NOTE 1. Billions is substituted for millions of millions: Trillions, for millions of millions of millions; Quatrillions, for millions of millions of millions of millions, and so on. These names of periods of figures, derived from the Latin numerals, may be continued without end. They are as follows, for twenty periods, viz. Units, Millions, Billions, Trillions, Quatrillions, Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredeciltions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, Octodecillions, Novemdecillions, Vigintillions. Fifteen. THE APPLICATION. Write down, in proper figures, the following numbers. Two hundred and seventy nine. Three thousand four hundred and three. Thirty seven thousand, five hundred and sixty seven. $ Four hundred, one thousand and twenty eight. f Fitty five millions, three hundred, nine thousand and nine. 15 279 37567 9072200 2513131702 Write down in words at length the following numbers. A less literal Lumber placed after a greater, always augments the value of the greater; if put before, it diminishes it. Thus, VI. is 6; IV. is 4; XI. is 11; IX. is 9, &c. The practice of counting on the fingers doubtless originated the method of Notation by Roman Letters. The letter I was taken for one finger, or one; and hence II, for two; III, for three; IIII, for four and V, as representing the opening between the thumb and fore-finger, and being also an easier combination of the marks for the fingers, was taken for five. As IV is a simpler expression for four than the above, it was doubtless adopted for this reason, and on the general principle too that a less literal number placed before a greater should diminish the greater so much, and, placed after a greater should augment it so much. Hence as IV, is four; VI is six; VIII is eight, and so on. Ten was expressed by X, because it is two Vs united, and twice five is ten. Fifty was expressed by L, because it is half of C of E, as it was anciently written, and C is the initial of the Latin centum, one hundred. Five hundred is expressed by D, because it is half of the Gothic CD or M., the initial of mille, one thousand. ADDITION Is the putting together of two or more numbers, or sums, to make them one total, or whole sum. SIMPLE ADDITION Is the adding of several integers or whole numbers together, which are all of one kind, or sort; as 7 pounds, 12 pounds, and 20 pounds being added together, their aggregate, or sum total, is 39 pounds. RULE. Having placed units under units, tens under tens, &c. draw a line underneath, and begin with the units; after adding up every figure in that column, consider how many tens are contained in their sum, and placing the excess under the units, carry so many as you have tens, to the next column, of tens: Proceed in the same manner through every column, or row, and set down the whole amount of the last row.* * This rule as well as the method of proof, is founded on the known axiom, "the whole is equal to the sum of all its parts." The method of placing the numbers, and carrying for the tens, is evident from the nature of notation; for, any other disposition of the numbers would alter their value; and carrying one, for every ten, from an inferiour to a superiour column, is, evidently, right; because one unit in the latter case is equal to the value of ten units in the former. Besides the method of proof here given, there is another, by casting out the nines; thus: 1. Add the figures in the upper row together, and find how many nines are contained in their sum. 2. Reject the nines, and set down the remainder, directly even with the figures in the row. 3. Do the same with each of the given numbers, and set all the excesses of nines in a column, and find their sum; then, if the excess of nines in this sum, found, as before, is equal to the excess of nines in the sum total; the question is supposed to be right. EXAMPLE. 57381 9156 8171 5324 286891 nines. This method depends upon a property of the number 9, which, except 3, belongs to no other digit whatever; viz, that any number divided by 9, will leave the same remainder, as the sum of its figures, or digits, divided by 9: which may be thus demonstrated.' Demonstration. Let there be any number, as 5432; this, separated into its several parts, becomes 5000+100+30+2; but 5000=5×1000=5×999+1=5×999+5. In like manner 400=4X994, and 30=3×9+3. Therefore, 5432=5×999+5, +4×99+1, +3×9+3+2=5×999+4×99+3×9+5+4+3+2. And 5432 5X993+4×99+3×9+5+4+3+2 ; but 5X999+4X99+3×9 is divisible by 9; therefore, 5432, divided by 9, will leave the same remainder, as 5 +4+3+2, divided by 9; and the same will hold good of any other number whatever. The same property belongs to the number 3: However, this inconveniency attends this method, that, although the work will always prove right, when it is so; it will not, always, be right, when it proves so; I have, therefore, given this demonstration more for the sake of the curious, than for any real advantage. In casting out the nines, proceed thus. Begin with the uppermost row of the example at the left hand; and 7 are 12, from which take out nine, and 3 remains: 3 added to 3 make 6, which must be added to the 8, because 6 is less than 9, and the sum is 14: cast out nine and 5 remains, which is to be placed at the right against the row, as in the example. In the next row, 9 the first figure, may be omitted because it is 9; then 1 and 5 make 6, which added to the 6, make 12, from which take out 9, and 3 remains to be placed on the right of the row as before. Proceed thus with all the rows and with the sum at bottom. Then add the remainders against the several rows, casting out 9 as often as it occurs, and, if the remainder PROOF. Begin at the top of the sum and reckon the figures downwards, in the same manner as they were added upwards, and, if it be right, this aggregate will be equal to the first. Or, cut off the upper line of figures, and find the amount of the rest; then, if the amount and upper line, when added, be equal to the sum total, the work is supposed to be right. Elel 10 11 12 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 11 12 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 When you would add two numbers, look one of them in the left hand coJun and the other at top, and in the common angle of meeting, or, at the right hand of the first, and under the second, you will find the sum-as, 5 and 8 is 13. When you would subtract: Find the number to be subtracted in the left hand column, run your eye along to the right hand till you find the number om which it is taken, and right over it at top you will find the difference-as 8, taken from 13, leaves 5. be the same as that against the sum, as it is in this example, the work is presumed to be right. An easier method of casting out the nines, is to begin as before, and when the sum exceeds nine, to add the figures themselves of this sum as before, and so proceed, and this new sum will always be equal to the remainder after nine is taken from the first sum. Thus, as before, 5 and 7 are 12,-now add the numbers of this sum, which, being 1 and 2, make 3, equal to the remainder after 9 is taken from 12; then 3 and 3 added to 8 make 14, -add the 1 and 4, and the sum is 5, the same as the remainder above. In the next row,-omitting the 9, the sum is 12, the numbers of which, 1 and 2, make 3, the remainder as above. The same will hold true in any case. NOTE. It should be noticed that the method of proof for this rule, and various others, depends upon the accuracy of both operations. It does not follow because the result is the same by both operations, that there can be no error. For both operations may be incorrectly performed, and the results, though alike, erroneous. The best proof that any result is right, is the correct performance of all the operations. In the first Example, the student adds together the several numbers, and finds the sum to be 45; and, as there is but one column, he must set down 45 for the answer. In the 4th Ex. the student will add the numbers of the column on the right hand, which he will find to be 38; he will set the 8 under the column, and carry 3 to the next column. The next column with the 3 to be carried, he will find to be 40; he must set down the 0, and carry 4 to the next column. This will be found to be 29; the nine is to be set under the column, and the 2 carried to the next column, which makes 40; the cipher is to be put under the column and the 4 will take the next higher place, for it is evident the whole must be set down. The same course must be pursued in each example. 11. What is the sum of 3406, 7980, 345 and 27? Ans. 11758. 12. A man borrowed of his neighbour, thirty dollars at one time, one hundred and 66 at another, and seventy-five at another: how much did he borrow in the whole? Ans. 271 dolls. 13. Four boys collected chesnuts; A. had 4096, B. 16784, C. 11590, and D. five hundred and 57; how many were there in the whole? Ans. 14. Four boys, on counting their apples, found that A. had 67, B. 11 more than A, C. had 101, and D. had sixteen more than C; how many had they all? 15. The Deluge happened 2348 years before the birth of our Saviour, and America was discovered 1492 years after it; how many years intervened? SUBTRACTION TEACHES to take a less number from a greater, to find a third, showing the inequality or difference between the given numbers. The greater number is called the Minuend. The less number is called the Subtrahend. The difference, or, what is left after th subtraction is made, is called the Remainder. SIMPLE SUBTRACTION Teaches to find the difference between any two numbers, which are of the same kind. RULE. Place the larger number uppermost, and the less underneath, so that units may stand under units, tens under tens, &c. then, drawing a line underneath, begin with the units, and subtract the lower from the upper figure, and set down the remainder; but if the lower figure be greater than the upper, add ten, and subtract the lower figure therefrom: To this difference, add the upper figure, which being set down, you must add one to the ten's place of the lower line, for that which you added before; and thus proceed through the whole.* PROOF. In either simple, or compound Subtraction, add the remainder and the less line together, whose sum, if the work be right, will be equal to the greater line: Or subtract the remainder from the greater line, and the difference will be equal to the less. * Dem. When all the figures of the less number are less than their correspondent figures in the greater, the difference of the figures, in the several like places, must, all taken together, make the true difference sought; because, as the sum of the parts is equal to the whole; so must the sum of the differences, of all the similar parts, be equal to the difference of the whole. 2. When any figure in the greater number is less than its correspondent figure in the less, the ten which is added by the Rule, is the value of an unit in the next higher place, by the nature of notation; and the one which is added to the next place of the less number, is to diminish the correspondent place of the greater, accordingly; which is only taking from one place and adding as much to another, whereby the total is never changed: and, by this means, the greater is resolved into such parts, as are, each, greater than, or equal to, the simifar parts of the less; and the difference of the correspondent figures, taken together, will, evidently, make up the difference of the whole. The truth of the method of proof is evident; for the difference of two numbers added to the less, is manifestly, equal to the greater. |