1 2 3 4 5 6 Table of Subtraction. 7 8 9 10 11 12 13 14 15 16 17 18 19 10 1 2 3 4 5 6 7 8 9 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 9 In this table, the uppermost horizontal column contains the greater numbers, and the left vertical column the less; also the excess of any number in the former over any number in the latter, is placed in the square formed by the intersection of a vertical column, descending from the larger numbers, with a horizontal column extending to the right of the smaller numbers. Thus, to find the difference of the numbers 12 and 5: tracing the vertical column 12 downwards to its intersection with the horizontal column 5, the number 7 is found to fill the square formed by their intersection. Whence, according to the table, 7 is the difference of the numbers 12 and 5. 34. Applying the principles which have been established to the solution of questions, let it be required to find the difference of the numbers 9 and 4. The subtraction of 4 from 9 may be made by the method of decomposition explained in Article 27 thus :— 9-1=8 = 7 4 = :6 ..945, the difference sought. Otherwise, the subtraction may be made by successive abstractions of unity from both minuend and subtrahend until the latter be exhausted, thus :— 9-4-8-3=7-2=6—1—5—0=5 Referring to the table of subtraction, it is found that 9—4—5. As an example involving larger numbers, let it be required to find the difference of the numbers 57 and 43. It is demonstrated (Art. 31) that the difference of large numbers is found by taking the units of the subtrahend from the units of the minuend, the tens from the tens, &c.; whence 7-3=6-2=5—1=4―0=4=difference of the units, The difference of the numbers 57 and 43 is thus composed of 1 ten and 4 units, consequently it is 14. Required the remainder? Subtracting, as in the last example, the units of the subtrahend from the units of the minuend, and the tens from the tens, the results are 7-4-3 difference of the units, 3-9, the subtraction cannot be effected. The method of subtracting a number by parts creates, in this instance, a difficulty; it requires 9 to be taken from 3, a greater number from a less; and yet, seeing that 237 exceeds 94, the subtraction of the whole number 94 from 237 can certainly be accomplished. The difficulty is removed thus: a convention made to facilitate the expression of numbers by means of cyphers is, that any figure placed to the left of another shall be held to express units of the order immediately superior to those of that other figure (Art. 5). Whence, in the example under consideration, one in the place of hundreds is equal to ten in the place of tens. Let one be taken, or (as the usual phrase is) borrowed, from the hundreds, this 1 hundred=10 tens, and 10 tens+3 tens=13 tens, from these 13 tens take 9 tens, then 13-9=4=difference of the tens. The subtrahend contains no units of the 3rd order; but, on account of the 10 tens, or 1 hundred, borrowed from the hundreds of the minuend to render possible the subtraction of 9 from 3, the 2 must be diminished by 1; .'.2—1=1=difference of the hundreds. Assembling the remainders of the partial subtractions, 3 units, 4 tens, and 1 hundred, or 143, compose the general result. Whence, from 237 taking 94, the remainder is 143. In figures the operation will stand thus, bund. tens units Required the remainder ? In order that the subtraction of the 8 units of the subtrahend from the 7 units of the minuend may be rendered possible, it becomes necessary to borrow 1 ten=10 units from the 6 tens of the minuend. The 6 tens and 7 units are, in consequence, to be taken as 5 tens and 17 units. Whence, proceeding with the subtraction of the units, 17-8-9-difference of the units. To subtract 7 from 5 a like arrangement is requisite, by which means the 5 in the second place becomes 15, and the 3 in the third place is reduced to 2. Then 15-7-8=difference of the tens, 2-2=0=difference of the hundreds, Consequently 9 units+8 tens +0 hundreds+2 thousands, that is 2089, is the remainder left by the subtraction of 3278 from 5367. C 36. To accomplish, in the last example, the subtraction of 8 units from 7, 1 ten or 10 units are borrowed from 6, the next figure to the left; and the 6 is, in consequence, reduced to 5. But, since the absolute difference of two numbers is not changed by the addition of unity to each of them, the result will be the same, whether the 1 borrowed be taken from the succeeding figure in the minuend, and the corresponding figure in the subtrahend be suffered to remain as it is, or the minuend be left as it is and 1 be added to the figure in the subtrahend, for this amounts simply to the increasing of each by unity. The second method, namely, that of carrying one, for ten borrowed, to the next higher figure of the subtrahend, instead of deducting one from the corresponding figure of the minuend, is usually adopted in practice, on account of its convenience. 4 being less than 6, 10 are borrowed; 10+4=14 and 14-6-8-difference of the units; carrying to 8, the next figure of the subtrahend, 1 for the ten borrowed, 8+1=9; the next figure to be subtracted, 5, being less than 9, 10 are borrowed, 10+5=15 and 15-9=6=difference of the tens ; carrying 1 for 10 borrowed, 9+1=10, and borrowing again, to render the subtraction possible, 10+9=19 and 19-10-9-difference of the hundreds ; carrying 1 for 10 borrowed, 1+1=2, 2 being less than 3, the subtraction is made at once; 3-2=1=difference of the thousands. By this example it appears that (carrying for loans to the subtrahend) it may be necessary to subtract 10 from 19; a contingency which has been provided for in the table of subtraction. Abridged detail of the operation: 10-2=8=units of the difference. 1+8=9 and 10-9=1=tens of the difference. 1+4 5 and 11-5=6=hundreds of the difference. 1+6=7 and 10-7=3=thousands of the difference. 37. In the preceding example, the places of several orders of units in the minuend are filled with zeros, the corresponding orders of the subtrahend being significant figures. If, instead of carrying one to the next figure of the subtrahend for every ten borrowed, the other method of taking one from the next figure of the minuend were followed, the details of the work would require some explanation. For instance, 2 units cannot be subtracted from 0: 1 ten 10 units, ought therefore to be borrowed from the tens of the minuend. But the place of tens (or units of the 2d order), is also occupied by 0; and from 0, 1 cannot be borrowed. In such a case it is necessary to proceed to the hundreds (or units of the 3d order) of the minuend, borowing 1 from the units of the 3d order: 1 hundred 10 tens 9 tens +10 units; whence 10-2-8=units of the difference, and 9-8=1=tens of the difference. The figure expressing units of the third order is 1; and 1 having been borrowed from it, this I is reduced to zero. 4, the corresponding figure of the subtrahend, cannot be taken from 0; therefore a necessity of borrowing recurs. The places of the units of the 4th and 5th order being both filled with zeros, the loan must be taken from the units of the 6th, that is, the hundred thousands: 1 hundred thousand=10 ten thousands. 9 ten thousands +10 thousand. = 9 ten thousands+9 thousand +10 hundred. Whence, 10-4-6-hundreds of the difference. 9-6-3 thousands of the difference. 9-2-7 ten thousands of the difference. The figure in the 6th place, from which 1 was taken, is 4: on account of the 1 borrowed it is reduced to 3. 3 being less than 5, the corresponding figure of the subtrahend, 1 must be borrowed, not from the 0, which fills the place of the units of the 7th order, but from 7 of the 8th order. 1 unit of 8th order=10 millions. = 9 millions+10 hundred thousand. Also 10 hundred thousand+3 hundred thousand=13 hundred thousand. Whence, 13-5=8=hundred thousands of the difference, and 9-3-6=millions of the difference. The 7 in the 8th place is reduced to 6 by the loan taken from it, and 6 being less 9, 1 is borrowed from the 3 in the 9th place. 1 unit of 9th order=10 units of 8th order. 10+6=16 and 16-9-7-ten millions of the difference. The 3 in the 9th place is reduced to 2 by the 1 borrowed, and 2—2=0. Whence, the remainder contains no units of the 9th order. Collecting the partial differences, the result, as before, is 76873618. 38. If every figure of the subtrahend is less than the corresponding figure of the minuend, the subtraction may be performed from right to left, or from left to right, at pleasure. Thus, from 7854 to take 3721, the remainders, beginning with the highest figure, are, 7-3-4-thousands of the difference. 8-7=1=hundreds of the difference. 4-1=3=units of the difference. And the whole remainder is composed of 4 thousands +1 hundred+3 tens +3 units, or 4133. Subtracting from right to left, the same results are obtained, but in a contrary order. When, however, a figure of any order of units in the subtrahend is greater than the corresponding figure in the minuend, and a necessity of borrowing arises, the course of the operation from left to right is encumbered with troublesome reductions. As affording an instance of such, let it be required from 5748 to subtract 3876. 5-3-2-thousands of the difference. 7-8; without a power of borrowing, the subtraction is impossible. The loan, besides, must be taken from the next higher figure, that on the left; and the operation, so far as regards this figure, is already performed. Ten, however, must be borrowed, and the thousands of the difference made less by 1. Proceeding, 7+10=17 and 17-8-9-hundreds of the difference. 4-7; again the subtraction is not possible without a loan. One must be borrowed from the hundreds of the difference, and the proper reduction made; 4+10=14 and 14-7=7=tens of the difference, 8-6=2=units of the difference. The result of the preceding operations may be put down as follows: If zero occupy any place in the unreduced remainder, and it be requisite, on account of a loan, to take 1 from this 0, a further reduction will be necessary. The conventions, that ten units of any order shall be considered as one unit of the next higher order, and that this shall be written immediately to the left of those units, from the decomposition of which into tens and units it has arisen, fix the direction in which the progression of numbers by tens is carried, namely, from right to left, and confer a facility on the performance in the same direction of those arithmetical operations in which reductions between units of different orders may be necessary. If in addition it were never requisite to carry, or in subtraction to borrow, the direction in which the work is performed would be of no importance. But when the units in the sum of a column exceed 10, the next figure to the left must be augmented; and when, in subtraction, a figure of the subtrahend is greater than the corresponding figure in the minuend, one must be borrowed from the next figure to the left. Proceeding from right to left, these operations fall into the work; but from left to right, they give rise to new additions and subtractions.* 39. To find the difference of two unequal numbers: General Rule. Write the minuend, and underneath it the subtrahend, in such a manner that the figure expressing the units of any order in the latter shall be under the figure expressing units of the same order in the former. Then, if all the figures of the subtrahend are less than the corresponding figures of the minuend, take the simple units of the subtrahend from the simple units of the minuend, and record the excess of the latter below the subtrahend, in the place of simple units. Proceed after the same manner to take the tens of the subtrahend from the tens of the minuend, the hundreds from the hundreds, &c., *The tens arising in addition from the sum of any column might be disposed of by an arrangement expressing the deficit of the column from, or its excess over, certain repetitions of ten; and in subtraction, whenever a figure of the minuend is less than that figure of the subtrahend which is to be taken from it, a means could be devised to express the deficient units. Thus in addition the operation could be performed from left to right, without a necessity of disturbing any part of the result by subsequent combination of the tens arising from collections of inferior units; and in subtraction, without borrowing from the units of that order immediately to the left of any deficient figure. But such contrivances, without being of any real utility, infringe the principles of decimal notation, and the results expressed by their help would require some reduction before they could be made to enter into combination with numbers expressed in the usual manner. 1 |