ARITHMETIC being the science of computing by numbers, is comprehended in the two general operations of increasing and diminishing. The fundamental rules applied to the former are, ADDITION and MULTIPLICATION, and to the latter, SUBTRACTION and DIVISION: Notation and Numeration being only preparatory to these. NOTATION Is the expressing of any number in writing, either by words or figures. The characters by which numbers are now generally denoted are the ten following, with their simple values. I, 2, 3, 4, 5, 6, 7, 8, 9, O. one, two, three, four, five, six, seven, eight, nine, cipher. Besides the values here assigned to the first nine of these characters or figures, they have other values, arising from the situations in which they stand relative to each other, when used to any number greater 2, express than nine, as in the following. 2 2 2 2 2 1, V, X, L, C, D, M. 1, 5, 10, 50, 100, 500, 1000. The intermediate and greater numbers they formed by different combinations of these seven. When the letter which denoted the greater value stood on the left-hand, the number was expressed by their sum; as LX, sixty; but when on the right, by their difference; as XL, forty. A line drawn horizontally over any letter, increased its value a thousand fold; as C expressed a hundred thousand; M, a million. For 500 they also wrote 10, and the addition of every Ɔ increased the former value in a decuple proportion; so I was 5000, and 10, 50,000. A thousand was likewise denoted by CIƆ, and its value increased as in the last method by adding C and Ɔ; thus CC13) equalled 10,000. taining this notation, by which chapters, sections, and dates of books are frequently expressed. X, 10, The following columns will be of service in at & Billious. From this it is evident, that the relative values of figures increase in a decuple or tenfold proportion for every place they are removed from that of units towards the left hand; for the first 2 on the right-hand expresses only its simple value, or two units, but the second denotes two tens, or ten times the value it would have expressed in the first place; the third signifies two hundreds, or its first value a hundred, and its second ten times repeated; the fourth is expressive of thousands, &c. Thus, 2222 taken-together, denote two thousand two hundred and twenty-two. The above number is sufficiently extensive for common purposes, but, as greater numbers sometimes occur, it may be proper to observe, that if they be divided into periods of six figures each, beginning at the right-hand, the first period is units, the second millions, the third billions, the fourth trillions, the fifth quadrillions, &c. When numbers are further divided into half periods of three figures each, it adds much to the facility of reading large numbers. The cipher has no value of its own, but increases that of other figures when placed on the right of them, by removing them farther from the unit's place; thus, 50 denotes fifty, and 500, five hundred. It is also used to fill up the place or places in a number where no value is expressed; so five bundred and five (there being no tens) is denoted by 505, and four thousand and seven (where neither hundreds nor tens occur) by 4007; and the number forty thousand six hundred and seven, is written 40,607. ROMAN NOTATION. The Romans made and recorded their calculations by means of seven numeral letters, which with their values are as follow. I, I, XX, 20, II, 2, XXX, 30, III, 3, XL, 40, IV, 4, V, 5, L, 50, УІ, 6, LX, 60, The intermediates of these are formed by writing each of the numbers in the left-hand column respectively on the right-hand side of each in the right-hand column; as XI, 11; XII, 12, &c. XXI, 21; XXII, 22, &c. NUMERATION Is the reverse of Notation, or the reading of numbers: and this is done by using the words which they denote, as shown above. Thus, 1,808 is read one thousand eight hundred and eight. The rules of addition,subtraction, multiplication, and division, are both simple and compound, as they relate to quantities of one or more denominations. SIMPLE ADDITION Consists in collecting two or more numbers of the same denomination into one sum or total; which is done as follows. The character denoting addition is +, named plus. RULE. Write the numbers under each other, units under units, tens under tens, &c. add up the column of units, and beneath it set the right-hand figure of the sum; carrying the rest, if any, to the next column, which add up and set down as before, and so on to the last column to the left band, the this new number be the sum of all the others, as whole sum of which must be put down. Then will was required. work be right; or begin at the top and add down- right of that multiplied, and to the tens in the prowards. SIMPLE SUBTRACTION Consists in finding the difference between two numbers of the same denomination. The character by which subtraction is denoted is called minus -. RULE. Write the numbers under each other as in addition, the less under the greater. Begin with the units, and take each figure in the lower number from that above it, and set down the remainder as in addition. When any figure in the lower line is greater than that above it, suppose ten to be added to the upper one, and having taken the bottom figure from this sum, and set down the remainder as before, carry one to the next lower figure on the left, and so on through the whole number, which will give the difference required. PROOF Add this difference to the less number, and the sum will be equal to the greater when the work is right. 37869473 EXAMPLES. 78674216 duct of the last figure, add the figure itself; and Multiplicand 93478604 478640473 Product CASE II. 8 747828832 18 8615528514 When the multiplier consists of several figures. RULE. Multiply by each figure separately, setting the first figure of each product under the figure multiplied by, and add the several products together, then their sum will be the whole pr duct. 314786 EXAMPLES. 4780483 5073 Consists in obtaining the amount of any simple NOTE. If there be ciphers on the right of either the multiplicand or multiplier, or both, find the product of the other figures by one of the preceding rules, and to the right of it annex as many o phers as were omitted, EXAMPLES. 8704600 × 720 9 20 22 24 748321 × 42 7 27 30 33 36 36 40 44 48 6 12 24 36 48 60728496 108 120 132 144 CASE I. When the multiplier does not exceed 12. RULE. Having set the multiplier under the unit's place of the multiplicand, multiply each figure of the latter by the former, set the righthand figure of each product under the figure multiplied, and carry the rest, if any, to the next product; the resuit will be the whole product required. NOTE. When the multiplier is any number between 12 and 20, multiply by the unit's figure, add to the product the figure which stands on the SIMPLE DIVISION Is a compendious method of finding how often less of two simple numbers is contained in the greater; and consequently is only a ready way performing subtraction. The character denoting division is. See Division in ALGEBRA. The number to be divided is called the divided; that to be divided by, the divisor; and the o ber of times the former contains the latter, the quotient. There is frequently a remainder after the division is finished, and which must always be less than the divisor. CASE I. When the divisor does not exceed 11. RULE. Write the divisor on the left-hand si of the dividend, separating them by a small curve line. Find how many times it is contained in as many of the left hand figures of the dividend s are necessary, and set the number under the right hand figure used: then carry the remainder, if any, as so many tens to the next dividend figure, and divide the sum as before, and so on to the unit's place of the dividend. Observe, that there must be a quotient figure under every one of the dividend to the right of that under which the first quotient figure stands, and if no significant ugw occur its place must be supplied by a cipher; also if there be any remainder it must be set on the right of the quotient, with the divisor under it and a small line between them. PROOF. Multiply the quotient by the divisor, and the remainder added to the product will give the dividend. Ex. I. 8)786349 CASE II. 98293 Ex. II. 12)38479041 3206586/9/2 When the divisor consists of several figures. RULE. Set the divisor on the left-hand of the dividend as before, and let the quotient be placed in a similar manner on the right. Multiply the divisor by the first figure of the quotient, found as above, and set the product under the left hand figures of the dividend, from which subtract it, and to the right of the remainder annex the next figure of the dividend; find another figure of the quotient, and so on until all the figures of the dividend are used. NOTE. If there be ciphers on the right hand of the divisor, cut them off, and omit the same number of figures on the right of the dividend: then divide as above, and annex the figures omitted to the right hand of those remaining after the division is finished for the true remainder. Ex. I. 374)763214(2040 from a lower to a higher by dividing by as many of the former as make one of the latter. Ex. I. Reduce 571. 75. 4d. into pence. 20 1147 shillings 12 13768 pence. II. Bring 497386 farthings into crowns. 4)497386 12)124346-2 5)10362-2 2072: 25. 2d. ans. COMPOUND ADDITION Is the collecting of several quantities of different denominations into one sum or total quantity. Set RULE. Place the numbers so that those of the same denomination may stand directly under each other, and draw a line below them. Add up the figures in the lowest denomination, and find, by reduction, how many units, of the next higher denomination, are contained in their sum. down the remainder below its proper column, and carry those units to the next denomination, which add up in the same manner as before. Proceed thus through all the denominations, to the highest, whose sum, together with the several remainders, will give the answer sought. The method of proof is the same as in simple addition. 34 15 24 27 12 94 30 9 111 51 18 9 79 15 44 59 19 71 COMPOUND SUBTRACTION Is the operation by which the difference of two quantities of several denominations is found. RULE. Place the less number under the greater, so that like denominations may stand under each other, and proceed as in simple subtraction; observing only that, when the lower number exceeds that above it, as many must be added to the latter as make one in the next column to the left. PROOF. The method of proof is the same as in simple subtraction. Ex. I. . s. d. 79 17 81 35 12 44 44 5 4 79 178 COMPOUND MULTIPLICATION Consists in finding the amount of any number, of different denominations, taken any assigned number of times; and like simple multiplication admits of several cases. £. 112 11 2 the quotient. product of small simple numbers. RULB. Divide by each of the numbers suc cessively, as in simple division. CASE I. When the multiplier does not exceed 12. RULE. Set it under the lowest denomination of the number to be multiplied, multiply the deno- CASE II. When the divisor exceeds 12, and is the mination immediately above it; and having ascertained the number of integers of the next higher denomination in the product, by the rule for reduction, set the remainder under the same denomination, and carry the integers to the product of the next higher denomination, and continue the same process unto the highest denumination; the whole product of which together with the several remainders, taken as one compound quantity, will be the whole product required. PROOF. The same as in simple multiplication. Is the operation by which any number, of different denominations, is divided into any required number of parts. Ex. What is sugar per cwt. if 32 cwt. cost 61. 17s. 4d? £. s. d. 8)61 17 4 4)7 14 8 £18 8 answer. Is that comprehensive branch of calculation which enables us to find a fourth quantity having a certain relation to three given quantities. Tais rule, on account of its great and extensive usefulness, is oftentimes called The Golden Rais of Proportion; for, on a proper application of it, and the preceding rules, the whole business of arithmetical, as well as every mathematical, enquiry depends, The rule itself is founded on this obvious principle, that the magnitude or quantity of any effect varies constantly in proportion to the varying part of the cause: thus, the quantity of goods bought is in proportion to the money laid out; the space gone over by an uniform motion is in proportion to the time, &c. It is usually divided into the three distinct parts or rules of DIRECT, INVERSE, and COMPOUND PROPORTION. The characters used to denote proportion are ::::, and the terms are written thus 2:4:8 : 16, and read as 2 is to 4, so is 8 to 16. DIRECT PROPORTION CASE I. When the divisor does not exceed 12. RULE. Place the divisor on the left of the dividend, as in simple division.-Begin at the lefthand, and divide each denomination by the diyiser, setting down the quotients under their re- Is employed in finding, from three given numbers, RULE. State the question; that is, place the terms so, that the first may be one of the terms of supposition, the second of the same nature as the fourth or term sought, and the third, the term of demand. Then bring the first and third terms into the same denomination, and the second to the lowest name mentioned. Multiply the second and third terms together, and divide the product by the first, and the quotient will be the answer, in the same name as the second term; and which may be brought to any other denomination required, PROOF. The method of proof is by inverting the question; and in this manner each question will furnish four, and thus supply a very useful exercise. Ex. If 12lb. of cheese cost 9s. 6d. what will 4 sheeses cost, each weighing 1gr. 6lb.? If alb.: 9s. 6d. :: 1qr. 5lb. x 4. 12 1. Set down the terms expressing the conditions of the question, in one horizontal line; separating the producing terms from the produced; that is to say, those which necessarily and jointly tend to produce or to modify any effect, and the constituent parts of such effect. 2. Under each conditional term, set its corresponding one in another line. 3. Multiply the producing terms of one line, and the produced terms of the other line, continually; and take the result for a dividend. 4. Multiply the remaining terms continually, and let the product be a divisor. 5. The quotient of this division, will be the term required. N. B. In a question where a term is only understood, and not expressed, it may always be represented by unity. The required term may be denoted either by Q or by an asterisk. Ex. 1. If 40 acres of grass be mowed by 8 men in 7 days: how many neres can be mowed by 24 men in 28 days equally long? 28 M. 48 Da. H. L. De. W. 6 12 24 2 3 24 180 10 X Q 3 24 x 180 x 10 x 24 × 2 × 3 48 × 6 × 12 × 3 × 5 5 #20 yards. Ans. Is a compendious method of performing such questions in direct proportion as have unity for the first term; it obtained its name from being in daily use among merchants and tradesmen, as a ready and concise manner of answering most questions that occur in business. An aliquot part of any number is such a part as being taken a certain number of times, will exactly make that number; thus is an aliquot part of 1, and 2 of 6; for the former being taken 4 and the latter times, make the numbers COMPOUND PROPORTION Is a rule in which more than three terms are given to find another, dependant upon them. This rule is of general use, extending to all arithmetical operations where proportions are concerned: it may be performed by means of the following directions. |