CHECKING. In arithmetic, one of the old est and best methods of checking the results of operations in decimal arithmetic is known as casting out nines. It originated at an early date among the Hindus, and from them it passed to the Arabs. Proofs for this rule appear in the works of Avicenna in the Tenth Century. Luca Pacioli (1494) adds this check to his work on division, pointing out eases in which it fails. Its use in elementary schools has been neglected more on the Continent of Europe than in Eng land, and not until recently has the method been seriously urged by American teachers. The pro cess may be best explained by an example.
Required to check the multiplication, 35 X 34 1190: (1) Dividing 35 by 9. the remainder is 8; (2) Dividing 34 by 9. the remainder is 7: (3) Dividing 50 (the product of 7 and 5) by 9. the remainder is 2: (4) Dividing 1190 by 9, the remainder is like wise 2.
But 2 = 2: therefore the product, 1190, is cor rect.
According to a proposition in the theory of number?, the remainder arising from dividing a number by 9 (called the exress) is the saute as that arising from dividing the sum of the digits by 9. Hence, the above remainders may be obtained thus: (1) 3 + 5 =S; (2) 3 + 4 = 7: (3) 5+6=11=9+2; (4) 1+1+9+ 0 = 9 + 2: but 2 = 2 as before.
In the case of addition, the excess in the sum is equal to the excess in the sum of the excesses of the addends. Thus, in 635 + 234 = S69. 6 4- 3 + 5 = 9 + 5. 2 + 3 + 4 = 9 + 0, 8 + 6 + 9 = 2 X 9 + 5, but 5 0 = 5: therefore the sum 869 is correct.
From the identity of division, dividend = di visor X quotient + remainder, it appears that the excess in the first member must equal that in the second. Hence, the cheek for division is made to depend upon that for addition and multiplication. Thus, in 8765 = 42 X 208 + 29.
8 + + 6 + 5 = 2 X 9 + 8; 4 4- 2 = 2 + 0+ 8 = 9 24- 9 = 9 + 2: (1 X I 2 = S: but S = 8, therefore the division is correct.
In practice, the sou of the digits is rarely found. As soon as the addition produces 9. this is rejected, and so on. Thus, in 150136. 6 + 3 = 1 + 0 + S = 9; hence. 1 is the excess.
If the result obtained from any operation dif fers from the true result by a multiple of 9. the check evidently fails, as is also the case if the result differs from the true result by having certain digits interchanged. These eases: how ever, rarely occur. Any number could be chosen for the purpose of checking, but the excess for 0 is easier found than that for any other tier not having more exceptions to its effiriency.
The method of casting out nines is only one of several important checks used in mathematical In algebra. one of the principal checks is that of arbitrary values. Thus. in the multiplication of a' + 2f(b b' by a + b, the product is o° :3a'b 3ab: This may be checked by substituting any arbitrary values for a and b, as a = 2, = 3, giving 5 X 25 = i25. There is also the check of homogeneity, likewise illustrated by the above multiplication. where the product of two homogeneous functions of degrees 2 and 3 respectively is also homoge neous of degree 2 + :3. Functions which are, as in the above ease, symmetric with respect to cer tain letters, also give rise in general to func tions symmetric with respect to those letters when one is operated upon by the other, thus sug gesting a simple check. The use of cheeks choir a•terizes the work of all who have to perform operations of various kinds in any of the branches of mathematics, and the importance of the subject can hardly be overestimated. Consult Ileman and Smith, ll igher .1 ri thin rt ic (Boston, 1895).