Owing to the correspondence between the finite cardinals and the finite ordinals, the proposi tions of cardinal arithmetic and ordinal arithmetic correspond point by point. But the definition of the cardinal number of a class applies when the class is not finite, and it can be proved that there are different infinite cardinal numbers, and that there is a least infinite cardinal, now usually denoted by
where ti is the Hebrew letter aleph. Similarly, a class of serial relations, called well-ordered serial relations, can be defined, such that their cor responding relation-numbers include the ordinary finite ordinals, but also include relation-numbers which have many properties like those of the finite ordinals, though the fields of the relations be longing to them are not finite. These relation-numbers are the infinite ordinal numbers. The arithmetic of the infinite cardinals does not correspond to that of the infinite ordinals. The theory of these extensions of the ideas of number is dealt with in the article NUMBER. It will suffice to mention here that Peano's fourth prem ise of arithmetic does not hold for infinite cardinals or for infinite ordinals. Contrasting the above definitions of number, cardinals and ordinals, with the alternative theory that number is an ultimate idea incapable of definition, we find that our procedure exacts greater attention and less credulity.
Rational numbers and real numbers in general can now be defined according to the same general meth od. If m and n are finite cardinal numbers, the rational number m/n is the relation which any finite cardinal number x bears to any finite cardinal number y when n X x= m X y. Thus the rational number one, which we will denote by a„ is not the cardinal num ber a ; for 1,. is the relation i /I as defined above, and is thus a relation holding between certain pairs of cardinals. Similarly, the other rational integers must be distinguished from the correspond ing cardinals. The arithmetic of rational numbers is now estab lished by means of appropriate definitions, which indicate the entities meant by the operations of addition and multiplication. But in order to obtain general enunciations of theorems without exceptional cases, mathematicians employ entities of ever-ascend ing types of elaboration. These entities are not created but are employed by mathematicians, and their definitions should show the construction of the new entities in terms of the old. The real numbers, including irrational numbers, have now to be defined. Consider the serial arrangement of the rationals in their order of magnitude. A real number is a class (a, say) of rational numbers which satisfies the condition that it is the same as the class of those rationals each of which precedes at least one member of a. Thus, consider the class of rationals less than 2
any member of this class precedes some other members of the class—thus 1/2 precedes 4/3,
and so on; also the class of predecessors of predecessors of 2, is itself the class of predecessors of 2,. Accordingly this class is a real number ; it will be called the real number 2R. Note that the class of rationals less than or equal to 2,. is not a real number. For 2, is not a predecessor of some member of the class. In the above example 2R is an integral real number, which is distinct from a rational integer, and from a cardinal number. Similarly, any
rational real number is distinct from the corresponding rational number. But now the irrational real numbers have all made their appearance. For example, the class of rationals whose squares are less than 2, satisfies the definition of a real number; it is the real number V2. The arithmetic of real numbers follows from appropriate definitions of the operations of addition and multi plication. Except for the immediate purposes of an explanation, such as the above, it is unnecessary for mathematicians to have separate symbols, such as 2, 2,. and 2R, or 2/3 and
Real numbers with signs (+ or — ) are now defined. If a is a real num ber, +a is defined to be the relation which any real number of the form x+a bears to the real number x, and —a is the relation which any real number x bears to the real number x--1-a. The addition and multiplication of these "signed" real numbers is suitably defined, and it is proved that the usual arithmetic of such numbers follows. Finally, we reach a complex number of the nth order. Such a number is a "one-many" relation which relates n signed real numbers (or n algebraic complex numbers when they are already defined by this procedure) to the n cardinal numbers I, 2 ... n respectively. If such a complex number is written (as usual) in the form xiei+x2e2+ +xnen, then this particular
plex number relates
to 1,
to 2, ...
to n. Also the "unit" e, (or e,) considered as a number of the system is merely a short ened form for the complex number (4- i)ei+oe2
This
last number exemplifies the fact that one signed real number, such as o, may be correlated to many of the n cardinals, such as 2 ... ?I in the example, but that each cardinal is only correlated with one signed number. Hence the relation has been called above "one-many." The sum of two complex numbers
+xnen and
is always defined to be the com plex number (xi-Fyi)el±(z2+312)e2-1-
But an indefinite number of definitions of the product of two complex numbers yield interesting results. Each definition gives rise to a corresponding algebra of higher complex numbers. We will con fine ourselves here to algebraic complex numbers—that is, to com plex numbers of the second order taken in connection with that definition of multiplication which leads to ordinary algebra. The product of two complex numbers of the second order—namely,
and y1ei+y2e2, is in this case defined to mean the com plex (xiyi+x2Y2) ei--1-(ziY2-1-x2yi)e2. Thus
X ei=ei, e2 X e2 = —
X
e, X
With this definition it is usual to omit the first symbol e,, and to write i or — z instead of
Accordingly, the
typical form for such a complex number is x+yi, and then with this notation the above-mentioned definition of multiplication is invariably adopted. The importance of this algebra arises from the fact that in terms of such complex numbers with this defini tion of multiplication the utmost generality of expression, to the exclusion of exceptional cases, can be obtained for theorems which occur in analogous forms, but complicated with exceptional cases, in the algebras of real numbers and of signed real numbers. This is exactly the same reason as that which has led mathematicians to work with signed real numbers in preference to real numbers, and with real numbers in preference to rational numbers.