NUMBER (OF., Fr. nombre. from Lat, ailme nts, number; connected with Gk. 1,44etp, acalein, to distribute, and ultimately with Goth.
OHG. acama. (;er. nehmen, obsolete Eng. flint, to take). Number is the result of counting or of the comparison of a magnitude with a stand ard unit. This is more precisely expressed by Newton'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 number, 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 extant are odd even, prime and composite, rational and irraiiowd, and figurate numbers. The last clas sification grew out of the Greek tendency to associate numbers with geometric ideas. This notion may be illustrated by arranging groups of dots corresponding to the numbers 3, 0, 10, 15, as shown in the figures.
These forms, being triangular, suggest the pro priety of calling the numbers 3, 6, 10, 15, tri angular numbers. In the same way the numbers 4, 9, 16, 25, eame to be called square numbers.
Since other series of numbers can be made to correspond to pentagons, and still others to various other polygons, the general term poly gonal numbers was applied to all numbers of this class. An arithmetic definition of polygonal numbers as old as Hypsicles reads, "If as ninny 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 the 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 pyramidal and cubical. For example, 4, 10, 20 are pyramidal numbers, and S. 27, 64 are cubical numbers. The fact that sortie of these numbers correspond to figures of two dimensions and others to those of three dimensions also gave 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 usu ally discussed in works on higher algebra under the title Figuratire or Polygonal Numbers. Among the obsolete classifications are amicable (q.v.). perfect, defective, redundant, and helero meeic numbers. A perfect number is one equal to the sum of its aliquot parts; e.g. 6 = 1 + ± 3. If the sum of the aliquot parts exceeds the number, it is called redundant; if it is less, de fective. A heteromecie number is a number of the form at (ma + 1 ).