ARITHMETIC. 1. A RITHMETIC is the science of numbers, or the art of computing by means of the ten numeral digits, or figures; O cipher, 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine. All numbers may be denoted by those figures variously combined. And the rule which teaches their different values according to their different places, is called NOTATION, or NUMERATION. LET the number 444444444444 be proposed: then the different values of the same figure 4 will be as follows: The first figure on the right stands for four units, being its simple value; the next for four tens, or forty, or ten times its simple value; the third for four hundreds, or a hundred times VOL. I. its simple value; the fourth for four thousands, or a thousand times its simple value, &c. and the four together or 4444 denote four thousand four hundred and forty four. Hence it appears that the values increase from the right to the left in a decuple proportion, each figure standing for ten times the value of the preceding one. It is also evident that in reading of numbers there is a constant repetition of hundreds, tens, and units, at every three figures: thus, the three first on the right denote four hundred and forty-four; the next three, four hundred and forty four thousands; the next three, four hundred and forty-four millions; and the next three, four hundred and forty-four thousands of millions, &c. Therefore in reading of large numbers, if we divide them into periods of six figures each, the first period to the right will be units, tens, hundreds, and thousands; the next period will be millions; the next millions of millions, or bi-millions, or billions; the next tri-millions or trillions, &c. &c. For example, let 12802410007815104906709 be a proposed number: Then dividing it into periods as above, it will be read thus: twelve thousand eight hundred and two trillions, four hundred ten thousand and seven billions, eight hundred and fifteen thousand one hundred and four millions, nine hundred and six thousand, seven hundred and nine. 3. The digits 1, 2, 3, 4, 5, 6, 7, 8, 9, are called significant figures, because each has a value by itself, but the cipher or zero o stands for nothing if alone; when annexed however, on the right hand to other figures, or any number, it increases the value ten times: thus 7 denotes only seven, but 70 is seven tens NUMERATION. or seventy; and 700 seventy tens or seven hundred; also 11, signifies only eleven, but 110 eleven tens, or one hundred and ten; 1100 eleven hundreds, or one thousand one hundred; &c. And therefore in setting down a proposed number, the places of significant figures must be supplied by ciphers when the former are wanting, as in the following example: 3 4. THE Romans made use of seven capital letters to express numbers, D. M. Namely I. V. X. L. C. The intermediate and other numbers are denoted by two or more of those letters joined or repeated till the sum of the whole make up the proposed number, the characters of the greatest value being set to the left; thus, VI is 6; VII, 7; VIII, 8; and MDCLXVI, 1666. Sometimes a less character is put to the left of a greater, and then it represents their difference as IV, 4; IX, 9; XL, 40; XC, 90; CD, 400. stands for D or 500; and CIO for M or 1000. Every C and annexed on each side increases the value ten times; thus Also IO CCIO is 10000. A bar or stroke over a letter increases the value 1000 times, as X is 10000, and C 100000, &c. This notation is frequently used for the dates, numbering the chapters or sections of books, &c. SIMPLE ADDITION. 5. SIMPLE ADDITION consists in finding the sum of two or more numbers of the same denomination. This is done in the following manner : Place the numbers under each other, so that units are exactly under units, tens under tens, hundreds under hundreds, &c. and draw a line under them. Then add the row of units together, and find how many tens are in the sum.-Set down exactly under the units what remains more than those tens, or when nothing remains, a cipher, and carry one for every ten to the second row.-Next, add up the second row, together with the number carried, then proceed with the sum as be fore. And in this manner continue the operation till the whole is finished. Examp. 1. Let the sum of 543 and 246 be required? 543 246 Sum. 789 Ex. 2. Required the sum of 57854, 480, and 769 ? 57854 769 Sum 59163 In this addition I proceed thus:-9 and 0 and 4 make 13 which is 1 ten and 3 over, therefore I put down the 3 and carry 1 to the rank of tens; next, 6 and 8 are 14 and 5 make 19 and 1 I carried make 20, which is 2 tens and 0 over, therefore I put down a cipher and carry 2; again, 7 and 4 make 11 and 8 are 19 and 2 that were carried make 21, which is 1 to put down and 2 to be carried; next, the 2 carried and 7 make 9; lastly, as there is nothing carried to the 5 it becomes the last figure in the sum. The reason for placing units under units, tens under tens, hundreds under hundreds, &c. and carrying the tens to the left, is manifest from Notation. But because the whole must be equal to the sum of all its parts, if we add together the units in one sum, the tens in another, the hundreds in a third, &c. and add the several sums together, it will prove the addition; and perhaps the reason for carrying the tens will appear more obvious. 6. Another method of proving addition, is to cut off the upper line, then having added all the other lines together, add the upper line to the sum. 7. When the numbers to be added are large, and consist of many ranks, divide them into two or more parts, and find the sum of each part separately, then add the several sums together. |