ARITHMETIC. This word has been and still is used in two quite distinct senses. It formerly signified merely the science of num bers (see ARITHMETIC, HISTORY or), and treated such numeral properties as seemed mys terious or peculiar. With the invention of al gebra it was often taken to include such por tions of that science as referred to the opera tions and to the number theory. In this sense it is still used in Germany and France to-day, the art of computation being indicated by the names Rechnung and Calcul. In English, how ever, the term early came to be applied to both the science of numbers and the art of compu tation. As the former branch developed the advanced portion was given the distinctive name of Theory of Numbers (q.v.), leaving the name Arithmetic to apply to calculation and its application to business problems. With the recent relegation of the progressions and the roots to algebra, this is the sense in which the word is used in the United States to-day. With this understanding of the term, the lead ing topics relating to the subject will be con sidered.
I. Notation and Numeration.—The former referring to the number of symbols is from the mediaeval Latin flake, meaning the numeral characters (see NuIdeams), and the latter, re ferring to number names, is from nunterus, number. The distinction between the terms is coming, however, to be less marked than for merly, the word numeration being used for both. The writing and reading of numbers generally refers to positive integers, common fractions (or vulgar fractions, so called to dis tinguish them from the fractiones physics or astronomic r, the old sexagesimal fractions still met in angle measure), decimal fractions, com pound numbers and surd numbers. Of these the positive integers are known as natural num bers, the others as artificial numbers. Negative numbers, also belonging to the artificial group, have until recently been excluded from arith metic. They have, however, so many practical applications that they are beginning to find a place, and in time they will probably be treated in arithmetic so far as necessary for cases in volving numbers of opposite nature, like debit and credit, opposed forces, and contrary direc tions.
The distinctive feature of our present nu meral system (see NUMERALS) is its place value.
The characters for 5 and 1, written in juxtapo sition, indicate addition in the Roman system (VI) ; but in the Arab-Hindu notation (51) they indicate 5 tens and 1 unit, the 5 having a place value showing that it represents tens. Thus by means of only 10 characters we are able to write numbers of any desired magni tude, and by means of the simple device of decimal fractions we are also able to represent any numbers, however small.
II. Scales.— Because man has a natural counting apparatus in his 10 fingers (see FINGER NOTATION) the world has come to write num bers on a scale of 10, and to give them names based upon a decimal system. We might use other scales, and the duodecimal (scale of 12) would be better on several accounts, although a change is not practicable. There has always been some tendency to use the scale of 12, as is seen in such tables as 12 inch=1 ft., 12 oz.=1 lb. troy. The superiority of the duodecimal over the decimal scale lies in the fact that 12 has more exact divisors than 10 has. Therefore the fractions most commonly employed could better be represented on the scale of 12, as is here shown: Scale of to Scale of zs 0.5 0.25 0.7$ 0.I2S o.6 ::1 0.3 0.9 0.15 0.I In the tables of denominate numbers the tendency formerly was to adopt a varying scale, but at present it is entirely toward a uniform scale, as in the metric system (q.v.) : Uniform scale Varying scale to mills=1 cent 3 pints = t quart to cent =z dime 8 quarts= z peck zo dimes=1 dollar 4 bushel . III. The Fundamental Operations.—These are now commonly considered as four in num ber, although formerly as many as nine species, ant, or passioni, as they were called, were given. They sometimes included doubling (duplicatio), because a common method of multiplication was by successive duplications. They also included halving (mediatio), this operation being often used in effecting a division. The Rule of Three, Evolution, and Progressions were also commonly included. The fundamental opera tions may more scientifically be classified as follows, each direct process having two in verses: Direct Inverse Addition: Subtraction: 5-3=2• Multiplication: 2 )03=46. Division: S6+ 2=S3.