SUBTRACTION. and to write the partial remainders in their proper places; the result thus obtained is the difference sought. sum; But if any figure of the minuend is less than the corresponding figure of the subtrahend, add ten to the units expressed by that figure of the minuend, subtract the figure of the subtrahend from this write the remainder in its proper place, carry 1 to the next figure of the subtrahend as an equivalent for the 10 combined with the next lower units of the minuend; take the augmented figure of the subtrahend from the corresponding figure of the minuend, borrowing and carrying, if requisite; and thus proceed till all the partial subtractions are effected. The result is the difference of the given numbers. 4th. 35417840 and 8709864?..... .Ans. 73. ..Ans. 5886. ..Ans. 6680. ..Ans. 26707976. ..Ans. 7504590. ...Ans. 19008. ..Ans. 37677. ..Ans. 1889089. ...Ans. 59870713. .....Ans. 99185959127. ....Ans. 82808495. ..Ans. 9753086421. ...Ans. 864197532. .Ans. 20093788636. ....Ans. 679876505. 8th. 5807854 and 3918765?...... ..Ans. 1476548996. ..Ans. 5062881059. ..Ans. 971756717. ....Ans. 9399102882. 41. In the preceding examples it is required to subtract one number only from any given minuend. It may be required, however, to determine the remainder which is left after several numbers have been taken from a given minuend, and the successive remainders. The most obvious manner of proceeding in such a case is, to take one subtrahend from the minuend, a second from the 1st remainder, a third from the 2d remainder, &c., till all the numbers are subtracted. The order of making the subtractions cannot, it is evident, affect the last remainder so long as the absolute number of individuals taken away is the Whence the subtractions may be made in any order. same. Since the last remainder is the difference between the minuend and the sum of all the subtrahends, if the latter are added together, and their sum at once taken from the minuend, the result must be equal to that obtained by the method of repeated subtractions; for thus also the absolute number of individuals subtracted is the same. 42. Any given number can be subtracted from 1, followed by as many zeros as the given number contains figures. The remainder is termed the Arithmetical Complement of the given number. c 3 Thus the arithmetical complement of 7 is 10-7=3; of 84 is 100-84=16; Whence, generally, the sum of a number and its arithmetical complement is equal to 10, considered as having the relative value of the highest figure of the number. When the difference of two numbers is to be found, if the less is taken from the greater, the remainder is the difference required. But if, instead of taking the less number from the greater, the arithmetical complement of the less is added to the greater, the result must contain ten units, of the same order as the highest figure of the subtrahend, more than the remainder obtained by the common process of subtraction. Taking away these ten units, the results are equal. Whence the arithmetical complement of a number supplies the means of changing subtraction into addition. Examples of subtraction made by adding the arithmetical complement of the subtrahend to the minuend: Example 1. From 795484 take 389685. Solution by subtraction. From 795484 Take 389685 Rem. 405799 Solution by addition of arith. compt. To the minuend 795484 Add arith. compt. of 389685 610315* From this sum subtract 1405799 - - 10 When in the same calculation it is requisite to add several numbers and to subtract several others, the arithmetical complements of the latter may sometimes be employed with advantage. But it is chiefly in calculating with logarithms that this employment of the arithmetical complement is of much utility. 43. The subtraction of one number from another is indicated by writing the minuend to the left and the subtrahend to the right of the symbol-. Thus, the expression 25-18 indicates that 18 is to be subtracted from Replacing particular by general expressions of number, the subtraction of b from a is indicated by the expression a-b. *If the subtraction of any given subtrahend from 10, followed by the proper number of zeros, is made from right to left, it is evident that ones, for tens borrowed, must either be carried to the figures of the subtrahend or taken from the corresponding figures of the minuend; since the figures of the minuend are zeros, excepting the last, if the second method is adopted (see Article 35) the last significant figure of the subtrahend must be taken from 10 and the others from 9. Whence the subtraction, which gives the arithmetical complement of a number, may be performed from left to right, by taking the figures of the subtrahend each from 9, excepting the last significant figure, and this from 10. The arithmetical complement of the subtrahend in Example 1. may be found thus: 9-369-8-19-9-09-6=3:9-8=1:10-5=5; and 610315, the number composed of these remainders, is the arithmetical complement required. With a little practice the arithmetical complement of any number may be found in this manner mentally, and with very little trouble. SUBTRACTION. If two quantities, b, c, are both to be taken from a, the expression of the remainder is a-b-c or, a-(b+c) Art. 41. If the excess of a over the sum of b and c is equal to d, the subtraction is expressed thus: a-(b+c)=d. When the quantities whose difference is required are similar, that difference is expressed in terms of the common quantity. Thus 5a-3a=2a: And, ma-na=(m—n)a. For 5a and 3a express respectively 5 repetitions and 3 repetitions of the common quantity a. From 5a=a+a+a+a+a. Take 3a=a+a+a. The remainder=a+a=2a. Therefore 5a-3a=2a=(5-3)a. Again, ma is equal to a repeated as often as m contains simple units; and na is equal to a repeated as often as n contains simple units. Therefore, if n repetitions of a are taken from m repetitions of a, the Whence remainder must contain a as often as the excess of m over n contains simple units. This excess is represented by the expression m―n. ma-na =(m—n)a. Exercises in the subtraction of general expressions of quantity: rem.=3a. 1st. from 4a subtract a 12a 9a+6b 6th. 15x a+b (m-12)a. Verification of Addition and Subtraction. 44. The composition and decomposition of numbers, as explained under the titles of Addition and Subtraction, depending partly on the conventions of numeration, and partly on axioms or first principles of indisputable certainty, the accuracy of the methods can require no further proof. But as those combinations of number which constitute the operations of Addition and Subtraction are made, chiefly or altogether, through means of the memory, and are on this account liable to the suspicion of error, it becomes necessary to find some test, by the application of which the accuracy of any given result may be tried. Such process may, with more propriety, be termed verification than (what it is commonly called) proof. The process of verification ought, for an obvious reason, to be of, at the utmost, not greater difficulty than that of the operation whose accuracy is tested; for if the verification be the more difficult, so also will the chance of error in performing it be greater, and the coincidence of result less probable. 45. In addition the accuracy of the result may be tried in several ways. 1st. If the operation has been performed by commencing with the last figure of each column and adding upwards, a repetition of the process, only beginning with the first figure of each column and adding downwards, will constitute a verification. For, the combinations being made in an order one the reverse of the sumed correct. The reason is plain: the result arises from the combination of all the addenda into one number; and if the whole of these be again taken away there can be no remainder. 3d. Another verification is obtained by combining the preceding methods. The manner of effecting this may be, perhaps, most clearly explained through the medium of an example. Let, therefore, the numbers to be added together be, To verify the result, begin with the left column, and proceed thus: 3+4=7; 7+8=15; 15+5=20. From the two highest figures in the sum, i. e. 23, subtract 20, the sum of the left column, and to the remainder 3 annex 0, the next figure of the sum. The sum of the second column=7+9+4+8=28. Take this from 30; and to the remainder, 2, bring down 5, the next figure of the sum. The sum of the third column is 23, and 25-23-2. To this remainder annex 3, and from the 23 subtract the sum of the fourth column, which is 22. To the remainder =1, annex 9, the last figure of the sum, and from 19. subtract the sum of the last column; this sum is also 19. 19-19=0. The last subtraction leaves no remainder, from which the inference is that the addition is correct. The subtraction of the sum of the numbers in each column from the corresponding figures in the entire result may be made mentally; in which case the work may stand thus, 37865 49736 84297 58641 230539 82210 3, 2, 2, 1, 0, are the remainders from the partial subtractions. As soon as each figure is cancelled, in succession, by these subtractions, a stroke is drawn through it to record the fact. These remainders, it will be observed, are the tens which in the course of the addition were carried from the sums of the lower columns respectively, and combined as units with the next higher. 46. The most simple and obvious verification of subtraction is made by reversing the operation. The remainder being obtained by taking the subtrahend from the minuend, it follows that, adding together the subtrahend and remainder, their sum ought to reproduce the minuend. If not, the subtraction ought to be revised as probably erroneous. Example: From 517846 Whence, since the sum of the remainder and subtrahend is equal to the minuend, the subtraction is to be presumed correct. Another mode of verifying subtraction is to take the remainder from the minuend. The number left ought in this case to be the subtrahend. Example: From 736942 } =547986=the subtrahend. Subtracting the remainder from the minuend, their difference is Remark. These verifications, as well as the results which they are employed to corroborate, are in a great measure performed mentally, and their accuracy must depend upon the strength of memory and closeness of attention which the computer brings to the performance of his task. It is unlikely that, the work and the verification starting from different points and following different routes, they should both agree, and yet be both in error. Such a coincidence is not, however, impossible. Hence the utmost which one is entitled to infer from the agreement of a verification with a given result is, that a strong probability, but still not an absolute certainty, of its accuracy exists. Multiplication and Division. 47. Addition and subtraction are, in effect, the only elementary operations of arithmetic. But when, in a case of addition, the addenda, instead of being different numbers, are all equal, the work admits of considerable abbreviation. The methods by which this is accomplished are constituted into an operation, also generally regarded as elementary, and which has been named Multiplication. The terms Addenda and Sum, respectively employed to distinguish the numbers which are to be added together, and the result of such addition are, like the name of the process itself, replaced by others. The addendum, or single number which is to be repeated a given number of times, is termed Multiplicand; the number which expresses the times of repetition, Multiplier; and the sum or result of these repetitions, Product. The multiplicand and multiplier are also termed Factors (that is, makers) of the product. The symbol, x (into, or multiplied by), placed between two numbers, indicates that the one is to be multiplied by the other. 48. A similar arrangement is made with respect to a particular case of subtraction. If, instead of taking the subtrahend only once from the minuend, it were required to take the subtrahend again from the first remainder, then from the 2d, 3d, . . . . until a remainder either less than the subtrahend or equal to zero were attained, for the purpose of ascertaining the exact number of subtractions which must be made in order to exhaust the minuend the repeated subtractions would be tedious. To avoid this an abridged process has been devised, and this is called Division. The number, from which the repeated subtractions are made, is termed Dividend; that by the repeated subtraction of which the dividend is exhausted, Division; and the number expressing the times of subtraction possible, Quotient. |