversely, the relation m/0 is always the same, whatever inductive number m may be. There is not any inductive cardinal to correspond to m/0. We may call it “the infinity of rationals.” It is an instance of the sort of infinite that is traditional in mathematics, and that is represented by “∞.” This is a totally different sort from the true Cantorian infinite, which we shall consider in our next chapter. The infinity of rationals does not demand, for its definition or use, any infinite classes or infinite integers. It is not, in actual fact, a very important notion, and we could dispense with it altogether if there were any object in doing so. The Cantorian infinite, on the other hand, is of the greatest and most fundamental importance; the understanding of it opens the way to whole new realms of mathematics and philosophy.

It will be observed that zero and infinity, alone among ratios, are not one-one. Zero is one-many, and infinity is many-one.

There is not any difficulty in defining greater and less among ratios (or fractions). Given two ratios m/n and p/q, we shall say that m/n is less than p/q if mq is less than pn. There is no difficulty in proving that the relation “less than,” so defined, is serial, so that the ratios form a series in order of magnitude. In this series, zero is the smallest term and infinity is the largest. If we omit zero and infinity from our series, there is no longer any smallest or largest ratio; it is obvious that if m/n is any ratio other than zero and infinity, m/2n is smaller and 2m/n is larger, though neither is zero or infinity, so that m/n is neither the smallest [page 66] nor the largest ratio, and therefore (when zero and infinity are omitted) there is no smallest or largest, since m/n was chosen arbitrarily. In like manner we can prove that however nearly equal two fractions may be, there are always other fractions between them. For, let m/n and p/q be two fractions, of which p/q is the greater. Then it is easy to see (or to prove) that (m+p)/(n+q) will be greater than m/n and less than p/q. Thus the series of ratios is one in which no two terms are consecutive, but there are always other terms between any two. Since there are other terms between these others, and so on ad infinitum, it is obvious that there are an infinite number of ratios between any two, however nearly equal these two may be.3 A series having the property that there are always other terms between any two, so that no two are consecutive, is called “compact.” Thus the ratios in order of magnitude form a “compact” series. Such series have many important properties, and it is important to observe that ratios afford an instance of a compact series generated purely logically, without any appeal to space or time or any other empirical datum.

Positive and negative ratios can be defined in a way analogous to that in which we defined positive and negative integers. Having first defined the sum of two ratios m/n and p/q as (mq+pn)/nq, we define +p/q as the relation of m/n+p/q to m/n, where m/n is any ratio; and −p/q is of course the converse of +p/q. This is not the only possible way of defining positive and negative ratios, but it is a way which, for our purpose, has the merit of being an obvious adaptation of the way we adopted in the case of integers.

We come now to a more interesting extension of the idea of number, i.e. the extension to what are called “real” numbers, which are the kind that embrace irrationals. In Chapter I. we had occasion to mention “incommensurables” and their [page 67] discovery by Pythagoras. It was through them, i.e. through geometry, that irrational numbers were first thought of. A square of which the side is one inch long will have a diagonal of which the length is the square root of 2 inches. But, as the ancients discovered, there is no fraction of which the square is 2. This proposition is proved in the tenth book of Euclid, which is one of those books that schoolboys supposed to be fortunately lost in the days when Euclid was still used as a text-book. The proof is extraordinarily simple. If possible, let m/n be the square root of 2, so that m2/n2 =2, i.e. m2= 2n2. Thus m2 is an even number, and therefore m must be an even number, because the square of an odd number is odd. Now if m is even, m2 must divide by 4, for if m=2p, then m2=4p2. Thus we shall have 4p2 =2n2, where p is half of m. Hence 2p2=n2, and therefore n/p will also be the square root of 2. But then we can repeat the argument: if n=2q, p/q will also be the square root of 2, and so on, through an unending series of numbers that are each half of its predecessor. But this is impossible; if we divide a number by 2, and then halve the half, and so on, we must reach an odd number after a finite number of steps. Or we may put the argument even more simply by assuming that the m/n we start with is in its lowest terms; in that case, m and n cannot both be even; yet we have seen that, if m2/n2=2, they must be. Thus there cannot be any fraction m/n whose square is 2.

Thus no fraction will express exactly the length of the diagonal of a square whose side is one inch long. This seems like a challenge thrown out by nature to arithmetic. However the arithmetician may boast (as Pythagoras did) about the power of numbers, nature seems able to baffle him by exhibiting lengths which no numbers can estimate in terms of the unit. But the problem did not remain in this geometrical form. As soon as algebra was invented, the same problem arose as regards the solution of equations, though here it took on a wider form, since it also involved complex numbers.

It is clear that fractions can be found which approach nearer [page 68] and nearer to having their square equal to 2. We can form an ascending series of fractions all of which have their squares less than 2, but differing from 2 in their later members by less than any assigned amount. That is to say, suppose I assign some small amount in advance, say one-billionth, it will be found that all the terms of our series after a certain one, say the tenth, have squares that differ from 2 by less than this amount. And if I had assigned a still smaller amount, it might have been necessary to go further along the series, but we should have reached sooner or later a term in the series, say the twentieth, after which all terms would have had squares differing from 2 by less than this still smaller amount. If we set to work to extract the square root of 2 by the usual arithmetical rule, we shall obtain an unending decimal which, taken to so-and-so many places, exactly fulfils the above conditions. We can equally well form a descending series of fractions whose squares are all greater than 2, but greater by continually smaller amounts as we come to later terms of the series, and differing, sooner or later, by less than any assigned amount. In this way we seem to be drawing a cordon round the square root of 2, and it may seem difficult to believe that it can permanently escape us. Nevertheless, it is not by this method that we shall actually reach the square root of 2.

If we divide all ratios into two classes, according as their squares are less than 2 or not, we find that, among those whose squares are not less than 2, all have their squares greater than 2. There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. There is no lower limit short of zero to the difference between the numbers whose square is a little less than 2 and the numbers whose square is a little greater than 2. We can, in short, divide all ratios into two classes such that all the terms in one class are less than all in the other, there is no maximum to the one class, and there is no minimum to the other. Between these two classes, where √2 ought to be, there is nothing. Thus our [page 69] cordon, though we have drawn it as tight as possible, has been drawn in the wrong place, and has not caught √2.

The above method of dividing all the terms of a series into two classes, of which the one wholly precedes the other, was brought into prominence by Dedekind,4 and is therefore called a “Dedekind cut.” With respect to what happens at the point of section, there are four possibilities: (1) there may be a maximum to the lower section and a minimum to the upper section, (2) there may be a maximum to the one and no minimum to the other, (3) there may be no maximum to the one, but a minimum to the other, (4) there may be neither a maximum to the one nor a minimum to the other. Of these four cases, the first is illustrated by any series in which there are consecutive terms: in the series of integers, for instance, a lower section must end with some number n and the upper section must then begin with n+1. The second case will be illustrated in the series of ratios if we take as our lower section all ratios up to and including 1, and in our upper section all ratios greater than 1. The third case is illustrated if we take for our lower section all ratios less than 1, and for our upper section all ratios from 1 upward (including 1 itself). The fourth case, as we have seen, is illustrated if we put in our lower section all ratios whose square is less than 2, and in our upper section all ratios whose square is greater than 2.

We may neglect the first of our four cases, since it only arises in series where there are consecutive terms. In the second of our four cases, we say that the maximum of the lower section is the lower limit of the upper section, or of any set of terms chosen out of the upper section in such a way that no term of the upper section is before all of them. In the third of our four cases, we say that the minimum of the upper section is the upper limit of the lower section, or of any set of terms chosen out of the lower section in such a way that no term of the lower section is after all of them. In the fourth case, we say that [page 70] there is a “gap”: neither the upper section nor the lower has a limit or a last term. In this case, we may also say that we have an “irrational section,” since sections of the series of ratios have “gaps” when they correspond to irrationals.

What delayed the true theory of irrationals was a mistaken belief that there must be “limits” of series of ratios. The notion of “limit” is of the utmost importance, and before proceeding further it will be well to define it.

A term x is said to be an “upper limit” of a class α with respect to a relation P if (1) α has no maximum in P, (2) every member of α which belongs to the field of P precedes x, (3) every member of the field of P which precedes x precedes some member of α. (By “precedes” we mean “has the relation P to.”)

This presupposes the following definition of a “maximum”:—

A term x is said to be a “maximum” of a class α with respect to a relation P if x is a member of α and of the field of P and does not have the relation P to any other member of α.

These definitions do not demand that the terms to which they are applied should be quantitative. For example, given a series of moments of time arranged by earlier and later, their “maximum” (if any) will be the last of the moments; but if they are arranged by later and earlier, their “maximum” (if any) will be the first of the moments.

The “minimum” of a class with respect to P is its maximum with respect to the converse of P; and the “lower limit” with respect to P is the upper limit with respect to the converse of P.

The notions of limit and maximum do not essentially demand that the relation in respect to which they are defined should be serial, but they have few important applications except to cases when the relation is serial or quasi-serial. A notion which is often important is the notion “upper limit or maximum,” to which we may give the name “upper boundary.” Thus the “upper boundary” of a set of terms chosen out of a series is their last member if they have one, but, if not, it is the first term after all of them, if there is such a term. If there is neither [page 71] a maximum nor a limit, there is no upper boundary. The “lower boundary” is the lower limit or minimum.

Reverting to the four kinds of Dedekind section, we see that in the case of the first three kinds each section has a boundary (upper or lower as the case may be), while in the fourth kind neither has a boundary. It is also clear that, whenever the lower section has an upper boundary, the upper section has a lower boundary. In the second and third cases, the two boundaries are identical; in the first, they are consecutive terms of the series.

A series is called “Dedekindian” when every section has a boundary, upper or lower as the case may be.

We have seen that the series of ratios in order of magnitude is not Dedekindian.

From the habit of being influenced by spatial imagination, people have supposed that series must have limits in cases where it seems odd if they do not. Thus, perceiving that there was no rational limit to the ratios whose square is less than 2, they allowed themselves to “postulate” an irrational limit, which was to fill the Dedekind gap. Dedekind, in the above-mentioned work, set up the axiom that the gap must always be filled, i.e. that every section must have a boundary. It is for this reason that series where his axiom is verified are called “Dedekindian.” But there are an infinite number of series for which it is not verified.

The method of “postulating” what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil.

It is clear that an irrational Dedekind cut in some way “represents” an irrational. In order to make use of this, which to begin with is no more than a vague feeling, we must find some way of eliciting from it a precise definition; and in order to do this, we must disabuse our minds of the notion that an irrational must be the limit of a set of ratios. Just as ratios whose denominator is 1 are not identical with integers, so those rational [page 72] numbers which can be greater or less than irrationals, or can have irrationals as their limits, must not be identified with ratios. We have to define a new kind of numbers called “real numbers,” of which some will be rational and some irrational. Those that are rational “correspond” to ratios, in the same kind of way in which the ratio n/1 corresponds to the integer n; but they are not the same as ratios. In order to decide what they are to be, let us observe that an irrational is represented by an irrational cut, and a cut is represented by its lower section. Let us confine ourselves to cuts in which the lower section has no maximum; in this case we will call the lower section a “segment.” Then those segments that correspond to ratios are those that consist of all ratios less than the ratio they correspond to, which is their boundary; while those that represent irrationals are those that have no boundary. Segments, both those that have boundaries and those that do not, are such that, of any two pertaining to one series, one must be part of the other; hence they can all be arranged in a series by the relation of whole and part. A series in which there are Dedekind gaps, i.e. in which there are segments that have no boundary, will give rise to more segments than it has terms, since each term will define a segment having that term for boundary, and then the segments without boundaries will be extra.

We are now in a position to define a real number and an irrational number.

A “real number” is a segment of the series of ratios in order of magnitude.

An “irrational number” is a segment of the series of ratios which has no boundary.

A “rational real number” is a segment of the series of ratios which has a boundary.

Thus a rational real number consists of all ratios less than a certain ratio, and it is the rational real number corresponding to that ratio. The real number 1, for instance, is the class of proper fractions.

[page 73]

In the cases in which we naturally supposed that an irrational must be the limit of a set of ratios, the truth is that it is the limit of the corresponding set of rational real numbers in the series of segments ordered by whole and part. For example, √2 is the upper limit of all those segments of the series of ratios that correspond to ratios whose square is less than 2. More simply still, √2 is the segment consisting of all those ratios whose square is less than 2.

It is easy to prove that the series of segments of any series is Dedekindian. For, given any set of segments, their boundary will be their logical sum, i.e. the class of all those terms that belong to at least one segment of the set.5

The above definition of real numbers is an example of “construction” as against “postulation,” of which we had another example in the definition of cardinal numbers. The great advantage of this method is that it requires no new assumptions, but enables us to proceed deductively from the original apparatus of logic.

There is no difficulty in defining addition and multiplication for real numbers as above defined. Given two real numbers μ and ν, each being a class of ratios, take any member of μ and any member of ν and add them together according to the rule for the addition of ratios. Form the class of all such sums obtainable by varying the selected members of μ and ν. This gives a new class of ratios, and it is easy to prove that this new class is a segment of the series of ratios. We define it as the sum of μ and ν. We may state the definition more shortly as follows:—

The arithmetical sum of two real numbers is the class of the arithmetical sums of a member of the one and a member of the other chosen in all possible ways.

[page 74]

We can define the arithmetical product of two real numbers in exactly the same way, by multiplying a member of the one by a member of the other in all possible ways. The class of ratios thus generated is defined as the product of the two real numbers. (In all such definitions, the series of ratios is to be defined as excluding 0 and infinity.)

There is no difficulty in extending our definitions to positive and negative real numbers and their addition and multiplication.

It remains to give the definition of complex numbers.

Complex numbers, though capable of a geometrical interpretation, are not demanded by geometry in the same imperative way in which irrationals are demanded. A “complex” number means a number involving the square root of a negative number, whether integral, fractional, or real. Since the square of a negative number is positive, a number whose square is to be negative has to be a new sort of number. Using the letter i for the square root of −1, any number involving the square root of a negative number can be expressed in the form x+yi, where x and y are real. The part yi is called the “imaginary” part of this number, x being the “real” part. (The reason for the phrase “real numbers” is that they are contrasted with such as are “imaginary.”) Complex numbers have been for a long time habitually used by mathematicians, in spite of the absence of any precise definition. It has been simply assumed that they would obey the usual arithmetical rules, and on this assumption their employment has been found profitable. They are required less for geometry than for algebra and analysis. We desire, for example, to be able to say that every quadratic equation has two roots, and every cubic equation has three, and so on. But if we are confined to real numbers, such an equation as x2+1=0 has no roots, and such an equation as x3−1=0 has only one. Every generalisation of number has first presented itself as needed for some simple problem: negative numbers were needed in order that subtraction might be always possible, since otherwise ab would be meaningless if a were less than b; fractions were needed [page 75] in order that division might be always possible; and complex numbers are needed in order that extraction of roots and solution of equations may be always possible. But extensions of number are not created by the mere need for them: they are created by the definition, and it is to the definition of complex numbers that we must now turn our attention.

A complex number may be regarded and defined as simply an ordered couple of real numbers. Here, as elsewhere, many definitions are possible. All that is necessary is that the definitions adopted shall lead to certain properties. In the case of complex numbers, if they are defined as ordered couples of real numbers, we secure at once some of the properties required, namely, that two real numbers are required to determine a complex number, and that among these we can distinguish a first and a second, and that two complex numbers are only identical when the first real number involved in the one is equal to the first involved in the other, and the second to the second. What is needed further can be secured by defining the rules of addition and multiplication. We are to have

(x+yi)+(x'+y'i)

= (x+x')+(y+y')i

(x+yi)(x'+y'i)

= (xx'yy')+(xy'+x'y)i.

Thus we shall define that, given two ordered couples of real numbers, (x, y) and (x', y'), their sum is to be the couple (x+x', y+y'), and their product is to be the couple (xx'yy', xy'+x'y). By these definitions we shall secure that our ordered couples shall have the properties we desire. For example, take the product of the two couples (0, y) and (0, y'). This will, by the above rule, be the couple (−yy', 0). Thus the square of the couple (0, 1) will be the couple (−1, 0). Now those couples in which the second term is 0 are those which, according to the usual nomenclature, have their imaginary part zero; in the notation x+yi, they are x+0i, which it is natural to write simply x. Just as it is natural (but erroneous) [page 76] to identify ratios whose denominator is unity with integers, so it is natural (but erroneous) to identify complex numbers whose imaginary part is zero with real numbers. Although this is an error in theory, it is a convenience in practice; “x+0i” may be replaced simply by “x” and “0+yi” by “yi,” provided we remember that the “x” is not really a real number, but a special case of a complex number. And when y is 1, “yi” may of course be replaced by “i.” Thus the couple (0, 1) is represented by i, and the couple (−1, 0) is represented by −1. Now our rules of multiplication make the square of (0, 1) equal to (−1, 0), i.e. the square of i is −1. This is what we desired to secure. Thus our definitions serve all necessary purposes.

It is easy to give a geometrical interpretation of complex numbers in the geometry of the plane. This subject was agreeably expounded by W. K. Clifford in his Common Sense of the Exact Sciences, a book of great merit, but written before the importance of purely logical definitions had been realised.

Complex numbers of a higher order, though much less useful and important than those what we have been defining, have certain uses that are not without importance in geometry, as may be seen, for example, in Dr Whitehead’s Universal Algebra. The definition of complex numbers of order n is obtained by an obvious extension of the definition we have given. We define a complex number of order n as a one-many relation whose domain consists of certain real numbers and whose converse domain consists of the integers from 1 to n.6 This is what would ordinarily be indicated by the notation (x1, x2, x3, … xn), where the suffixes denote correlation with the integers used as suffixes, and the correlation is one-many, not necessarily one-one, because xr and xs may be equal when r and s are not equal. The above definition, with a suitable rule of multiplication, will serve all purposes for which complex numbers of higher orders are needed.

We have now completed our review of those extensions of number which do not involve infinity. The application of number to infinite collections must be our next topic.

[Chapter VII notes]

1. Vol. iii. *300ff., especially 303.

2. Of course in practice we shall continue to speak of a fraction as (say) greater or less than 1, meaning greater or less than the ratio 1/1. So long as it is understood that the ratio 1/1 and the cardinal number 1 are different, it is not necessary to be always pedantic in emphasising the difference.

3. Strictly speaking, this statement, as well as those following to the end of the paragraph, involves what is called the “axiom of infinity,” which will be discussed in a later chapter.

4. Stetigkeit und irrationale Zahlen, 2nd edition, Brunswick, 1892.

5. For a fuller treatment of the subject of segments and Dedekindian relations, see Principia Mathematica, vol. ii. *210–214. For a fuller treatment of real numbers, see ibid., vol. iii. *310ff., and Principles of Mathematics, chaps. xxxiii. and xxxiv.

6. Cf. Principles of Mathematics, §360, p. 379.