S64-42.
Involution: Evolution: '— Logarithms: 3=logi 8.
Of these the primitive one is addition, mul tiplication by a positive integer arising when the addenda are equals, and involution to a positive integral power arising from multiplica tion when the factors are equal. Arbitrarily, elementary arithmetic has usually excluded evo lution beyond the cube root, and logarithms. It is now tending to relegate cube root to algebra on account of its difficulty and lack of applications. The exclusion of logarithms is due to their relatively late invention, since, if the theory i of their computation is ex cluded, the subject is simple of presentation and valuable in application.
From the primary operations with natural numbers have been derived operations, desig nated by the same names and subject to the same laws, involving the artificial numbers. For example, 2X$3-6 means that $3 is taken twice as an addendum, thus : $3+$3. But %X% cannot mean that is taken as an addend % of a time. It means that % of % is taken, or that % of % is taken 2 times. It is, however, con venient to broaden the definitions so as to use the same phraseology and symbols as in the case of positive integers. Similar considerations fix a meaning for-2X-3=+6, V2 X V3= V6, and V —2 X V —3=—V6. For the justifi cation of these usages, see ALGEBRA, DEFINITIONS and FUNDAMENTAL CONCEPT. In certain cases an operation is so difficult that it is more con venient to substitute for it another which gives the same result. This is seen in the case of the division of fractions, where to divide 36 by 3i it is easier to multiply 36 by % than to reduce to a common denominator as was for merly done, and then divide, thus : Of the four common operations, addition is the simplest of comprehension, although not in actual work. In fractions it is usually easier to multiply than to add, as in the case of % X 1 compared with 3i + With inte gers, both addition and multiplication require the learning of 45 combinations of numbers (1 + 2, 1 + 3, .... 1 X 2, 1 X 3 .... ), and the mere memorizing of these facts is as easy in one operation as the other. Subtraction does not require memorizing a table, since it is merely the inverse of addition, and if taught by the (rnalcing change)) method it uses the addition table, as division uses that of multiplication.
IV. Checks.— An important consideration in all computations is the checking of the work, to be reasonably sure that no error enters. Checks
should be applied at every opportunity so that an error may be discovered as soon as it is made, and not vitiate the further work. The most important check in addition is the repeat ing of the work in the opposite direction, adding downward if the first addition was upward. The psychological reason for this is that like stimuli tend to produce like reactions, and if an error has been made it is liable to be made again if the numbers are soon met in the same order. Hence the order is reversed to counter act this tendency. In subtraction the best check is that of adding the subtrahend and remainder. If the remainder was obtained by the or (making changes method, this addition should be performed in the oppo site direction as in the check for addition. The best check for multiplication and division is that of (casting out nines?) This ancient Oriental method was of especial value when the sand board form of the abacus (q.v.) was used, since the numbers were so frequently erased as to render a general review of the work impossible. This check has gone out of use in American schools, but it is so simple and valuable that it will probably be revived. The check depends upon two propositions: (1) The excess of 9's in a number (that is, the remainder arising from dividing a number by 9) is the same as the excess in the sum of the digits. In the case of 1247 the sum of the digits is 14, and this divided by 9 gives a remainder of 5. It is customary to cast out the 9's as the digits are added, thus: 7 + 4=11; cast out 9 and 2 is left ; 2 + 2 + 1=5, the excess. no (2) The excess of 9's in the prod- 21 uct equals the excess in the prod r247 uct of the excesses of the factors, 23 In the case here given, the ex- cesses in the factors are 5 and 3, indicated in the right and left angles of the cross. The excess in their product (15) is 6, indicated in the upper angle. The excess in the product, 26187 is 6, indicated in the lower angle. The upper and lower numbers in the cross are the same, showing that the result is probably correct. In division, the excess of 9's in the dividend equals the excess in the product of the excesses of the divisor and the quotient, plus that in the remainder. Of course, the check of 9's fails to detect an error involving a multiple of 9. There is a somewhat similar check by casting out 11's requiring slightly longer time, but in some re spects more liable to detect errors.