A NEW AND CONCISE SYSTEM OF ARITHMETICK. Arithmetick is that part of the mathematicks, which exhibits the doctrine of numbers, explains their properties, and teaches the mode of calculating them, by whole and broken numbers. 1. By whole numbers is meant entire quantities. 2. By broken numbers is understood the parts of some entire quantity, or number less than a whole, commonly called Fractions. All Arithmetical operations signify, that by having some numbers or quantities given, to find out others before unknown. Arithmetick is divided into five fundamental rules, or divisions; without a thorough knowledge of which, we cannot move one step, viz. NOTATION or NUMERATION, ADDITION, SUBTRACTION, MULTIPLICATION and DIVISION; of which we shall treat in due order. And 1st, of NUMERATION. Numeration teaches how to read or write any sum or number by figures or characters, which increase in a ten-fold proportion, from the right to the left hand. They are as follows: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. O is called a cypher, and the rest figures or digits. The names and significations of these characters, and their original generation of numbers they stand for, are as follow: 0, nothing; 1, one, usually called unit; 2, two; 3, three; 4, four; 5, five; 6, six ; 7, seven ; 8, eight; 9, nine; 10, ten; which has no single character. N. B. By the continual addition of unity or one, all numbers are generated. Besides the simple value of figures, as before mentioned, they have each a local value, according to the following law, viz. In a combination of figures, counting from the right to the left hand, the figure in the first place represents its primitive simple value; that in the second place ten times its simple value, and so on; the value of the figure in each succeeding place, being ten times the value of the figure in that immediately preceding it. The values of the places are estimated according to their situation. The first is denominated the place of units; the second, tens; the third, hundreds; and so on, as in the following table. Thus, in the number 5293467, 7, in the first place, means only seven; 6, in the second place, signifies 6 tens, or sixty; 4, in the third place, four hundred ; 3, in the fourth place, three thousand; 9, in the fifth place, ninety thousand; 2, in the sixth place, two hundred thousand; 5, in the seventh place, is five millions; and the entire taken together is read thus : five millions, two hundred and ninety-three thousand, four hundred and sixty s -seven. A cypher, though it is of no signification itself, yet it possesses a place, and when set on the right hand of figures, in whole numbers, increases their value in the same ten-fold proportion thus, 6 signifies only six, but if a cypher be placed at the right hand, (thus, 60) it then becomes sixty; and if two cyphers be placed on the right hand, (thus, 600) it is changed to six hundred, &c. Now, to enumerate any given parcel of figures, observe the following universal rule, as appears in the Numeration Table at large : NUMERATION TABLE. Billions. Hund. of Thous. of Mill. Tens of Thous. of Mill. Millions. Hundreds of Thous. Thousands. Tens. Units. 67 EXPLANATION. The words at the head of the Table shew the significa tion of the figures against which they stand; and the figures shew how many of that denomination are meant. Thus units in the first place signify ones, and 6 being against them, shews that 6 ones or units are here meant; tens in the second place shew that every figure in this 4 2 1 8 3 6 place signifies so many tens, and 3 stan3 4 0 7 6 2 1 4 6 3 1 2 ding against them, shews that three tens 1 3 0 2 5 0 3 7 6 4.5 are meant,equal to thirty, what the figure 4 13 9 8 2 1 0 6 4 really signifies. Hundreds, in the third 2 7 0 2 1 3 6 7 5 place, shew the figures in this place to 4 6 3 2 7 2 9 1 be Hundreds, and 8 shews the number 1 2 3 4 6 3 2 of hundreds meant, which is eight hun2 3 4 5 6 7 dred. Having proceeded through in 8 90 9 8 the same order, the sum of the top line 76 5 4 of figures will read thus: Two Billion, 1 2 3 one hundred sixty-seven thousand, two 4 5 hundred and thirty five Million, four 7 hundred twenty-one thousand, eight In like manner all the remaining numbers in hundred and thirty-six. the able may be read. N. B. It is requested, that the learner will commit the words at the top of the Table to memory, before he proceeds further. It is recommended to the learner, and men of publick business in general, to observe the following mode of dividing numbers into periods and half periods, as it will be convenient in reckoning and practice : N. R. Counting six places from the unit's place, or right hand, the unit period is pointed out; the next six the million period; then the trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, undecillion, duodecillion periods, &c. follow. This is the Arabick mode of counting: the following is the Roman method, (by letters :) 1. One. II. Two. III Three. IV. Four. V. Five. VI. Six. VII. Seven. XI. Eleven. XVII. Seventeen. XVIII. Eighteen. XIX. Nineteen. DC. Six hundred. M. or Cl. One I housand. MDCCCXXI. One thousand eight hundred and twenty one ADDITION, Teaches how to bring several numbers, either of one or more denominations, into one aggregate or total sum. Addition is of two kinds, viz. SIMPLE and COMBINED; and each of these subdivided into that of whole numbers and mixed quantities. General rule for setting down several numbers or quantities, so as to prepare them for the work of Addition; called the sum or amount. 1. Place all the numbers to be added in this order, viz. units under units, tens under tens, hundreds under hundreds, &c. and draw a line under the lowest number. 2. Add the right hand column, and if the sum be less than ten, write it under the column; but if it be ten, or any even number of tens, write down a cypher; and if it be not an exact number of tens, write the excess above tens at the bottom of the column, and for every ten the sum contains, carry one to the next column, and add in the same manner as before. 3. Proceed in like manner to add each column respectively, carrying for the tens of each to the next column, and set down the full sum of the last column. PROOF. Begin at the top of the sum, and count the figures downwards, in the same manner as they were added upwards, and if the sum total be equal to the first, the addition is allowed to be right; or, it is commonly proved by cutting off the upper line of figures, and finding the sum of the rest; then, if the amount and upper line, when added, be equal to the sum total, the work is allowed to be right. Another method to prove addition, is to cast the nines out of each line, beginning at the top line, and carry the excess above the nines to the next line, and so proceed from line to line, from top to bottom, and note down the last excess; then cast the nines out of the aggregate sum, and if the excesses in both be equal, the work is supposed to be right; which depends on a property of the number 9, which, except the number 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 is thus demonstrated :— : Let the number 5432 be given this separated into its several parts will stand thus: 5000+400+30+2: but 5000=5×1000=5×999+ 1=5X999+5. In like manner, 400=4X99+4, and 30=3×9+3, therefore. 5432=5X999+5,+4x99+4,+3x9+3+2=5×999+4 99+3×9+5+4×3+2, and 5432-9= 5X999+4X99+3x9+5+4+3+2; 9 but 5X999+4x99+3x9 is divisible by 9; therefore, 5432 divided by 9, will leave the same remainder as 5+4+3+2 divided by 8 ; and the same will hold good of any other number whatever. The same property belongs to the number 3. However, this inaccuracy 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 therefore, this demonstration is more for the sake of the curious, than it is for any real advantage. 2 5 7 5 Sum total. 4=7+18+2=10-9 leaves 1, which I set down at the top of the cross; and then begin with the sum total, and say 2+5=7+7=14—9—5+5=10—9—1, the same as before found in the parts; the nature of which, and addition in particular, is founded on the well known axiom, "the whole is equal to the sum of its parts taken together." ADDITION AND SUBTRACTION TABLE. 6 | 7 | 8 | 9 | 10 11 12 1 2 3 4 | 5 | 2 4 5 6 7 13 14 When you would add two numbers look for one of them in the left hand column, and the other at the top, and 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 are 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 from which it was taken, and right over it at the top you will find the difference: as, 8 taken from 13 leaves 5. EXAMPLES. 1. What will be the amount of 3612 dollars ; 8043 dollars; 651 doland of 3 dollars, when added together? lars |