Page:EB1911 - Volume 19.djvu/882

 may say that defines the irrational number $$\sqrt{2}$$. The theory of cuts, in fact, provides a logical basis for the treatment of all finite numerical irrationalities, and enables us to justify all arithmetical operations involving the use of such quantities.

17. Since the aggregate of cuts ( say) has an order of magnitude, we may construct cuts in this aggregate. Thus if is any element of, and  is the aggregate which consists of  and all anterior elements of , We may write = +′, and (, ′) is a cut in which  has a last element. It is a remarkable fact that no other kind of cut in is possible; in other words, every conceivable cut in  is defined by one of its own elements. This is expressed by saying that is a continuous aggregate, and itself is referred to as the numerical continuum of real numbers. The property of continuity must be carefully distinguished from that of close order (§ 11); a continuous aggregate is necessarily in close order, but the converse is not always true. The aggregate is not countable.

18. Another way of treating irrationals is by means of sequences. A sequence is an unlimited succession of rational numbers a1, a2, a3. . . am, am+1. .. (in order-type ) the elements of which can be assigned by a definite rule, such that when any rational number, however small, has been fixed, it is possible to find an integer m, so that for all positive integral values of n the absolute value of (am+n − am) is less than. Under these conditions the sequence may be taken to represent a definite number, which is, in fact, the limit of am when m increases without limit. Every rational number a can be expressed as a sequence in the form (a, a, a, . . .), but this is only one of an infinite variety of such representations, for instance— 1 ＝ (·9, ·99, ·999, . . .) ＝ $\left({1\over 2}, {3\over 4}, {7\over 8}, . . . {2^n - 1\over 2^n} . . . \right)$ and so on. The essential thing is that we have a mode of representation which can be applied to rational and irrational numbers alike, and provides a very convenient symbolism to express the results of arithmetical operations. Thus the rules for the sum and product of two sequences are given by the formulae

from which the rules for subtraction and division may be at once inferred. It has been proved that the method of sequences is ultimately equivalent to that of cuts. The advantage of the former lies in its convenient notation, that of the latter in giving a clear definition of an irrational number without having recourse to the notion of a limit.

19. Complex Numbers.—If is an assigned number, rational or irrational, and n a natural number, it can be proved that there is a real number satisfying the equation $$x^n=a$$, except when n is even and a is negative: in this case the equation is not satisfied by any real number whatever. To remove the difficulty we construct an aggregate of polar couples {x, y}, where x, y are any two real numbers, and define the addition and multiplication of such couples by the rules

We also agree that {x, y} < {x′, y′}, if x<x′ or if x=x′ and y<y′. It follows that the aggregate has the ground element {1, 0}, which we may denote by ; and that, if we write for the element {0, 1}, 2＝{–1, 0} ＝ –.

Whenever m, n are rational, {m, n} = m+n, and we are thus justified in writing, if we like, x+y for {x, y} in all circumstances. A further simplification is gained by writing x instead of x, and regarding as a symbol which is such that 2= −1, but in other respects obeys the ordinary laws of operation. It is usual to write i instead of ; we thus have an aggregate of complex numbers x+yi. In this aggregate, which includes the real continuum as part of itself, not only the four rational operations (excluding division by {0, 0}, the zero element), but also the extraction of roots, may be effected without any restriction. Moreover (as first proved by Gauss and Cauchy), if a0, a1 an an are any assigned real or complex numbers, the equation $a_0 z^n + a_1 z^{n-1} + ... + a_{n-1} z + a_n = 0$, is always satisfied by precisely n real or complex values of z, with a proper convention as to multiple roots. Thus any algebraic function of any finite number of elements of is also contained in, which is, in this sense, a closed arithmetical field, just as  is when we restrict ourselves to rational operations. The power of is the same as that of.

20. Transfinite Numbers.—The theory of these numbers is quite recent, and mainly due to G. Cantor. The simplest of them,, has been already defined (§ 4) as the order-type of the natural scale. Now there is no logical difficulty in constructing a scheme u1, u2, u3. .. indicating a well-ordered aggregate of type immediately followed by a distinct element v1: for example, we may think of all positive odd integers arranged in ascending order of magnitude and then think of the even number 2. A scheme of this kind is said to be of order-type (+1); and it will be convenient to speak of (+1) as the index of the scheme. Similarly we may form arrangements corresponding to the indices +2, +3 . . . +n, where n is any positive integer. The scheme u1, u2, u3. .. is associated with + = 2; u11, u12, u13. . . &#124; u21, u22, u23. . . &#124; . . . &#124; un1, un2,. . . &#124; . .. with. or 2; and so on. Thus we may construct arrangements of aggregates corresponding to any index of the form ＝an+bn−1+. . . +k+l, where n, a, b,. . . l are all positive integers.

We are thus led to the construction of a scheme of symbols— $\begin{matrix} \mbox{I}. & \ 1,\ 2,\ 3,\ .\ .\ .\ n\ .\ .\ .\ \qquad \qquad \qquad \qquad \quad \\ \mbox{II}. & \begin{cases}\omega,\ \omega+1,\ .\ .\ .\ \omega+n,\ .\ .\ .\ \\ 2\omega,\ 2\omega+1,\ .\ .\ .\ 2\omega+n,\ .\ .\ .\ \\ \vdots \\ \omega^2,\ \omega^2 + 1,\ \omega^2 + 2,\ .\ .\ .\ \omega^2 + n,\ .\ .\ .\ \\ \vdots \\ \phi(\omega),\ \phi(\omega)+1,\ .\ .\ .\ \phi(\omega)+n,\ .\ .\ .\ \\ \vdots\end{cases} \\ \mbox{III}. & \begin{cases} \omega^\omega,\ \omega^\omega+1,\ .\ .\ .\ \omega^\omega+n,\ .\ .\ .\ \\ \vdots \\ \omega^{\phi(\omega)},\ \omega^{\phi(\omega)}+1,\ .\ .\ .\ \omega^{\phi(\omega)}+n,\ .\ .\ .\ \\ \vdots \end{cases} \end{matrix}$

The symbols form a countable aggregate: so that we may, if we like (and in various ways), arrange the rows of block (II.) in a scheme of type : we thus have each element  succeeded in its row by (+1), and the row containing  succeeded by a definite next row. The same process may be applied to (III.), and we can form additional blocks (IV.), (V.), &c., with first elements $$\omega_4 = \omega^{\omega^\omega,}$$ $$\omega_5 = \omega^{\omega_4,}$$ &c. All the symbols in which occurs are called transfinite ordinal numbers.

21. The index of a finite set is a definite integer however the set may be arranged; we may take this index as also denoting the power of the set, and call it the number of things in the set. But the index of an infinite ordinable set depends upon the way in which its elements are arranged; for instance, ind. $$(1,\ 2,\ 3,\ .\ .\ .\ )=\omega$$, but ind. $$(1,\ 3,\ 5,\ .\ .\ .\ |\ 2,\ 4,\ 6,\ .\ .\ .\ )=2\omega$$. Or, to take another example, the scheme— $\begin{matrix} 1,\ 3,\ 5,\ .\ .\ .\ (2n - 1)\ .\ .\ .\ \quad \\ 2,\ 6,\ 10,\ .\ .\ .\ 2\ (2n - 1)\ .\ .\ .\ \\ \vdots \ \ \ \vdots \ \ \ \ \ \vdots \qquad \qquad \qquad \qquad \quad \  \\ \quad \qquad \qquad 2^m,\ 2^m\ .\ 3,\ 2^m\ .\ 5,\ .\ .\ .\ 2^m\ (2n-1)\ .\ .\ .\  \\ \vdots \qquad \qquad \qquad \qquad \qquad \qquad \end{matrix}$ where each row is supposed to follow the one above it, gives a permutation of (1, 2, 3, . . . ), by which its index is changed from to 2. It has been proved that there is a permutation of the natural scale, of which the index is, any assigned element of (II.); and that, if the index of any ordered aggregate is , the aggregate is countable. Thus the power of all aggregates which can be associated with indices of the class (II.) is the same as that of the natural scale; this power may be denoted by a. Since a is associated with all aggregates of a