If great accuracy be not required, in any practical question connected with the circle, it may be sufficient to consider the 7 diameter as of the circumference. But should greater ex 22 actness be requisite, then the fraction will yield a more accurate result. From the statement and examples now given, the manner of forming continued fractions, and likewise of obtaining a series of fractions expressed in lower terms, which continually approximate to the true value of the original fraction, will be easily apprehended. To facilitate the formation of these approximate fractions, the following general rule may be given: The first fraction will have unity for its numerator, and for its denominator the first quotient obtained by dividing, as in (No. 44.) The second fraction will have for its numerator the second quotient, and for its denominator the product of the two quotients increased by unity. And each successive approximation will be found by multiplying both terms of the last approximate fraction by the next quotient, and increasing these products respectively by the terms of the second last fraction. We may often approximate to the value of an irreducible fraction, by making such a small change on its terms as shall render it reducible. EXERCISES ON CONTINUED FRACTIONS. 1. Required the successive approximations to the fraction 2. Required the successive approximations to the fraction Ans. 1 11 12 59 71 2' 23' 25' 123' 148 DECIMAL FRACTIONS. 57. Decimal fractions are those which have for their denominator 10, or 100, or 1000, or in general some power of 10. The denominator of a decimal fraction is not generally expressed, but is indicated by a point prefixed to the numera tor. Thus 5 57 345 10' 100' 1000' are respectively written .5, .57, .345; or sometimes 0.5, 0.57, 0.345, when the cipher shows that there is no integral digit. &c. it is obvious that the figure next the decimal point will express tenths, the second hundredths, the third thousandths, and so on, diminishing in a tenfold proportion from left to right, the same as integer numbers. 58. From this remark, it will be easy to point the numerator, so that it shall express any decimal fraction whatever. For, suppose we have the fraction then as it is the second 5 100' place from the decimal point which expresses hundreths, we prefix a cipher to the 5, in order that it shall occupy that 5 1000 place thus .05. In the same manner will be written thus .005, because it is the third figure from the decimal point which expresses thousandths; 7 100000' 57 are equal to 100000 5 10000 + fourth place from the decimal point being that of ten thousandths, and the fifth that of hundred thousandths. From these examples it is easy to perceive, that in every instance where the value of a decimal fraction is to be expressed by means of its numerator, that numerator must contain as many figures as there are ciphers in the denominator. If the numerator of itself does not contain so many figures, then the deficiency must be made up by prefixing ciphers to it. Thus in the fraction there are five ciphers in the denominator and only two figures in the numerator, and hence three ciphers must be prefixed, that the digits may occupy their proper rank. and will, therefore, be written thus, .00057, the 57 100000 59. The converse of the problem enunciated in the preceding number will also be easily managed, vix. given a decimal fraction, expressed in the entire form, to express it in the ordinary fractional form. From what is there stated, it is obvious that the denominator of a decimal fraction will be unity, with as many ciphers annexed as there are figures in the given numerator. Thus .5, .043, .00307, will be changed into 5 43 10' 100' and 60. Since the value of any figure of a decimal fraction depends entirely on the rank which it holds in regard to the decimal point; any number of ciphers may either be effaced or written on the right of a decimal without affecting its value. Thus, .5, for example, expresses the same value as .50 or .500; for .5, .50, .500, are respectively, 50 10' 100' and 500 1000' (No. 59,) each of which is equal to according to No. 42. The annexing of ciphers on the right of a decimal, corresponds to the multiplication of the terms of the fraction by the same number, and the suppression of ciphers, to the division of the terms by the same number, in either of which cases the value of the fraction is not changed. 61. From what has been stated, it appears that decimal fractions are written exactly in the same way as integer numbers. Thus, for example, the number five hundred and forty-nine units, or which is the same thing, 5 hundreds, 4 tens, and 9 units, is written 549. In like manner, five hundred and forty-nine thousandths, or which is the same thing, 5 tenths, 4 hundredths, and 9 thousandths, is written .549, or with the cipher, 0.549. An expression, in which there are both integers and decimals, is read by first enunciating the integral part, and then the fractional. Thus, 73.685, is read 73 units and 685 thousandths, or, simply, for the sake of conciseness, 73 decimal 685. 62. The great utility of decimal fractions, consists in their enabling us to express any numerical quantity, however small, on the decimal scale, and thus to subject fractional quantities to the same operations as integer numbers. It is in Logarithmic calculation, however, that their convenience is particularly felt. 63. The addition of decimals is performed exactly in the same way as that of integer numbers. The fractions to be added, are written in such a manner, that quantities of the same kind may stand under one another, and then their sum is obtained as in Simple Addition. Ex. 1. Required the sum of 0.67, 0.087, and 0.0054. 0.67 0.0054 0.7624 The sum. Ex. 2. Required the sum of 4.86, 28.075, 52.904, and 0.0083. 4.86 28.075 52.904 0.0083 85.8473 The sum. It is hardly necessary to add one remark in illustration of the preceding examples. In the last, the sum of the thousandths is 17, which are equal to 1 hundredth and 7 thousandths. The 1 hundredth being carried to its proper column, the sum will there be 14, which, in like manner, are equal to 1 tenth and 4 hundredths. The sum of the tenths, with the addition of the 1 carried, will be 18; now, 18 tenths are obviously equal to 1 unit and 8 tenths, the unit is of course carried to its proper column, &c. With regard to the place of the decimal point in the sum, it is manifest that it will be exactly under those of the several quantities to be added. 64. The method of performing subtraction in decimals, is also precisely the same as that in integer numbers. Ex. 1. Required the difference of 0.76 and 0.4579. The value of the first number will not be altered if two ciphers be annexed to it, (No. 60.). Let this be done, then the operation will be as follows: 0.7600 0.3021 The difference. The object of annexing ciphers to the minuend, is to reduce it to a fraction, having the same denominator as the sub trahend. Thus, 0.76 = 76 100 and 0.4579 4579 10000 76 100 (No. 59.) Now, if the terms of the fraction be both multiplied by 100, (by which its value will not be altered, No. 42,) we get 7600 10000' and hence 7600 10000 The ciphers are generally omitted in practice. 47.042 37.33516 The difference. The same remark may be made here as in Addition, with respect to the place of the decimal point. 65. Since the decimal point separates the integral units from the decimal parts, its change of place will necessarily be accompanied by a change of value of the total number. In moving the decimal point towards the right, the number of integer units will be increased, and consequently the total value of the proposed number. In moving it towards the left, the number of integer units will be diminished, and hence also the value of the proposed number. 66. If the decimal point be moved one place toward the right, then the value of every figure will be increased 10 times, and consequently the value of the whole number will now be 10 times as great as at first. If it be moved two places toward the right, then, since by one removal, the value of the whole number is increased 10 times, another removal will increase this last value also ten times, and hence, by two removals, the original value will be increased 100 times. In like manner, by moving the decimal point three places to the right, the original value will be increased 1000 times, and so on. From this it follows, that the multiplication of a number containing decimals, by 10, 100, 1000, or, in general, any power of 10, will be effected by removing the decimal point as many places toward the right, as there are ciphers in the given multiplier. Thus 54.8674, when multiplied by 10, 100, 1000 respectively, will become 548.674, 5486.74, and 54867.4. In the same manner, 0.00784 when multiplied by 10, 100, 1000, will become 0.0784, 0.784, and 7.84 respectively. 67. Conversely, the division of any number by any power of 10, will be effected by removing the decimal point as many places toward the left as the given divisor contains ciphers. Thus 8764.7, when divided by 10,100 and 1000 respectively, will become 876.47, 87.647, and 8.7647. In the same manner, 0.845, when divided by 10,100 and 1000, will become 0.0845, 0.00845, and 0.000845 respectively. |