§ 2. Simple Suvtraction.

SIMPLE SUBTRACTION is the taking a lefs number from a greater of the fame denomination, fo as to fhew the difference or remainder; as 5 taken from 8, there remains 3.

The greater number (8) is called the Minuend, the lefs number (5) the Subtrahend, and the difference (3) or what is left after subtraction, the Remainder.

66

RULE.

"Place the less number under the greater, units under units, tens under 66 tens, and fo on. Draw a line below; then begin at the right hand, and " fubtract each figure of the lefs number from the figure above it and place the remainder directly below. When the figure in the lower line "exceeds the figure above it, fuppofe 10 to be added to the upper figure; "but in this cafe you must add 1 to the under figure in the next column "before you fubtract it. This is called borrowing ten."

PROOF.

Add the remainder and fubtrahend together, and if the fum of them correfpond with the minuend, the work is fuppofed to be right.

Subtrahend 5 27 1

EXAMPLES.

Minuend 8 6 5 3 The numbers being placed with the larger uppermoft, as the rule directs, I begin with the unit or right hand figure in the fubtrahend, and fay, 1 from 3 and there remain 2, which I fet down, Remainder 3 3 8 2 and proceeding to tens, or the next figure, 1 fay 7 from 5 I cannot, I therefore borrow, or fuppofe 10 Proof to be added to the upper figure (5) which make 15, then I fay, 7 from 15, and there remain 8, which I fet down; then proceeding to the next place, I fay, 1 which I borrowed to 2 is 3, and 3 from 6 and there remain 3; this I fet down, and in the next place I fay 5 from 8 and there remain 3, which I fet down, and the work is done.

8653

PROOF. I add the remainder to the fubtrahend; on finding the fum juft equal to the minuend, and fuppofe the work to be right.

1

NOTE. The reaf of borrowing ten will appear if we confider, that, when two numbers are equally increased by adding the fame to both, their difference will be equal. Thus, the difference between 3 and 5 is 2; add the number 19 to each of these figures (3 and 5) they become 13 and 15, ftill the difference is 2. When we proceed as above directed, we add or fuppofe to be added, 10 to the minuend, and we likewife add 1 to the next higher place of the fubtrahend, which is just equal in value to 10 of the lower place.

2. From 3 2 7 8 6 5 3 2 1 4 6 06 79 3 6 1 2

Take 1

5 the minuend,

3

4

2 the fubtrahend.

Remainder.

6. From 3 7 5 1 dollars, take 1 67 4 dollars. Write the lefs number under the greater, with units under units, &c. as the rule directs.

The distance of time fince any remarkable event, may be found by fub. tracting the date thereof from the present year.

Supplement to Subtraction.

QUESTIONS.

1. What is Simple Subtraction?

2. How many numbers must there be given to perform that operation? 3. How muft 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 lefs number from a greater, the occafional 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 ufe to a man engaged in the purfuits of life?

EXERCISES.

1. What is the difference between 78360 and 5841 ?

Ans. 72939

2. From a piece of cloth that meafured 691 yards, there were fold 278 yards; how many yards fhould there re main? Ans. 413

NOTE. In cafe of borrowing ten, it is a matter of indifference, as it refpects the operation, whether we fuppofe 10 to be added to the upper figure, and from the fum fubtract the lower figure and fet down the difference; or as Mr. PIKE directs, first, fubtract the lower figure from 10, and adding the difference to the figure above, fet down the fum of this difference and the upper figure. The latter method may, perhaps, be thought more eafy, but it is conceived, that it does not lead the understanding of youth fo directly into the nature of the operation as the former.

§3. Simple Multiplication.

SIMPLE MULTIPLICATION teaches, having two numbers given of the fame 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) fpoken of together are called Factors. Spoken of separately, the firft or largest number, (8) or number to be multiplied, is called the Multiplicand; the lefs number, (5) or number to multiply by, is called the Multiplier, and the amount, (40) the Product.

8

8

This operation is nothing else than the addition of the fame number feveral times repeated. If we mark 8 five times underneath each other and add them, the fum 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 fhorten the operation, we mark down the number only once, and conceive it to be repeated as often as there are units in the multiplier.

8

8

8

Before any progress can be made in this rule, the following Table 40 must be committed perfectly to memory.

MULTIPLICATION TABLE.

|12|24|36| 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144

By this Table the product of any two figures will be found in that fquare 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: fo 2 times 2 is 4; 3 times 3 is 9, &c.—In this way the table must be learned and re membered.