5) 165369

9) 330734

36743 +45

Having divided by 5, the quotient is 330734.

Again, in dividing the integral part of this quotient, which is 33073, by 9, we have 36743; and, in dividing the fractional part by 9, (165,) we have 4. We have, therefore, 3674% for the final quotient.

45

+ 4
45

45'

But, to have only one fraction, we add the two fractions and together. Now the denominator of is the last divisor, and the denominator of 45, the product of the two divisors: consequently, as 9 is contained in 45 as often as expressed by the first divisor; to reduce to a fraction, the denominator of which shall be 45, we must multiply the numerator 7 by the first divisor 5, which gives 35; then, adding the two fractions The numerator 25 and together, we have 3 for the sum.

45

39 of this sum is composed of the last remainder 7 multiplied by the first divisor 5, plus the first remainder 4: the denominator 45, being the product of the two divisors, or whole number divided by; and it is easy to see that the same kind of reasoning will hold, whatever may be the two remainders or divisors.

Hence, we have the following general rule: When there are two remainders, multiply the last remainder by the first divisor, and to the product add the first remainder; under this sum write the product of the two divisors, and reduce the fraction, if necessary, to its lowest terms.

The above example, performed by this rule, will stand thus:

5) 165369

9) 33073

3674

17X5+4 39 13

45

45 15 367413 quotient.

It is evident that 367413 is the quotient of 165369 divided by 45; because (144) in dividing by 5, we take ; and, in dividing by 9, we take of, which (211) is

When there is but one remainder, it is easy to perceive that if it belongs to the first division, it will take the product of the two divisors for a denominator; because it is a part of the dividend which has not yet been divided by either divisor, and that,

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.

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 14661940383, to 169281735?

Answer 2245

2592

The ratios of each of two numbers to their sums are the com

plements of each other.

10. What is the ratio of 169281735 to 22662331 31?

864

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 quantity.

11*

BOOK III.

PRELIMINARY

IDEAS-NUMERATION-ADDITION,

SUBTRAC

TION, 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 frac tion, (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

least (158) its approximate value, within any assignable quantity.

9

9

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, 100, 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 9 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.

224. The complements (205) of the fractions 90 99 999 &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.

9

then S=10180 + 1000 + 10800, &c., ad inf. Multiply both sides by 10. Then

108=9+ (1% + 180 + 1000 + 10800, &c, ad inf.) substitute S for its equivalent (o, TU, TOOO, TOOOO; &c.

ad inf.)

Then we have 10S 9+ S.

Subtract S from both sides.

10009

Then 989; and,

dividing by 9, we have S1. Therefore,
the series,9999, &c. ad inf. = 1.

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 100, 100, 10000, 18350, and 100000

5

25 1250 625

These in their lowest terms are

1 1 1

20, 4, 16, and go

875

In the same manner the student will find that ,5=1;,75 = .,375 ;,1875;,03125 = = and ,0015625810.

1

321

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.

25

250

100 1000

2500 10000

100000

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.