SIMPLE SUBTRACTION is taking a less number from a greater of the same denomination, so as to shew the difference or remainder; as 5 taken from 8, there remains 3. The greater number (8) is called the Minuend, the less number (5) the Subtrahend, and the difference (3) or what is left after Subtraction, the Remainder. "Place the less number under the greater, units under units, tens under tens, and so on. Draw a line below; then begin at the right hand, and subtract each figure of the less number from the figure above it, and place the remainder directly below. When the figure in the lower line exceeds the figure above it, suppose 10 to be added to the upper figure; but in this case you must add 1 to the under figure in the next column before you subtract it. This is called, borrowing ten." PROOF. Add the remainder and subtrahend together, and if the sum of them cor respond with the minuend, the work is supposed to be right. Minuend 8 6 5 3 Subtrahend 5 2 7 1 The numbers being placed with the larger uppermost, as the rule directs, I begin with the unit or right hand figure in the subtrahend, and say, 1 from 3 there remain 2, which I set Remainder 3 3 8 2 down, and proceeding to tens, or the next figure, 7 from 5 I cannot, I therefore borrow, or supProof 8 6 5 3 pose ten to be added to the upper figure (5) which make 15, then I say 7 from 15 and there remain 8, which I set down: then proceeding to the next place, I say, 1 which I borrowed to 2 is 3, and 3 from 6 and there remain 3; this I set down, and in the next place I say 5 from 8 and there remain 3, which I set down and the work is done. PROOF. I add the remainder to the subtrahend, and finding the sum just equal to the minuend, suppose the work to be right. NOTE. The reason of borrowing ten, will appear if we consider, that, when two numbers are equally increased by adding the same to both, their difference will be equal. Thus the difference between 3 and 5 is 2; add the number 10 to each of these figures (3 and 5) they become 13 and 15, still the difference is 2. When we proceed as above directed, we add or suppose to be added, 10 to the minuend, and we likewise add one to the next higher place of the subtrahend, which is just equal in value to 10 of the lower place. Note.-In case of borrowing ten, it is a matter of indifference, as it respects the operation, whether we suppose ten 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. 1. From 1 0 2 3 6 7 4 2 3 1 7 9 8 10 6 2 8 7 9 1 2 8 4 5 0 6 7 0 3 2 8 1 Take Rem. 2. From Take Rem. 1 0 2 3 6 7 4 2 3 1 7 9 8 1 0 6 2 3. From 21468317012101, take 568497067382. Rem. 20399819944719 Rem. 85306959447 4. From 364710825193, take 279403865746. 5. From 168012372458, take 89674807683. 6. From 100610528734, take 99874197867. 7. From 628103570126, take 248976539782. 8. From 10000, take 9999. Rem. 1. 9. From 10000, take 1. Rem. 9999. The distance of time since any remarkable event, may be found by subtracting the date thereof from the present year. EXERCISES. So, likewise, the distance of time from the occurrence of one thing to that of another, may be found by subtracting the date of the thing first happening, from that of the last. EXAMPLE. 1. How long from the discovery of America by Columbus, 1492, to the commencement of the war, 1775, which gained our Independence? 1 7 7 5. Ans. 2 8 3 years. 2. How long from the termination of the war in 1783, which gained our Independence, to the commencement of the last war between the United States and Great Britain in 1812? Ans, 29 years. C 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 occasional borrowing of ten does not affect the difference between these two numbers ? 5. Suppose a man to have been born in the year 1745, how old was he in 1799? Ans, 54 years. 6. What number is that to which if you add 789 it will become 6350 ? Ans. 5561. 7. Supposing a man to have been 63 years old in the year 1801; in what year was he born? Ans. in the year 1738. 8. At the census in 1800, the number of inhabitants in the New-England States was 1233011; at the late census in 1810, the number was 1471937 What was the increase of the population in the New-England States in the ten years between 1800 and 1810? Ans. 238926. 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. Before any progress can be made in this rule, the following Table mus be committed perfectly to memory. 8 | 16 | 24 | 32 | 40 | 48|56|64| 99 | 108 10 | 20 30 | 40 | 50 | 60 | 70|80| 90 | 100 | 110 | 120 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. and 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. |