ARITHMETIC MODERNISED. MODERN ARITHMETIC, is the art of calculation, by means of the nine digits and cipher. All Arithmetical calculations are performed by Addition, Subtraction, Multiplication, or Division ;-but, that the learner may fully understand these operations, he must be previously acquainted with the method of representing numbers by proper characters, which is called Notation; and that of expressing them in words, when they are so represented, which is called Numeration. NOTATION. ARITHMETICAL NOTATION is the art of expressing any number, by means of the nine digits, or figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, and the cipher O. The principles of Arithmetical Notation may be thus stated: 1. When any digit stands alone, its name expresses its value; but when there are several figures arranged in a line, each of them derives its value from the place it occupies. Thus, in the number 5555, the digit 5, in the first, or right hand place, signifies 5 units; in the second place, 5 tens; in the third place, 5 hundreds, &c. always rising in value 10 times, by every removal to the left. 2. Since any number below 10 is expressed by some of the digits in the first place, and the scale rises in a tenfold proportion, B it is evident, that any number of tens, of hundreds, of thousands, &c. may be expressed by varying the place of the digits: but all numbers whatever may be resolved into units, tens, hundreds, &c.; consequently all numbers whatever can be represented by the 9 digits, when the place of each is determined. 3. When a number consists exactly of so many units, so many tens, &c. the digits themselves show their respective places. When there are vacant places, the vacancy is supplied by the cipher; which, without possessing any value in itself, is used merely as a mark to point out the proper places, and, consequently, to fix the value of the digits on its left. Hence, it is obvious, 1st, That one or more ciphers placed be fore a number, can neither augment nor diminish its value. 2dly, That one cipher annexed to the right hand of a number, increases its value 10 times; two ciphers, 100 times; and so on. 3dly, That a cipher interposed in the middle of a number, augments the value of the figure or figures to the left 10 times; but in no degree affects the value of any figure or figures to the right hand. The following TABLE exhibits a general view of the doctrine of Notation: Which number is read thus: 489 thousand 60 trillions, 980 thousand 500 billions, 700 thousand 604 millions, 345 thousand, 678.* *The Arithmeticians on the Continent divide numbers into classes of three places, to each of which they assign an appropriate name, These names are, units, thousands, millions, billions, trillions, quadrillions, quintillions, sextillions, septillions, octillions, nonillions, &c. According to their scheme, the number in the above Table is read, 489 sextillions, 60 quintillions, 980 quadrillions, 500 trillions, 700 billions, 604 millions, 345 thousand, 678. TO READ ANY NUMBER. RULE. Divide the given number, as in the preceding Table, into as many parcels of three figures each, as it will admit of. Then (remembering that the first figure of each parcel is named singly, HUNDRED, and that the other two figures are named together,) begin at the left hand, and read, in each class, as many hundreds, tens, and units, as the figures in those places express; pronouncing the name THOUSAND, after reading the three left hand figures of each class; and NAMING the CLASS, after reading the three right hand figures of any class, except the first, the name of which (i. e. UNITS) is not expressed. EXAMPLE. Express in words 341,775,360. Three hundred and forty one millions, seven hundred and seventy five thousand, three hundred and sixty. TO EXPRESS ANY NUMBER IN FIGURES. RULE. Write as many ciphers as there are places in the given number, and point them as in the preceding Table; then consider what place each figure of the given number should occupy; expunge the cipher there, and write the proper figure in its place. EXAMPLE. Set down in figures two hundred and nine thousand, one hun-' dred and fifty. 606,90 EXERCISES. Express in figures the following numbers: 1. Six hundred and seventy three. 2. Four thousand and nine. 3. One thousand three hundred and one. 4. Thirty nine thousand and seventy three 5. Six hundred and thirty two thousand. 7. Fifty five millions thirteen thousand eight hundred and sixty six. 8. Ninety five millions. 9. One thousand two hundred and ninety three millions, eight hundred and eighteen thousand, seven hundred and fifty one. 10. Twenty three billions, seventy millions, and forty nine. ROMAN NOTATION. The ancient Romans represented numbers by certain letters of their alphabet. The letters employed for this purpose were, I for 1, V for 5, X for 10, L for 50, C for 100, I or D for 500, and CI or M (and sometimes ) for 1000. Other numbers were denoted by the repetition or combination of these, as follows: 1. As often as the same letter is repeated, so often is its value repeated: thus, II=2; XXX=30; CCCC=400. 2. When a letter of a less value is placed before one of a greater value, it represents their difference: thus, IV=4; IX=9; XL=40; XC=90; CM 900. 3. When a letter of a less value is placed after one of a greater value, it represents their sum: thus, VI=6; XI=11; LX=60; DC or IC=600; MC or CIOC-1100; MD=1500. 4. The value of a number is increased 1000 times, when a line is drawn over its top: thus, X=10,000; C=100,000; M— 1,000,000. 5. One annexed to the number I increases its value 10 times; two 's 100 times; and so on: thus, IƆ=5000; 100=50,000. 6. One C prefixed, and at the same time one annexed to the number CI, increases its value 10 times; two, 100 times; and so on: thus, CC=10,000; C=100,000. 7. A smaller letter placed over another, or at the upper corner on the right hand of a number, denotes their product: thus, M=500,000; IV =400. ID C 8. In expressing large numbers, points were sometimes interposed: thus, XVI.XX.DCCCXXIX, in Pliny, denotes 1,620,829. ADDITION. ADDITION is that process by which we collect two or more numbers into one sum. RULE. Write the given numbers below each other, so that UNITS may stand below UNITS, TENS below TENS, and so on; and draw a line below them. Then, beginning with units, add up the several rows successively, setting down the right hand figure of the sum of each row at the bottom of that row; carry the rest to the next row, and put down the complete sum of the last row. PROOF. The best test of the accuracy of the work, is to begin at the top of the columns, and reckon them downwards, in the same manner as you added them upwards; then, if no error has been committed, the sum will be the same as before. Note. In this rule, it is presumed, that the learner knows the sum of any two digits. If he does not, he should commit to memory Table I. annexed to this tract, before he attempts to add the following exercises. 423 567 385 1375 EXAMPLE. Illustration of the process. 5 and 7 are 12, and 3 are 15; I put down 5, and carry 1 to 8 are 9, and 6 are 15, and 2 are 17; I put down 7, and carry 1 to 3 are 4, and 5 are 9, and 4 are 13, which I put down. REMARKS. I. The rule for addition is founded on the following axiom: "The sum of two or more numbers is equal to the sum of all the units of those numbers, and the sum of all the tens, and the sum of all the other places, taken together:"—this is evident, because the whole is equal to all the parts of that whole taken together. The following process will illustrate the reason of carrying forward the left hand figure of the sum of any column to the next: 66 Begin at any column, and write down its complete sum; do the same with all the other columns, placing the unit's figure of their respective sums under the column whence it arises. Then, if these partial sums be added together, the amount will evidently be the total amount of the given numbers." From this process, it is plain, that the carrying forward of the left hand 423 423 567 567 figure of the sum of any column to the next, is only a method of doing at once what is here done at several steps. II. As no process of arithmetic occurs in business more frequently than Addition, the learner should practise it till he can add with great accuracy and readiness. He should take care to write the figures distinctly, and to arrange the columns regularly, before he begins to add them; as errors frequently arise, from in |