if it belongs to the last division, it will take the last divisor only, for a denominator; because it has no more division to undergo. Examples. = 1. 23754325 950118. 6. 9939274535=28397925. 7. Divide 169281735, separately, by each of the numbers 24, 32, 36, 48, 81, and find the difference between the sum of the quotients and the dividend. Answer 146619403833. 8. In the preceding question, what is the ratio of the sum of the quotients to the dividend; or, in other words, what part of the dividend is the sum of the quotients? Answer 347 2592 9. What is the ratio of 146619403883, to 169281735? Answer 2245 2592' The ratios of each of two numbers to their sums are the complements of each other. 10. What is the ratio of 169281735 to 2266233134 Here the dividend divided by the sum of the quotients gives its ratio to that sum. Hence, it is the product of the sum of the quotients and that ratio. Now, if the product of two numbers be divided by either, the quotient will be the other. Therefore, if we divide the dividend by its ratio to the sum of the quotients, we shall have that sum for the result. That is to say, any quantity divided by its ratio to another quantity, will give that other quantity. But to divide by a fraction is the same as to multiply by its reciprocal. Wherefore, any quantity multiplied by the reciprocal of its ratio to another quantity, will give that other quan tity. 11* BOOK III. PRELIMINARY IDEAS-NUMERATION-ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF DECIMAL FRACTIONS, AND FRACTIONAL NUMBERS-REPEATING OR CIRCULATING DECIMALS. SECTION X. PRELIMINARY IDEAS-NUMERATION OF DECIMALS. 220. The word decimal, derived from the Latin word decem, ten, is applied to that which is numbered by tens. Hence, the scale of Natural Numbers (Sec. I.) is properly termed, the decimal scale of Natural Numbers. If, on the right of this scale, we place a comma, and consider a single unit, or unit of its right-hand order, to be an alt unit of the same scale continued downwards, that is, towards the right, it is evident (37) that this unit is greater than any definite number of figures which can stand on the right of the comma; consequently, those figures constitute a fraction, properly termed a decimal fraction, (often simply a decimal.) 221. As the whole scale is decimal, subject to a universal law, that ten units of any order constitute one unit of the next order on the left, or, inversely, that an alt unit of any order is equal to ten units of the next order on the right, it is plain, that every natural number, whole, fractional, or mixed, is a decimal number; notwithstanding which, the figures on the right of the comma are, by way of distinction, exclusively called decimal figures, or decimals; and those numbers alone, which contain, or, by reduction, are supposed to contain, orders on the right of the comma, are called decimal numbers. 222. Considering the scale of Natural Numbers to extend, in both directions, from the comma to infinity, it is now complete; the part on the left of the comma being integral, that is, (36,) capable of expressing any whole number, and that on the right fractional, that is, capable of expressing any fraction, or at 126 least (158) its approximate value, within any assignable quantity. 223. Beginning at the comma, if, on the right of it, we suppose a series of nines, thus, ,9999, &c., to extend ad inf., these (221) according to the universal law, are successively 9 tenths, 9 hundredths, 9 thousandths, 9 ten-thousandths, &c.; as vulgar fractions, 10, 180, 1000, 10800, &c.; that is, the denominator of each is a unit followed by as many ciphers as there are decimal places, counting from the comma to the figure inclusive. But any number of consecutive figures on the right of the comma, as well as in any other part of the scale, may (48) be read as a number of units of the order of the right-hand figure of the number taken. Thus, if we take, consecutively, one, two, three, four, &c. nines, we read, 9 tenths, 99 hundredths, 999 thousandths, 9999 ten-thousandths, &c. These, as vulgar fractions are 0 100 1000 100002 99 999 9999. &c. Wherefore (48) the denominator of any decimal number, expressed as a vulgar fraction, is a unit followed by as many ciphers as there are places of decimals in the number, counting from the comma to the right-hand place inclusive. 9 99 999 224. The complements (205) of the fractions 10, 100, 1900, &c. are 10, 100, 1000, &c; that is, the complement of a decimal series of nines, beginning at the comma, is a unit of the lowest order of the series; also, (87,) the integral unit, on the left of the comma, is the alt of the series, and is first, second, third, fourth, &c.; that is, it is 10, 100, 1000, 10000, &c., according to the number of nines in the series. Hence, it is plain that a definite number of these nines can never equal a unit; but because the part lacking, or complement, is ten times less at each remove from the comma, it is easy to see that the ultimatum or limit of the series is a unit. Let S=,9999, &c. ad inf. then S=10180 + 1000 + 10000, &c., ad inf. Multiply both sides by 10. Then 1089+%+ 100 + 1000 + 10800, &c, ad inf.) substitute S for its equivalent (0, 180, TOO, Toboo; &c. ad inf.) Then we have 1089+ S. Subtract S from both sides. Then 989; and, dividing by 9, we have S1. Therefore, 225. The alt, therefore, of a decimal is its denominator, when expressed as a vulgar fraction; and is a unit followed by as many ciphers as there are places of figures in the decimal, whether those places are supplied with significant figures or ciphers. Wherefore, to express a decimal as a vulgar fraction, we first write the decimal as a whole number, for the numerator, and underneath we write its alt for denominator. When there are ciphers between the comma and the highest order of the decimal, we omit them in writing the numerator. Thus, ,05; 25;,1250;,0625, and,00875, expressed as vulgar fractions, are 180, 100, 150, 18350, and 10000 25 625 These in their lowest terms are 20, 4, 18, and go 875 In the same manner the student will find that ,5=1;,75 = 2.,375 = ;,1875=;,03125 and ,0015625610. 226. Ciphers, placed on the right of a decimal, do not alter its value, because, for every additional cipher in the decimal, or numerator, we have one more cipher in the alt, or denomitor, (165 and 225.) Thus : ,25,250,2500=,25000=,250000, &c., that is (165) 25000 250000, &c. 100000 25 250 2500 100-1000 10000 1000000 Hence, if the number of decimals in several decimal numbers be different, it may be rendered the same in each, without altering their value, by placing ciphers on the right of those which have an inferior number, till they all have as many as that which has the greatest number. Thus, instead of 25,3,,416,,7854, and 3,141592 we may write 25,300000;,416000; ,785400, and 3,141592 227. To write, in its natural form, a decimal number, given in the form of a vulgar fraction, we first write the numerator as a whole number; then, beginning on its right-hand figure, we count towards the left as many figures, for decimals, (225,) as there are ciphers in the denominator, and, placing the comma on the left, we have the required decimal number. When the number of figures is not sufficient, we make it so, by placing ciphers on the left. 83 Thus, to express 100 in its natural form, we first write 83. Then, as there are four ciphers in the denominator, beginning on 3, we count towards the left, one, two, three, four, placing a cipher, as we say three, and another as we say four; and, prefixing the comma, we have ,0083 for the required decimal. 228. In the scale of Natural Numbers, (222,) the value of any figure is determined by its distance from the comma. Therefore, in any finite number, to remove the comma one, two, three, &c. places to the right, is the same as to advance each figure one, two, three, &c. places towards the left; that is, (36,) it is the same as to multiply the number by 10, 100, 1000, &c. Also, to remove the comma one, two, three, &c. places to the left, is to advance each figure one, two, three, &c. places towards the right; that is, (223,) to divide the number by 10, 100, 1000, &c. Wherefore, to multiply a decimal number by 10, 100, 1000, &c., we remove the comma one, two, three, &c. places towards the right. Also, to divide the number by 10, 100, 1000, &c., we remove the comma one, two, three, &c. places towards the left. Thus : ,7854 × 107,854;,7854 × 1000=785,4. If, in either direction, we do not find a sufficient number of places, we supply the deficiency with ciphers. Thus : 785,4 X 10078540; 1,6 100,016;,7854 ÷ 1000 =,0007854; ,7854 1000000 785400 Numeration of Decimals. 229. The denominator of a decimal figure is (223) a unit followed by as many ciphers as there are places from the comma to the figure inclusive; and is also (225) the alt of a whole number containing the same number of places. We have, therefore, two methods by which to determine the order of any decimal figure; namely, we can point off the places of figures from the figure, considered as units, to the comma. Then, having read the figure, we pronounce the name of the alt or unit of the next higher grade with the usual denominating termination, ths. Or, beginning at the comma, we can pronounce on the places successively, tenths, hundredths, thousandths, tenthousandths, hundred-thousands, millionths, ten-millionths, hundred-millionths, billionths, &c., till we reach the figure. 230. In the scale of Natural Numbers, a number of consecu |