Page:The New International Encyclopædia 1st ed. v. 14.djvu/793

NUMA POMPILIUS. 675 NUMBER. can regai'd as pmbably historical is that which indicates the infusion of a Sabine religious ele- ment into Roman bistory at some remote period. NUMA ROUMESTAN, nii'mii rUTJ'mO.s'taN'. A romance by Alplionse Daudet (ISSl), and the name of its chief figure, a typical Provencal who has been identified as (ianibetta. The thread of the story is the opposition of the northern and southern temperaments, as shown in the light- hearted and irresponsible Nunia and his Parisian wife, Rosalie, and is the antithesis in develop- ment of the author's happy relations with his northern wife. NUMBER (OF., Fr. nombrc. from I.at. mime- rus, number; connected with (ik. v^nei.i', ucinein, to distribute, and ultimately with Gotli. nimaii, OHG. neman. Ger. iielimcii, obsolete Eng. nim, to take). Xumber is the result of counting or of the comparison of a magnitude with a stand- ard unit. This is more precisely expressed hy Xewton"s definition — the abstract ratio of one quantity to another of the same kind. If a name is attached to the abstract number to indicate the nature of the quantity measured, the result- ing number is said to be concrete. Thus, the ratio of the length of a room to one yard may be the abstract number 5 ; but 5 yards, the meas- ure of the length of the room, is a concrete num- ber. In the evolution of number through the application of the fundamental operations to positive integers, there have arisen the fraction, the irrational number, the negative nmnber, and the complex number. All these kinds of number may be found described in special articles. Various classifications of numbers, some of which have become obsolete, date from the time of Pythagoras. Among those e.xtant are odd and even, prime and composite, rntional and irrational, and figurate nundiers. The last clas- sification grew out of the Greek tendency to associate numtiers with geometric ideas. This notion may be illustrated by arranging groups of dots corresponding to the numbers 3, 6, 10, 15, as shown in the figures. •• ••• ••• •••• ••••• These forms, being triangular, suggest the pro- priety of calling the numbers 3, fi, 10, 15, tri- anf/ular numbers. In the same way the numbers 4. 0, 10, 25, came to be called square numbers. Since other series of numbers can be ma<le to correspond to pentagons, and still others to various other polygons, the general term poli/- yonal numbers was applied to all numbers of this class. An arithmetic definition of polj'gonal numbers as old as Hypsicles reads, "If as many numbers as you please are set out at equal in- tervals from 1, and the interval is 1, their sum is a triangular number ; if the interval is 2, a square; if 3, a pentagonal; and generally the number of angles is greater by 2 than tlie in- terval." Spherical shot piled in the form of tri- angular pyramids, or square pyramids, or held in cubical boxes suggest numbers which were called pfirnmidal and cubical. For example, 4, 10, 20 are pyramidal numbers, and 8. 27. 64 arc cubical numbers. The fact that sonle of these numbers correspond to figures of two dimensions and others to those of three dimensions also gaA'e rise to the classifications plane and solid. The numbers in each of these groups belong to a series which has special properties and which is usjj- ally discussed in works on higher algebra under the title Fiyuralicc or I'olijyonal Numbers. Among the obsolete classifications are amicable (q.v.), perfect, defective, redundant, and hetero- mecic numbers. A perfect number is one equal to the sum of its aliquot parts ; e.g. 6=1 + 2 4- 3. If the sum of the aliquot parts exceeds the number, it is called redundant; if it is less, de- fective. A heteromeeic number is a number of the form m ( »!. + 1 ) . Theory of Nimbers. This is one of the most intricate and extensive branches of mathematics. It treats princi|)ally of the forms and properties of numbers. Thus, many indeterminate problems of the Diophantine type (see Diophantine .x.Lysi.s) belong to this subject; e.g. to find two numbers the sum of whose squares shall be a square number is a condition satisfied by 5 and 12, 8 and 15, 9 and 40. . . . To find three square nund)ers in arithmetical progression is a condition satisfied by 1, 25, and 49, or by 4, 100, and 196. Various algebraic formulas serve to express all integers by assigning proper values to the letters involved. Thus by giving to m the successive values 0, I, 2, 3, 2m any of the following groups of formulas: '2m, 2)» + 1 ; 3m, 3m + I, 3m. + 2; 4m, 4m + 1, 4to -f- 2, 4m, + 3, the natural series of numbers results. This is evident since there is one number between every two consecutive even numbers, there are two nunibers between every two consecutive multiples of three, three between every two consecutive mul- tiples of four, and so on. By means of such for- mulas, many properties of numbers may easily be exhibited. E.g. the product of two consecutive numbers is divisible by 2. Let 2m be one of the numbers ; then the other is either 2m + 1 or 2m — I; the product, 2m (2h! + 1), contains 2 as a factor, and hence is divisible by 2. The product of three consecutive numbers is divisible by 6. For, let 3m be one of the numbers (as in every triad of consecutive numbers one must be a mul- tiple of 3), then the others are 3m — 2 and 3m — 1, or 3m — I and 3m + 1, or 3m + 1 and 3»n + 2. Each of the three possible products, 3m (3m — 2) (3m — 1), or 3m (3m — 1) (3m -f 1), or 3m(3m^- 1) (3m + 2), is obviously divisible by 3; and as, besides, at least one of each pair of factors by which 3m is multiplied is an even number, the product must also be divisible by 2; l)ut being divisil)le by 3 and by 2, it is divisible by 6. It may similarly bo shown that, in general, the product of n consecutive integers is divisible by 1-2-3 n, called factorial n. These propositions form the basis of proof for many properties of numbers, such as : the ditTerenee of the squares of any two odd numbers is divisible by 8 ; the difference between a number and its cube is the product of three consecutive numbers, and is consequently always divisible by 6; any ])rime number which, when divided by 4, leaves a remainder nnitv, is the sum of two square num- bers ( thus, 41 = 25 -f 16 = 5= + 4% 233 = 169 + 64 = 13= + S". etc.) . Besides these there are a great many interesting properties of numbers which do not come under any of the common classifications. Thus, the .sum of the first n odd numbers equals n'; e.g, 1 + 3 + 5 = 3"; 1 + 3 + 5 + 7 = 4^, etc. History. No essential advance was made in the theory of numbers beyond the knowledge of