SUPPLEMENT to Subtraction.

QUESTIONS.

1. WHAT is Simple Subtraction?

2. How many numbers must there be given to perform that operation? 3. How must the given numbers be placed?

4. WHAT are they called?

5. WHEN the figure in the lower number is greater than that of the upper number from which it is to be taken, what is to be done?

6. How does it appear, that in subtracting a less number from a greater, the occasionally borrowing of ten, does not affect the difference between these two numbers?

7. How is subtraction proved?

8. WHEN, and how may Subtraction be of use to a man engaged in the pursuits of life?

NOTE. In case of borrowing ten, it is a matter of indifference, as it respects the operation, whether we suppose 10 to be added to the upper figure, and from the sum subtract the lower figure and set down the difference ; or, as Mr. PIKE directs, first, subtract the lower figure from 10, and adding the difference to the figure above, set down the sum of this difference and the upper figure. The latter method may, perhaps, be thought more easy, but it is conceived, that it does not lead the understanding of youth so directly into the nature of the operation as the former.

§ 3. Simple Multiplication.

SIMPLE MULTIPLICATION teaches, having two numbers given of the same. denomination, to find a third which shall contain either of the two given numbers as many times as the other contains a unit. Thus, 8 multiplied by 5, or 5 times 8 is 40.-The given numbers ( 8 and 5) spoken of together are called Factors. Spoken of separately, the first or largest number (8) or number to be multiplied, is called the Multiplicand; the less number, (5) or number to multiply by, is called the Multiplier; and the amount, (40)the product. THIS operation is nothing else than the addition of the same number several times repeated. If we mark 8 five times underneath each other and add them, the sum is 40, equal to the product of 5 and 8 multiplied together. But as this kind of addition is of frequent and extensive use, in order to shorten the operation we mark down the number only once, and conceive it to be repeated as often as there are units in the multiplier.

BEFORE any progress can be made in this rule, the following Table must be committed perfectly to memory.

MULTIPLICATION TABLE.

1 2 3 4 5 6 6 7 8 9 10 11

8

8

8

8

8

40

| |

12

816 24 32 40 48 56 64 72 80¦ 88 96

12 24 36 48 60 72 8496 108 120 132 144

By this Table the product of any two figures will be found in that square which is on a line with the one and directly under the other. Thus, 56 the product of 7 and 8, will be found on a line with 7 and under 8 : so 2 times 2 is 4; 3 times 3 is 9, &c. in this way the Table must be learned and remembered,

RULE.

1. PLACE the numbers as in Subtraction, the larger number uppermost with units under units, &c. then draw a line below.

2. WHEN the Multiplier does not exceed 12; begin at the right hand of the multiplicand, and multiply each figure contained in it by the multiplier, setting down all over even tens and carrying as in addition.

3 WHEN the multiplier exceeds 12; multiply by each figure separately,first by the units of the multiplier, as directed above, then by the tens, and the other figures in their order, remembering always, to place the first figure of each product directly under the figure by which you multiply; having gone through in this manner with each figure in the Multiplier, add their several products together, and the sum of them will be the product required.

In this example, as the Multiplier exceeds 12, therefore, you must multiply by each figure, separately. First, by the units (7) just in the manner of the other examples. Secondly, by the tens (3) in the same way, excepting only, that the first figure of the product in the multiplication by 3, must be placed under the 3, that is, under the figure by which you multiply. Lastly, add these two products together, the sum of them is the answer.

PROOF.

MULTIPLICATION may be proved by Division, but a method more concise

and easy, often practised by accountants, and which I shall recommend, is called

Casting out the 9's.

CASTING out the 9's from any sum or number, is the exhausting of that number by the figure 9, till there is nothing left of it but a remainder, or excess over even nines which remainder or excess is the thing sought.

How to cast out the 9's.

WHATEVER method may be adopted, this in effect, is nothing else than dividing the number by 9. The operation, however, would be tedious as naturally practised by division; besides, as yet, we do not suppose the learner acquainted with it. A shorter and more successful way is the following

METHOD.

BEGINNING at the right hand of the number, add the figures, and when the sum exceeds 9, drop the sum and begin anew by adding, first, the figures, which would express it. Pass by the nines, and when the sum comes out exactly 9, neglect it; what remains after the last addition will be the remainder sought.f

EXAMPLES.

Ir it be required to cast the 9's out of 576394, proceed thus ;-5 to 7 is 12 which sum (twelve) as it exceeds 9 you must drop, and begin anew, first add the figures (12) which would express twelve, saying 1 to 2 is 3 and (proceeding with the other figures, which remain to be added)6 is 9, being

†This Method of Casting out the 9's succeeds on a

PRINCIPLE.

THAT every Sfigure, in rising from the place of units to that oftens, takes to itself the addition of 9 times its value. The same from tens to hundreds, &c. CONSEQUENTLY, if any figure for instance *4 be removed from units place and divided by 9, it will leave a remainder of 4 ; the same of any other figure, removed and divided by 9 it will leave a remainder of ITSELF, and that only.

THEREFORE, if any †† number be divided by 9; or, the figures which express that number be added together and the sum of them divided by 9, the remainders will be equal.

§ MADE evident thus ;-1 in the place of units is the expression of an individual or one, in the place of tens (10) it is the expression of ten individuals, or ones; therefore taking 1 (one) its signification in units place, from 10 (tén)its signification in tens place, leaves 9, the increase of 1, or 9 times its value, in rising from the place of units to that of tens.

*4 removed from units place by a cypher is 40, which divided by 9 leaves 4 (4 times 9 is 36.J

6 removed by a cypher is 60 which divided by 9 leaves a remainder of 6; or 600 divided by 9 still the remainder is 6, the remainder always begin the same figure whatever may be the place of its removal if divided by 9.

‡‡ Thus, 5683 divided by 9 the remainder is 4; let the figures which express the number 5683 be added together-5 to 6 is 11 and 8 is 19 and 3 is 22, which number (22) divided by 9 leaves a remainder of 4, the same as when the number 5683 was divided by 9. THESE properties of the figure 9 belong to none other of the Digits, excepting to the figure 3, and this figure (3) possesses them in consequence only of being an even part of 9.

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