Arithmetic

ARITHMETIC, originally the science or theory of numbers; at present, as commonly understood in the English language, the art of computation and the applications of this art (Gr. &pi6,usirircs, from apeOµos number). In certain other languages the word still retains some of its early meaning and applies not only to a certain amount of theoretical work with numbers, but to the study of the fundamental operations with polynomials, such words as Rechnung (German) and calcul (French) being used for cal culation and its simple applications. In this article the word will be used with the common Anglo-American meaning. The subject will be treated in an elementary way with respect to its bearing upon the school curriculum. The advanced theory will be con sidered under special topics to which reference is hereinafter made.

Using the term in the sense already mentioned, arithmetic is little concerned with the genesis of the concept of number, this being a philosophical question. Arithmetic takes number as it is found, dividing it into the general classes of cardinal and ordinal. With regard to the numerical measure of a group, as the result of counting or of computing, the term cardinal number is used, as when we say that there are five persons in a room. With respect to number as designating position in a sequence, the term ordinal number is used, as when we speak of the third page of a book. The cardinal number is that upon which arithmetic turns (Lat. cardo, a hinge) ; it is the important type. (See NUMBER.) Of an assemblage of objects, the number that can be told at a glance is very limited. Unless the eye is aided by having the ob jects arranged symmetrically in some familiar order or so as to be divided readily into subgroups, the eye grasp is usually lim ited to four or five. Similarly in counting, the mind does not easily grasp the significance of more than a few numerical terms. For example, we may count up to ten or twelve, but by that time we find it desirable to combine the names of smaller numbers, as when we say thirteen (three ten), rather than invent a wholly new term. Similarly in writing numbers, the world finds it necessary, by grouping, to make a few selected characters serve to represent any number however large. At first the groups were small, and there are numerous evidences that the "couple," "pair," "brace," "span," and the like are relics of early groupings made in part for convenience in counting.

Notation.

Peoples who developerd independent alphabets also tended to develop independent numerical notations. The spread of Greek culture and commerce, however, carried the Greek nu merals of the pre-Christian period into all the leading ports of the Mediterranean sea, and the still more extensive development of the Roman civilization made the Roman numerals dominant in the Occident for many centuries. In the loth century the entry into Europe of the Indo-Arabic numerals, 1, 2, 3 . . . 9, together with the zero, was followed by a slow acceptance of the convenient system of place value by which, with only ten numerals but with an indefinite number of "places" (units, tens, hundreds, etc.), any number could conveniently be written. The symbols went by the Latin names characteres and notae, and at a later period by the English names figures, numerals and cyphers, and by similar names in other countries. From note came notatio—notation, or the writing of numbers. Thus we have the Egyptian, Greek, Roman, Indian, Arabic and other notations, each of the early written languages tending to have its own numerals. (See NUMERALS.) The grouping of objects for purposes of counting led to the use of the same device in the writing of numbers. A grouping by fives is called a quinary system and is said to be based upon the scale of five, or to have five as the radix. Since man has five fingers on each hand and five toes on each foot, he has a natural counting abacus (q.v.) arranged on a scale of five, ten or twenty. While there are traces of the early use of each of these and other scales, the predominant one has been the denary (or decimal) scale— the one with the radix ten. (See NUMERALS.) A familiar relic of grouping by twenties, for example, is seen in the English score, as in expressions like "seven score acres" in relatively late legal documents, and the French quatre-vingts (four twenty) for 8o and (formerly) other multiples like quinze-vingts (fifteen twenty) for 300, as in the Hospice des Quinze-Vingt in Paris, dating from the 13th century. On this system ninety-six appears in French as quatre-vingt-seize (four-twenty-sixteen). On the pure scale of ten the English counting proceeds as follows: one, two, three . . . nine, ten, oneteen (one and ten), twoteen, thirteen (three and ten) • • • twenty (two tens). The fact that twelve is scientifically a more convenient radix than ten (having its half, third, and fourth easily expressible), seems to have led to the use of "eleven" and "twelve" instead of "oneteen" and "twoteen," after which the denary scale was followed. The tendency to favour twelve as a radix appears in the number of inches (Lat. unciae, twelfths) in a foot, of lines in an inch, of ounces (also from unciae) in the Roman and Troy pounds, and of pence in a shilling. The scale of twenty seems to have had considerable standing in certain bare foot countries, the native Mexicans counting to "man finished" and then beginning again. One evidence of the tropical origin of the Greenlanders is that their system of counting also uses twenty as a radix.

A perfect decimal scale requires ten primary word-forms below "hundred"—"one," "two," "three," "ten," the other num bers being named by combining these, as in "fifty-seven" (five ten-seven). It also requires ten characters, o, 1, 2 ... 9. The numbers from 1 to I o corresponding to the ten fingers, seem to have been called by the early Latins digiti ("fingers"), whence our digits. With the coming of the Indo-Arabic numerals, however, it became convenient to use the word to designate the numbers expressed by 1, 2, 3 ... 9, and also to designate the characters (XapaKTflpEs) (numeral figures) themselves. Since "one" was, by various early writers, spoken of as the fons et origo numerorum ("source and origin of numbers"), this was often excluded, the digits being then considered as the eight numbers (or characters) 2,3,4...9.

Similarly, a scale of eight would need eight primary word-forms and eight characters (o, 1, 2 ... 7), and so for other scales. On the scale of 2, the number designated by the English word "eleven" would be represented by 'oil, that is, 1 X ± o X + I X 2 ± I, where (in denary symbols) 8 + o 2 -I- 1=11. Evi dently, therefore, the smaller the radix the more times the char acters must be written to express a given number; the larger the radix the more basic number names must be memorized. Either ten or twelve is a medium radix and tradition is too power ful to admit of change from the former to the latter, even though the duodecimal (Lat. duodecim, two-ten, whence the French douzaine and the English dozen) has a slight advantage over the decimal.

Numeration and Classification of Numbers.

The Latin ,iumeratio comes from numerus, a number, and refers to the names of numbers. These number names have little etymological significance; in fact, their original meaning is quite speculative.

The various kinds and properties of numbers are considered elsewhere (see NUMBER; NUMBERS, THEORY oF). A brief reference will, however, be made to those used in elementary arithmetic. With the child as with the race, the first need is for the integer (Lat., "whole") or whole number in the do main limited by the number of his fingers. He learns that "three" is the word to be used with a certain group, just as he learns the names of objects. With a group that is beyond his eye grasp, like six, he learns how to find the number name by memor izing a sequence—"one, two, three, four, five, six," pointing to each object as he counts. He thus subconsciously combines car dinal numbers with ordinal numbers, and thenceforth uses each as the need arises, learning the words "first," "second," and so on, as part of his everyday vocabulary.

His next step leads him to such unit fractions as 2 and 4, and he subconsciously learns that has a variety of meanings, as of an object, of a group, of a weight, i as large, as light, j as loud, as good, and so on, some of which suggest precision, as in the case of an object, while others are merely rhetorical. It is a considerable step from the notion of a to that of and the world probably required many centuries in which to take it, and thou sands of years to devise a satisfactory symbol for the fraction it self. To broaden the concept so as to include among fractions such cases as and 4, and especially such fraction forms as have fractions for numerator or denominator, did not occur to arith meticians until modern times. (See FRACTIONS.) The distinction between abstract numbers, like 4, and concrete numbers, like 4ft., is an inheritance that serves no important pur pose. A number that has a label attached to it, indicating the unit of measure to which it refers, is called a denominate number. Formerly it included such cases as "3 fourths" (as in Trenchant's arithmetic, 1566), but it is now usually limited to such "concrete" numbers as 3ft. and 2 lb. 3 oz. Numbers with more than a single denomination, such as £3 6s. 4d. and 3yd. 27in., are sometimes called compound numbers.

Denominate Numbers.

Denominate numbers in general and compound numbers in particular trace their origin to the difficulty which the ancients had in developing a satisfactory notation for either integers or fractions. In the days when one of the ways of writing 7291ft. was MMMMMMMCCLXXXXI pedes, it be came convenient to reduce the length of the written number. It was simpler to write this as Imi. 2oIIft.; and then as Imi. 3fur. 2i ft.; and then as 'mi. 3fur. 1 rd. 4ft.; and then, to avoid the fraction, to write 4ft. as 'yd. 'ft. 6in. Different trades, modes of life, and languages, brought in different units, and so arith metic inherited such a variety of units of length as the league, mile, stadium, furlong, rod, fathom, yard, cubit, ell, perch, foot, palm, hand, inch and line. The force of tradition kept all of them in the schools till the latter part of the 19th century. In the United States all but four or five have been generally discarded, and in Great Britain there is a tendency in the same direction.

The principal tables of denominate numbers needed for gen eral information in the British empire and the United States, as distinct from those needed in special trades and industries, and in countries where the metric system is used, are indicated as inch, foot, yard, rod (agriculture chiefly), furlong (Great Britain), mile, metre (meter), kilometre; area: square units (inch, foot, yard, mile, meter, kilometer), acre; capacity and volume: cubic units (I cu.in., I cu.f t., icu.yd., I cu. metre), gill, pint, quart, gallon, peck, bushel, litre; time: second. minute, hour, day, week, month, year, decade, century; value: the British and American units, with some knowledge of the mark and the franc (lira, peseta, and other gold equivalents) ; weight: ounce, pound (avoirdupois), stone (Great Britain), hundredweight (Great Britain), ton, gramme (gram), kilogramme (kilogram). These tables are constructed upon varying scales; that is, although 12 inches make I foot, it does not follow that 12 feet make the next denomination; in fact, the next item changes the scale and states that 3 feet make 1 yard. (See MENSURA TION.) Partly to avoid the inconvenience of a varying scale and partly to establish an international standard, French scientists at the close of the 18th century developed a decimal system of measures, this being known as the metric system. It is based upon the stand and metre, originally intended to be one ten-millionth of the dis tance from the equator to the pole, a degree of accuracy impos sible of exact attainment. The legal International metre (meter) is the length of the standard kept in the Bureau International des Poids et Mesures at the entrance of the Parc de Saint-Cloud, Sevres, near Paris, of which the various civilized countries have copies.

Rational and Irrational Numbers.—The nature of rational and irrational numbers is discussed in the articles on NUMBER; NUMBERS, THEORY OF; and ALGEBRA. With the transfer to al gebra of the theory of and operations with these numbers, there was left in elementary arithmetic only the topic of roots (evolu tion), this being limited to square root and cube root. With the increased use of tables, of logarithms, and of the slide rule and other calculating machines (q.v.), even these two operations have recently tended to receive little attention. As a result, cube root has disappeared from most of the arithmetics in the United States and square root will probably do the same, the theory of each being given a moderate amount of attention in algebra, and the practical finding of roots being dependent upon tables or mechan ical aids. In Great Britain there is a similar trend. This leads to the arithmetical treatment (finding of approximate values) of irrational numbers in connection with decimal fractions, the nature of such numbers being considered in algebra and the number theory.