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Originally appearing in Volume V19, Page 850 of the 1911 Encyclopedia Britannica.
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NUMA POMPILIUS, second legendary king of Rome (715-672 B.C.), was a Sabine, a native of Cures, and his wife was the daughter of Titus Tatius, the Sabine colleague of Romulus. He was elected by the Roman people at the close of a year's interregnum, during which the sovereignty had been exercised by the members of the senate-in rotation. Nearly all the early religious institutions of Rome were attributed to him. He set up the worhip of Terminus (the god of landmarks), appointed the festival of Fides (Faith), built the temple of Janus, reorganized the calendar and fixed days of business and holiday. He instituted the flamens (sacred priests) of Jupiter, Mars and Quirinus; the virgins of Vesta, to keep the sacred fire burning on the hearth of the city; the Salii, to guard the shield that fell from heaven; the pontifices and augurs, to arrange the rites and interpret the will of the gods; he also .divided the handicraftsmen into nine gilds. He derived his inspiration from his wife, the nymph Egeria, whom he used to meet by night in her sacred grove. After a long and peaceful reign, during which the gates of Janus were closed, Numa died and was succeeded by the warlike Tullus Hostilius. Livy (xl. 29) tells a curious story of two stone chests, bearing inscriptions in Greek and Latin, which were found at the foot of the Janiculum (181 B.C.), one purporting to contain the body of Numa and the other his books. The first when opened was found to be empty, but the second contained fourteen books relating to philosophy and pontifical law, which were publicly burned as tending to under-mine the established religion. No single legislator can really be considered responsible for all the institutions ascribed to Numa; they are essentially Italian, and older than Rome itself. Even Roman tradition itself wavers; e.g. the fetiales are variously attributed to Tullus Hostilius and Ancus Marcius. The supposed law-books, which were to all appearance new when discovered, were clearly forgeries. See Livy i. 18-21; Plutarch, Numa; Dion. Halic. ii. 58-76; Cicero, De republica, ii. 13-15. For criticism: Schwegler, Romische Geschichte, bk. xi.; Sir G. Cornewall Lewis, Credibility of early Roman History, ch. xi.; W. Ihne, Hist. of Rome, i.; E. Pais, Storia di Roma, i. (1898), where Numa is identified with Titus Tatius and made out to be a river god, Numicius, closely connected with Aeneas; J. B. Carter, The Religion of Numa (1906); O. Gilbert, Geschichte and Topographic der Stadt Rom tint Aitertum (1883—1885) ; and RoME : Ancient History. NUMBER' (through Fr. nombre, from Lat. numerus; from a root seen in Gr. vEµeu' to distribute), a word generally expressive of quantity, the fundamental meaning of which leads on analysis to some of the most' difficult problems of higher mathematics. I. The most elementary process of thought involves a distinction within an identity—the A and the not-A within the sphere throughout which these terms are intelligible. Again A may be a generic quality found in different modes Aa, Ab, Ac, &c.; for instance, colour in the modes, red, green, blue and so on. Thus the notions of " one," " two," and the vague " many " are fundamental, and must have impressed themselves on the human mind at a very early period: evidence of this is found in the grammatical distinction of singular, dual and plural which occurs in ancient languages of widely different races. A more definite idea of number seems to have been gradually acquired by realizing the equivalence, as regards plurality, of different concrete groups, such as the fingers of the right hand and those of the left. This led to the invention of a set of names which in the first instance did not suggest a numerical system, but denoted certain recognized forms of plurality, just as blue, red, green, &c., denote recognized forms of colour. Eventually the conception of the series of natural numbers became sufficiently clear to lead to a systematic terminology, and the science of arithmetic was thus rendered possible. But it is only in quite recent times that the notion of number has been submitted to a searching critical ' See also NUMERAL. analysis: it is, in fact, one of the most characteristic results of modern mathematical research that the term number has been made at once more precise and more extensive. 2. Aggregates (also called manifolds or sets).— Let us assume the possibility of constructing or contemplating a permanent system of things such that (1) the system includes all objects to which a certain definite quality belongs; (2) no object without this quality belongs to the system; (3) each object of the system is permanently recognizable as the same thing, and as distinct from all other objects of the system. Such a collection is called an aggregate: the separate objects belonging to it are called its elements. An aggregate may consist of a single element. It is further assumed that we can select, by a definite process, one or more elements of any aggregate A at pleasure: these form another aggregate B. If any element of A remains unselected, B is said to be a part of A (in symbols, B < A) : if not, B is identical with A. Every element of A is a part of A. If B< A and C-< B, then C< A. When a correspondence can be established between two aggregates A and B in such a way that to every element of A corresponds one and only one element of B, and conversely, A and B are said to be equivalent, or to have the same power (or potency) ; in symbols, A B. If A w B and B co C, then A co C. It is possible for an aggregate to be equivalent to a part of itself: the aggregate is then said to be infinite. As an example, the aggregates 2, 4, 6, . . . an, &c., and 1, 2, 3, . . . n, &c., are equivalent, but the first is only a part of the second. 3. Order.—Suppose that when any two elements a, b of an aggregate A are taken there can be established, by a definite criterion, one or other of two alternative relations, symbolized by ab, subject to the following conditions: (r) If a>b, then ba; (2) If a>b and b>c, then a>c. In this case the criterion is said to arrange the aggregate in order. An aggregate which can be arranged in order may be called ordinable. An ordinable aggregate may, in general, by the application of different criteria, be arranged in order in a variety of ways. According as ab we shall speak of a as anterior or posterior to b. These terms are chosen merely for convenience, and must not be taken to imply any meaning except what is involved in the definitions of the signs > and < for the particular criterion in question. The consideration of a succession of events in time will help to show that the assumptions made are not self-contradictory. An aggregate arranged in order by a definite criterion will be called an ordered aggregate. Let a, b be any two elements of an ordered aggregate, and suppose a b' or b< b' according as a > a' or a < a'. For example, (1, 3, 5, ... )" (2, 4, 6, ... ), because we can make the even number an correspond to the odd number (2n—1) and conversely. Similar ordered aggregates are said to have the same order-type. Any definite order-type is said to be the ordinal number of every aggregate arranged according to that type. This somewhat vague definition will become clearer as we proceed. 4. The Natural Scale.—Let a be any element of a well-ordered aggregate A. Then all the elements posterior to a form an aggregate A', which is a part of A and, by definition, has a first element a'. This element a' is different from a, and immediately succeeds it in the order of A. (It may happen, of course, that a' does not exist; in this case a is the last element of A.) Thus in a well-ordered aggregate every element except the last (if there be a last element) is succeeded by a definite next element. The ingenuity of man has developed a symbolism by means of which every symbol is associated with a definite next succeeding symbol, and in this way we have a set of visible or audible signs r, 2, 3, &c. (or their verbal equivalents), representing an aggregate in which (r) there is a definite order, (2) there is a first term, (3) each term has one next following, and consequently there is no last term. Counting a set of objects means associating them in order 'with the first and subsequent members of this conventional aggregate. The process of counting may lead to three different results: (r) the set of objects may be finite in number, so that they are associated with a part of the conventional aggregate which has a last term; (2) the set of objects may have the same power as the conventional aggregate; (3) the set of objects may have a higher power than the conventional aggregate. Examples of (2) and (3) will be found further on. The order-type of i, 2, 3, &c., and of similar aggregates will be denoted by w; this is the first and simplest member of a set of transfinite ordinal numbers to be considered later on. Any finite number such as 3 is used ordinally as representing the order-type of r, 2, 3 or any similar aggregate, and cardinally as representing the power of r, 2, 3 or any equivalent aggregate. For reasons that will appear, w is only used in an ordinal sense. The aggregate r, 2, 3, &c., in any of its written or spoken forms, may be called the natural scale, and denoted by N. It has already been shown that N is infinite: this appears in a more elementary way from the fact that (r, 2, 3, 4,. . • )` (2, 3, 4, 5,. . . ), where each element of N is made to correspond with the next following. Any aggregate which is equivalent to the natural scale or a part thereof is said to be countable. 5. Arithmetical Operations.—When the natural scale N has once been obtained it is comparatively easy, although it requires a long process of induction, to define the arithmetical operations of addition, multiplication and involution, as applied to natural numbers. It can be proved that these operations are free from ambiguity and obey certain formal laws of commutation, &c., which will not be discussed here. Each of the three direct operations leads to an inverse problem which cannot be solved except under certain implied conditions. Let a, b denote any two assigned natural numbers: then it is required to find natural numbers, x, y, z such that a=b+x, a=by, a=zb respectively. The solutions, when they exist, are perfectly definite, and may be denoted by a—b, a/b and 4j a; but they are only possible in the first case when a>b, in the second when a is a multiple of b, and in the third when a is a perfect bth power. It is found to be possible, by the construction of certain elements, called respectively negative, fractional and irrational numbers, and zero, to remove all these restrictions. 6. There are certain properties, common to the aggregates with which we have next to deal, analogous to those possessed by the natural scale, and consequently justifying us in applying the term number to any one of their elements. They are stated here, once for all, to avoid repetition; the verification, in each case, will be, for the most part, left to the reader. Each of the aggregates in question (A, suppose) is an ordered aggregate. If a, /3 are any two elements of A, they may be combined by two definite operations, represented by + and X, so as to produce two definite elements of A represented by a+/3 and aX(3 (or 0); these operations obey the formal laws satisfied by those of addition and multiplication. The aggregate A contains one (and only one) element t, such that if a is any element of Aalso t<2c<3c . . . We may express this by saying that A contains an image of the natural scale. The element denoted by c may be called the ground element of A. 7. Negative Numbers.—Let any two natural numbers a, b be selected in a definite order a, b (to be distinguished from b, a, in which the order is reversed). In this.way we obtain from N an aggregate of symbols (a, b) which we shall call couples, or more precisely, if necessary, polar couples. This new aggregate may be arranged in order by means of the following rules:—Two couples (a, b), (a', b') are said to be equal if a+b'=a'+b. In other words (a, b), (a', b') are then taken to be equivalent symbols for the same thing. If a+b'>a'+b, we write (a, b)>(a', b'); and if a+b' (a, b) and (a, b) Xc= (2a+b, a+2b) = (a, b). Hence e is the ground element of Sr. By definition, 2t=c+c=(4, 2) = (3, r): and hence by induction mt=(m+r, r), where m is any natural integer. Conversely every couple (a, b) in which a>b can be expressed by the symbol (a—b)c. In the same way, every couple (a, b) in which b>a can be expressed in the form (b—a)i , where t' = (I, 2). 8. It follows as a formal consequence of the definitions that L+i = (2, r)+(r, 2) = (3, 3) = (r, r). It is convenient to denote (r, r) and its equivalent symbols by o, because (a, b) +(I, r)=(a+r, b+r) =(a,b) (a, b)X(r, r)=(a+b, a+b)=(r, r); hence c+i = o, and we can represent N by the scheme—... 31", 2t', L', 0, c, 2t, 3L .. . in which each element is obtained from the next before it by the addition of t. With this notation the rules of operation may be written (m, ii, denoting natural numbers) mr+nt =(m+n)c mt'+nt' =(m+n)i m4+nc' = (m —n)L if m > n =(n—m)L' ,n a, and cu= a. Thus A contains the are [a, bl +[a', b'] = [ab'-J-a'b, bb'] elements c, t+t, c+t+c, &c., or, as we may write them, t, 2c, [a, b] X [a', b'] = [aa', bb']. 3L, . . . me . . . such that me+nc=(m+n)t and mmXnc =mnc; All the couples [a, a] are equivalent to [r, r], and if we denote this by v we have [a, b]-Fu= [a+b, b]> [a, b], [a, b] X v = [a, b], so that u is the ground element of the new aggregate. Again 2v=v+v=(2, I), and by induction mu=[m, r]. More-over, if a is a multiple of b, say mb, we may denote [a, b] by ma. I r. The new aggregate of couples will be denoted by R. It differs from N and N in one very important respect, namely, that when its elements are arranged in order of magnitude (that is to say, by the rule above given) they are not isolated from each other. In fact if [a, b]=a, and [a', b']=a', the element [a+a', b+b'] lies between a and a'; hence it follows that between any two different elements of R we can find as many other elements as we please. This property is expressed by saying that R is in close order when its elements are arranged in order of magnitude. Strange as it appears at first sight, R is a count-able aggregate; a theorem first proved by G. Cantor. To see this, observe that every element of R may be represented by a " reduced " couple [a, b], in which a, b are prime to each other. If [a, b], [c, d] are any two reduced couples, we will agree that [a, b] is anterior to [c, d] if either (I) a+b< c+d, or (2) a+b= c+d, but aa. 12. The division of one element of R by another is always possible; for by definition [c, d] X [ad, bcj = [acd, bcd] = [a, b], and consequently [a, b]= [c, d] is always interpretable as [ad, bc]. As a particular case [m, I]= [n, 11= [m, n], so that every element of R is expressible in one of the forms mu, my/nv. It is usual to omit the symbol v altogether, and to represent the element [m, n] by min, whether m is a multiple of n or not. Moreover, m/r is written m, which may be done without confusion, because m/r+n/r=(m+n)/I, and m/IXn/I=mn/r, by the rules given above. 13. Within the aggregate R subtraction is not always practicable; but this limitation may be removed by constructing an aggregate R related to R in the same way as N to N. This may be done in two ways which lead to equivalent results. We may either form symbols of the type (a, 0), where a, (3 denote elements of R, and apply the rules of § 7; or else form symbols of the type [a, 0], where a, 13 denote elements of N, and apply the rules of §ro. The final result is that R contains a zero element, o, a ground element v, an element v' such that v+v' = o, and a set of elements representable by the symbols (m/n)v, (m/n)v'. In this notation the rules of operation are mu+nz'u= (mn'+m'n) m , m' ,_ (mn'+m'n\ ,; n n' \ nn u n v +n' nn' Ju m m' ,mn'-m'n m'n-mn' = , n n, nn v, or nn' v', as inn >or . In the same way, if o denotes the zero element of R, and any other element, the symbol o/o is indeterminate, and l;/o in-admissible, because, by the formal rules of operation, /o+v = /o, which conflicts with the definition of the ground element v. It is usual to write +n m (or simply n) for n v, and -n for mv'. Each of these elements is said to have the absolute value m/n. The criterion for arranging the elements of R in order of magni- tude is that, if , .q are any two elements of it, >q when t-ti is positive; that is to say, when it can be expressed in the form (m/n)v. - 15. The aggregate R is very important, because it is the simplest type of a field of rationality, or corpus. An algebraic corpus is an aggregate, such that its elements are representable by symbols a, 0, &c., which can be combined according to the laws of ordinary algebra; every algebraic expression obtained by combining a finite number of symbols, by means of a finite chain of rational operations, being capable of interpretation as representing a definite element of the aggregate, with the single exception that division by zero is inadmissible. Since, by the laws of algebra, a-a = o, and a/a = I, every algebraic field contains R, or, more properly, an aggregate which is an image of R. to. Irrational Numbers.—Let a denote any element of R; then a and all lesser elements form an aggregate, A say; the remaining elements form another aggregate A', which we shall call complementary to A, and we may write R=A+A'. Now the essence of this separation of R into the parts A and A' may be expressed without any reference to a as follows: I. The aggregates A, A' are complementary; that is, their elements, taken together, make up the whole of R. II. Every element of A is less than every element of A'. Every separation R=A+A' which satisfies these conditions is called a cut (or section), and will be denoted by (A, A'). We have seen that every rational number a can be associated with a cut. Conversely, every cut (A, A) in which A has a last element a is perfectly definite, and specifies a without ambiguity. But there are other cuts in which A has no last element. For instance, all the elements (a) of R such that either a o, or else a> o and a2 <2, form an aggregate A, while those for which a> o and a2> 2, form the complementary aggregate A'. This separation is a cut in which A has no last element; because if p/q is any positive element of A, the element (3p+4q)/(2p+3q) exceeds p/q, and also belongs to A. Every cut of this kind is said to define an irrational number. The justification of this is contained in the following propositions: (I) A cut is a definite concept, and the assemblage of cuts is an aggregate according to definition; the generic quality of the aggregate being the separation of R into two complementary parts, without altering the order of its elements. (2) The aggregate of cuts may be arranged in order by the rule that (A, A') < (B, B') if A is a part of B. (3) This criterion of arrangement preserves the order of magnitude of all rational numbers. (4) Cuts may be combined according to the laws of algebra, and, when the cuts so combined are all rational, the results are in agreement with those derived from the rational theory. As a partial illustration of proposition (4) let (A, A'), (B, B') be any two cuts; and let C' be the aggregate whose elements are obtained by forming all the values of a'+/3', where a' is any element of A' and /3' is any element of B'. Then if C is the complement of C', it can be proved that (C, C') is a cut; this is said to be the sum of (A, A') and (B, B'). The difference, product and quotient of two cuts may be defined in a similar way. If n denotes the irrational cut chosen above for purposes of illustration, we shall have n2 = (C, C') where C' comprises all the numbers a'/3' obtained by multiplying any two elements, a', /3' which are rational and positive, and such that a 2> 2, 13'2> 2. Since a'2/3'2> 4 it follows that a'/3' is positive and greater than 2; it can be proved conversely that every rational number which is greater than 2 can be expressed in the form a'/3'. Hence n2=2, so that the cut n actually gives a real arithmetical meaning to the positive root of the equation x2 = 2 ; in other words we may say that n defines the irrational number d 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 (ZT say) has an order of magnitude, we may construct cuts in this aggregate. Thus if a is any element of ZT, and £4 is the aggregate which consists of a and all anterior elements of U, we may write n=a+ a', and (a, a') is a cut in which a has a last element a. It is a remarkable fact that no other kind of cut in U is possible; in other words, every conceivable cut in 27 is defined by one of its own elements. This is expressed by saying that ZT is a continuous aggregate, and ZT itself is referred to as the numerical continuum of real numbers. The property of continuity must be carefully distinguished from that of close order (§ II); a continuous aggregate is necessarily in close order, but the converse is not always true. The aggregate IT is not countable. 18. Another way of treating irrationals is by means of sequences. A sequence is an unlimited succession of rational numbers al, as, as . . . am, a",+1 .. . (in order-type w) the elements of which can be assigned by a definite rule, such that when any rational number e, 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—a,,) is less than e. Under these conditions the sequence may be taken to represent a definite number, which is, in fact, the limit of a", 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 = (.9, •99, .999, . . .) = I 4' 2I . ~2'$,. 2n and so on. The essential thing is that we have a mode of re-presentation 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 (al, as, a3, . . .) + (bl, b2, b3, . . .) _ (a,+bl, a2+b2, as+ba . . . ) (al, as, a3, . . .) X (bl, b, b3, . . .) _ (albs, a2b2, a3b3 . . . ) 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 a 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 xn=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 {x, yl+(x', y'}={x+x', y ;-y i; (x, yl X x', y'l = xx'—yy', xy'+x'y}. We also agree that {x, y} < {x', y'}, if x End of Article: NUMA POMPILIUS

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