and 14, and then subtract the 6 units from 14, the 8 tens from 15. tens, and the 2 hundreds from 4 hundreds; and thus, when any digit of the minuend is less than its corresponding digit of the subtrahend conceiving an unit prefixed to it and performing the subtraction, we may proceed to the next left-hand figure of the subtrahend, and conceive it to be less by one, on account of 1 already borrowed from it: but it affords the same result in practice, to conceive the next digit of the subtrahend increased by one, and the digit of the subtrahend unaltered; as it gives the same remainder to subtract 9 from 16, as to subtract 8 from 15, and hence appears the reason of what is called carrying, and proceeding from right to left.* An Example in figures will render the above as plain as can be required. 178884-100000+ 70000 + 8000+ 800+ 80+ 4 See Walker's Philosophy of Arithmetic, page 7 NOTE.-In compliance with custom, I have placed the figures in the above Examples, as is generally practised; but a child should be taught to subtract a less number from a greater, however placed, whether above or below, or in any manner, which 1 have found of advantage. In the 14th Example, the difference is 100; the stroke or vinculum signifying, that the total of the sums after the sign minus, is to be subtracted from the total of the three preceding sums. MULTIPLICATION. MULTIPLICATION is a compendious kind of Addition, consisting of three terms, called Multiplicand, Multiplier, and Product. The first is the number given to be multiplied, the second that by which the work is performed, and the third the result of the operation or answer. × This sign, or a period, interposed between two numbers, shows that one is to be multiplied by the other; that is, the number contained in one, shall be expressed as often as there are units in the other. Thus, 5X4=20, or 5·4=20, shows that 5 is to be multiplied by 4; the product of which is 20. Here 5 is the multiplicand, 4 the multiplier, and 20 the product arising from the multiplication of those numbers. The multiplier and multiplicand are also called factors. Before the learner proceeds farther, it will be necessary that he shall commit to memory the following To multiply one number by another, both of which shall be of the same name or denomination, take the following RULE. Place the multiplicand, and multiplier, if they consist of significant figures, or have ciphers intermixed under each other, units under units, tens under tens, &c.; but if they have ciphers annexed to one or both, proceed in this manner with the other figures only; after which, annex the ciphers in their proper places, without regard to the former limitation. Then commencing at the first right-hand significant figure of the multiplier, multiply every digit or figure of the multiplicand successively thereby, and set down the right-hand or unit's figure of each product, directly under the figure then multiplying, carrying the other figure as so many ones to the next product in succession, and set down the entire of the last product, as is done with the sum of the last column in Addition. Multiply, in like manner, by the next significant figure of the multiplier, taking care to put the first and each succeeding first figure of every product directly under the figure then multiplying, as before directed, placing each product in a line under the former; thus proceed, until all the figures in the multiplier are gone through, when there will or ought to be, as many rows of figures as there are significant figures in the multiplier; add up the several products, and annex to the sum as many ciphers (if there were any) as were at the right of the multiplier or multiplicand, or both, as the case may be, for the total product required.* *The above rule is best illustrated by Examples. 1. Let 5736 be multiplied by 6. = 600 × 20 × 4. Then from the above Example. Here 624 192852 × 100 19285200 |