HE numerals now in use, with the mode of causing them by peculiar situation to express any number, and whereby the processes of arithmetic have been rendered so highly convenient, have heretofore been supposed to be of Indian origin, transmitted through the Persians to the Arabs, and by them introduced into Europe in the tenth century, when the Moors invaded and became masters of Spain. Such in reality appears to have been in a great measure the true history of the transmission of these numerals; but as it has been lately found that the ancient hieroglyphical inscriptions of Egypt contain several of them, learned men are now agreed that they originated in that early seat of knowledge, between which and India there exist more points of resemblance, and more traces of intercourse, than is generally supposed. In the eleventh century, Gerbert, a Benedictine monk of Fleury, and who afterwards ascended the Papal throne under the designation of Sylvester II., traveled into Spain, and studied for several years the sciences there cultivated by the Moors. Among other acquisitions, he gained from that singular people a knowledge of what are now called the Arabic numerals, and of the mode of arithmetic founded on them, which he forthwith disclosed to the Christian world, by whom at first his learning caused him to be accused of an alliance with evil spirits. The knowledge of this new arithmetic was about the same time extended, in consequence of the intercourse which the Crusaders opened between Europe and the East. For a long time, however, it made a very slow and obscure progress. The characters themselves appear to have been long considered in Europe as dark and mysterious. Deriving their whole efficacy from the use made of the cipher, so called from the Arabic word tsaphara, denoting empty or veid, this term came afterwards to express, in general, any secret mark. Hence, in more troublous times than the present, a mode of writing was practiced, by means of marks pre viously concerted, and called writing in cipher. The Arabic characters occur in some arithmetical tracts composed in Eng. land during the thirteenth and fourteenth centuries, particularly in a work by John of Halifax, or De Sacrobosco; but another century elapsed before they were generally adopted. They do not appear to have settled into their present form till about the time of the invention of printing. It would be impossible to calculate, even by their own transcendent powers, the service which the Arabic numerals have rendered to mankind. HE Arabic numerals take the following wellknown forms :-1, 2, 3, 4. 5. 6, 7, 8, 9, 0. The first nine of these, called digits, or digital numbers, represent each one of the numbers between one and nine, and when thus employed to represent single numbers, they are considered as units. The last (0), called a nought, nothing or cipher, is, in reality, taken by itself, expressive of an absence of number, or nothing; but, in connection with other numbers, it becomes expressive of number in a very remarkable manner. The valuable peculiarity of the Arabic notation is the enlargement and variety of values which can be given to the figures by associating them. The number ten is expressed by I and o put together-thus, 10; and all the numbers from this up to a hundred can be expressed in like manner by the asso ciation of two figures-thus, twenty, 20; thirty, 30; eighty five, 85; ninety-nine, 99. These are called decimal numbers, from decem, Latin for ten. The numbers between a hundred and nine hundred and ninety-nine inclusive are, in like manner, expressed by three figures-thus, a hundred, 100; five hundred, 500; eight hundred and eighty-five, 885; nine hundred and ninety-nine, 999. Four figures express thousands; five, tens of thousands; six, hundreds of thousands; seven, millions; and so forth. Each figure, in short, put to the left hand of another, or of several others, multiplies that one or mere numbers by ten. Or if to any set of figures a nought (0) be added towards the right hand, that addition multiplies the number by ten; thus 999, with o added, becomes 9990, nine thousand nine hundred and ninety. Thus it will be seen that, in notation, the rank or place of any figure in a number is what determines the value which it bears. The figure third from the right hand is always one of the hundreds; that which stands seventh always expresses millions; and so on. And whenever a new figure is added towards the right, each of the former set obtains, as it were, a promotion, or is made to express ten times its former value. Units. Tens. ∞ Hundreds. Thousands. Tens of thousands. Hundreds of thousands. Millions. Tens of millions. N Hundreds of millions. Thousands of millions. A large number is thus expressed in the Arabic numerals, every set of three from the right to the left hand being divided by a comma for the sake of distinctness. The above number is therefore one thousand two hundred and thirty-four millions, five hundred and sixty-seven thousands, eight hundred and ninety. Higher numbers are expressed differently in France and England. In the former country, the tenth figure expresses billions, from which there is an advance to tens of billions, hundreds of billions, trillions, etc. In our country, the eleventh figure expresses ten thousands of millions, the next hundreds of thousands of millions, the next billions, etc. The two plans will be clearly apprehended from the following arrangement : ADDITION. 27 5 536 352 DDITION is the adding or summing up of several numbers, for the purpose of finding out their united amount. We add numbers together when we say, I and I make 2; 2 and 2 make 4; and so on. The method of writing numbers in addition, is to place the figures under one another so that units will stand under units, tens under tens, hundreds under hundreds, etc. Suppose we wish to add together the following numbers27, 5, 536, 352, and 275; we range them in columns one under the other, as in the margin, and draw a line under the whole. Beginning at the lowest figure of the right-hand column, we say 5 and 2 are 7-7 and 6 are 13-13 and 5 are 18 -18 and 7 are 25; that is, 2 tens and 5 units. We now write the five below the line of units, and carry or add the 2 tens, or 20, to the lowest figure of the next column. In carrying this 20, we let the cipher go, it being implied by the position or rank of the first figure, and take only the 2; we therefore proceed thus-2 and 7 are 9-9 and 5 are 14-14 and 3 are 17-17 and 2 are 19. Writing down the 9, we proceed with the third column, carrying I, thus-I and 2 are 3-3 and 3 are 6-6 and 5 are II. No more figures remaining to be added, both these figures are now put down, and the amount or sum of them all is found to be 1195. Following this plan, any quantity of numbers may be summed up. Should the amount of any column be in three figures, still only the last or right-hand figure is to be put down, and the other two carried to the next column. For example, if the amount of a column be 127, put down the 7 and carry the other two figures, which are 12; if it be 234, put down the 4 and carry 23. 275 1195 For the sake of brevity in literature, addition is often denoted by the figure of a cross, of this shape +. Thus, 7+6 means 7 added to 6; and in order to express the sum resulting, the sign, which means equal to, is employed, as 7+6 =13; that is, 7 and 6 are equal to 13. The Sign of Dollars is $. It is read dollars. Thus, $64 dollars is read 64 dollars; $5 is read 5 dollars. When dollars and cents are written, a period or point (.) is placed before the cents, or between the dollars and cents. Thus, $4.25 is read 4 dollars and 25 cents. Since 100 cents make $1.00, cents always occupy two places, and never more than two. If the number of cents is less than 10 and expressed by a single figure, a cipher must occupy the first place at the right of the point. Thus, 3 dollars 6 cents are written $3.06; I dollar 5 cents are written $1.05. When cents alone are written, and their number is less than 100, either write the word cents after the number, or place the dollar sign and the point before the number. Thus, 75 cents may be expressed, $.75. In arranging for addition, dollars should be written under dol lars, and cents under cents, in such order that the points stand in a vertical line. The sign $, and the point (.) should never be omitted. M ULTIPLICATION is a short method of addition under certain circumstances. If we wish to ascertain the amount of twelve times the number 57, instead of setting down twelve rows of 57, and adding them together, we adopt a shorter plan by which we come to the same conclusion. For ascertaining the amount of all simple numbers as far as 12 times 12, young persons commit to memory the following Multiplication Table, a knowledge of which is of great value, and saves much trouble in after-life :: When the multiplier consists of two or more figures, place it so that its right-hand figure comes ex 5463 34 21852 16389 actly under the right-hand figure of the multiplicand; for instance, to multiply 5463 by 34, we proceed as here shown. Here the number is multiplied, first by the 4, the product of which being written down, we proceed to multiply by 3, and the amount produced 185742 is placed below the other, but one place farther to the left. A line is then drawn, and the two products added together, bringing out the result of 185742. We may, in this manner, multiply by three, four, five, or any number of figures, always placing the product of one 76843. figure below the other, but shifting a place far4563 ther to the left in each line. An example is here given in the multiplying of 76843 by 4563. 230,529 4,610,58 38,421,5 307.372 350,634,609 Multiplication is denoted by a cross of this shape x thus 3 x 8 = 24, signifies, that by multiplying 8 by 3, the product is 24. A number which is produced by the multiplication of two other numbers, as 30 by 5 and 6, leaving nothing over, is called a composite number. The 5 and 6, called the factors (that is, workers or agents), are said to be the component parts of 30, and 30 is also said to be a multiple of either of these numbers. The equal parts into which a number can be reduced, as the twos in thirty, are called the aliquot parts. A number which cannot be produced by the multiplication of two other numbers, is called a prime number. When the mul tiplicand and multiplier are the same, that is, when a number is multiplied by itself once, the product is called the square of that number: 144 is the square of 12. 8|16|24|32 9|18|27|36| 45 | 54 | 63 | 72 10 | 20 30 40 50 60 | 70 | 80 | 90 100 110 120 11 | 22 | 33 44 | 55 | 66 | 77 | 88 99 | 110 121 132 12 24 36 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 This table is so well known, that it is almost superfluous to explain that, when any number in the top row is multiplied by any number in the left-hand side row, the amount is found in the compartment or square beneath the one and opposite the other. Thus, 2 times 2 are 4; 5 times 6 are 30; 12 times 12 are 144. The multiplying of numbers beyond 12 times 12 is usually effected by a process of calculation in written figures. The rule is to write down the number to be multiplied, called the multiplicand; then place under it, on the right-hand side, the number which is to be the multiplier, and draw a line under them. For example, to find the amount of 9 times 27, we set down the figures thus 27 (Multiplicand.) 9 (Multiplier.) 243 (Product.) Beginning with the right-hand figure, we say 9 times 7 are 63; and putting down 3 we carry 6, and say 9 times 2 are 18, and 6, which was carried, makes 24; and writing down these figures next the 3, the product is found to be 243. UBTRACTION is the deducting of a smaller number from a greater, to find what remains, or the difference between them. The Sign of Subtraction is It is read minus, and signifies less. When placed between two numbers, it indi cates that the one after it is to be subtracted from the one before it. Thus, 12 7 is read 12 minus 7, and means that 7 is to be subtracted from 12. A Parenthesis () is used to include within it such numbers as are to be considered together. A Vinculum has the same signification. Thus, 25 - (127), or 25 12 + 7, signifies that from 25 the sum of 12 and 7 is to be subtracted. PRINCIPLES.-I. Only like numbers and units of the same order can be subtracted. 2. The minuend must be equal to the sum of the subtrahend and remainder. 537 325 We subtract when we say, take 3 from 5, and 2 remains. To ascertain what remains, after taking 325 from 537, we proceed by writing the one under the other as here indicated, and then subtracting. Commencing at 5, the right-hand figure of the lower and smaller number, we say, 5 from 7, and 2 remains; setting down the 2, we say next, 2 from 3, and I remains; and setting down the I, we say, 3 from 5, and 2 remains; total remainder, 212. 212 8432 6815 1617 To subtract a number of a higher value, involving the carrying of figures and supplying of tens, we proceed as in the margin. Commencing as before, we find that 5 cannot be subtracted from 2, and therefore supply or lend 10 to the 2, making it 12; then we say, 5 from 12, and 7 remains. Setting down the 7, we take I, being the decimal figure of the number which was borrowed, and give it to the 1, making it 2, and taking 2 from 3, we find that I remains. Setting down the 1, we go to the 8, and finding it cannot be taken from the 4 above it, we lend 10 to the 4, making it 14, and then we say, 8 from 14, and 6 remains. In the same manner as before, adding the first figure of the borrowed number (1) to the 6, we say, 7 from S, and I remains; thus the total remainder is found to be 1617. From these explanations, which apply to all calculations in subtraction, it will be observed, that when the upper figure is less than the figure directly under it, 10 is to be added, and for this one is carried or added to the next under figure. A man having $15, paid $4 for a hat, and $2 for a vest. How many dollars had he left? ANALYSIS. The difference between $15, and the sum of $4 and $2, which is $9. DIVISION. IVISION is that process by which we discover how often one number may be contained in another, or by which we divide a given number into any proposed number of equal parts. By the aid of the Multiplication Table, we can ascertain without writing figures how many times any number is contained in another, as far as 144, or 12 times 12; beyond this point notation is employed. There are two modes of working questions in division, one long and the other short Let it be required to divide 69 by 3: according to the long method, write the figures 69 as annexed, with a line at each side, and the divisor, or 3, on the left. The question is wrought out by examining how many times 3 is in 6, and finding it to be 2 times, we place 2 on the right side; then placing 6 below six, we draw a line and bring down the 9, and proceed 3)69(23 6 9 9 6 19 18 15 321 with it in the same manner. The quotient is found to be 23. But we take a more difficult question-the division of 7958 by 6. In commencing we find. that there is only one 6 in 7, and I over; we 6)7958(1326 therefore place the 6 below the 7, and subtract it, in order to bring out the 1. The I being written, we bring down the 9 to it, and this makes 19. There being 3 times 6 in 18, we place the 3 to the product (which in division is called the quotient, literally, How many times?) and 18 below the 19, leaving I over as before. To this I we bring down the 5, and trying how many sixes there are in 15, it appears there are only 2. We place 2 to the quotient, and 12 below the 15. This leaves 3 over, and bringing down 8 to the 3, we have 38, in which there are 6 sixes. sixes make 36; therefore, placing 6 to the quotient, and 36 below the 38, we find that there are 2 over. Here the account terminates, it being found that there are 1326 sixes in 7958, with a remainder of 2 over. In this question, 6 is called the divisor; the 7958 is the dividend, and 1326 is the quotient. 38 36 2 6)7958 Six Skillful arithmeticians never adopt this long method of division; they pursue a plan of working out part of the question in the mind, called short division. They would, for example, treat the above question as here shown. The over number of I from the 7 is carried in the mind to the 9, making 19; the I from 19 is in the same manner carried to the 5; and the 3 from it is carried to the 8, leaving the overplus of 2. 1326-2 In Short Division the quotient only is written, the opera tions being performed mentally. It is generally used when the divisor does not exceed 12. Divide $48.56 by 8 cents. OPERATION. $.08)$48.56 607 times. Eight cents may be written $.08. When the divisor and dividend are like numbers, the quotient is an abstract number. Hence, 8 cents are contained in $48.56, 607 times. Division is denoted by the following character÷; thus, 7525, signifies that 75 is to be divided by 25. These explanations conclude the subject of simple or abstract numbers. On the substructure of the few rules in Addition, Multiplication, Subtraction, and Division, which we have given, whether in reference to whole numbers or fractions, every kind of conventional arithmetic is erected, because these rules are founded in immutable truths. Mankind may change their denominations of money, weights, and measures, but they can make no alteration in the doctrine of abstract numbers. That 2 and 2 are equal to 4, is a truth yes terday, to-day, and forever. H FRACTIONS.** ITHERTO we have spoken only of whole parts into which integers may be broken. more ordinary fractions of any single article or number are a half, third, quarter, etc.; but a number admits of being divided into any quantity of equal parts. All such fractions are called vulgar fractions, from their being common. It is the practice to write vulgar fractions with two or more small figures, one above the other, with a line between, as follows: (onehalf), (one-third,) (one-fourth or quarter), (one-eighth), (four-fifths), (nine-tenths), and so on. In these and all other instances, the upper number is called the numerator, the lower the denominator. GENERAL PRINCIPLES OF FRACTIONS. 1. Multiplying the numerator, or or} Multiplies the fraction. Dividing the denominator, 2. Dividing the numerator, or Multiplying the denominator, 3. Multiplying or dividing both) numerator and denominator by the same number, Divides the fraction. Does not change the value of the fraction. These three principles may be embraced in one GENERAL LAW. A change in the numerator produces a like change in the value of the fraction; but a change in the denominator produces an opposite change in the value of the fraction. It may happen that it is necessary to add together different fractions to make up whole numbers. In working all such questions, we must, in the first place, bring all the fractions into one kind; if we have to add, and together, we make all into eighths, and see how many eighths we have got: thus is then is, that is 2 and 4. which make 6, and makes a total of . The same plan is to be pursued in the subtraction of vulgar fractions. It is necessary sometimes to speak of the tenths, hundredths, or thousandths of a number, and for this arithmetic has provided a system of decimal fractions. Where great exactness of expression is required, decimals are indispensable. It has been already shown that, in writing common numbers, the value of a figure increases by ten times as we proceed from right to left; in other words, we ascend by tens. Now, there is nothing to prevent us in the same manner descending by tens from unity. This is done by decimal fractions. We place a dot after unity, or the unit figure, which dot cuts off the whole number from its fractional tenths; thus 120.3 means 120 and 3-tenths of a whole; if we write 120.31, the meaning is 120 and 31-hundredths of a whole, that is, 31 parts in 100 into which a whole is supposed to be divided. If we go on adding a figure to the right, we make the fraction into thousandths; as, for instance, 120.315, which signifies 120 and 315 out of a thousand parts. Tables of specific gravities, population, mortality, and many matters of statistics, are greatly made up of decimal fractions, and therefore it is proper that all should comprehend the principle on which they are designed. In many cases, it would answer the purpose to write the fractions as vulgar fractions; but there is a great advantage in reducing all broken parts to the decimal notation, for it allows of adding up columns of decimals all of the same denomination. Their great excellence, indeed, consists in the uniformity which they give to calculations, and the easy methods which, by these means, they present of pursuing fractional numbers to any degree of minuteness. The method of reducing a vulgar to a decimal fraction is a simple question in division. For instance, to reduce to a decimal, we take the 3, and putting two ciphers after it, divide SERIES of numbers is a succession of numbers that increase or decrease according to some law. Of the two kinds of series usually treated of in arithmetic, the simpler is one whose terms increase or decrease by some constant number called the common difference. This common difference or rate of increase is only one, when we say, 4, 5, 6, 7, 8; it is two, when we say 7, 9, II, 13; and four, when we say 6, 10, 14, 18, and so on. Every advancement of this nature, by which the same number is added at every step, is called arithmetical progression. There is a different species of advancement, by which the last number is always multiplied by a given number, thus causing the series to mount rapidly up. Suppose 4 is the multiplier, and we begin at 2, the progression will be as follows: 2, 8, 32, 128, 512, 2048, and so on. It is here observed, that multiply ing the 2 by 4 we have 8; multiplying the 8 by 4, we have 32; and multiplying the 32 by 4. we have 128, etc., till at the fifth remove we attain 2048. This kind of advancement of numbers is called geometrical progression. The very great difference between the two kinds of progression is exemplified in the following two lines, the number 3 being added in the one case and being used as the multiplier in the other. |