Intermediate Decimals.
\[ \begin{align*} \frac{1}{2} &= 0.5 \\ \frac{1}{3} &= 0.333 \\ \frac{1}{4} &= 0.25 \\ \frac{1}{5} &= 0.2 \\ \frac{1}{6} &= 0.1666 \\ \frac{1}{7} &= 0.142857 \\ \frac{1}{8} &= 0.125 \\ \frac{1}{9} &= 0.1111 \\ \frac{1}{10} &= 0.1 \\ \frac{1}{11} &= 0.0909 \\ \frac{1}{12} &= 0.08333 \\ \frac{1}{13} &= 0.076923 \\ \frac{1}{14} &= 0.0714285 \\ \frac{1}{15} &= 0.06666 \\ \frac{1}{16} &= 0.0625 \\ \frac{1}{17} &= 0.0588235294117647 \\ \frac{1}{18} &= 0.05555 \\ \frac{1}{19} &= 0.052631578947368421 \\ \frac{1}{20} &= 0.05 \\ \frac{1}{21} &= 0.047619 \\ \frac{1}{22} &= 0.0454545 \\ \frac{1}{23} &= 0.0434782608695652173913 \\ \frac{1}{24} &= 0.0416666 \\ \frac{1}{25} &= 0.04 \\ \frac{1}{26} &= 0.0384615 \\ \frac{1}{27} &= 0.037037 \\ \frac{1}{28} &= 0.03571428 \\ \frac{1}{29} &= 0.0344827586206896551724137931 \\ \frac{1}{30} &= 0.0333 \\ \end{align*} \]
Rules for reducing intermediate decimals to vulgar fractions.
I. "If the decimal be a pure repeater, place the repeating figure for the numerator, and 9 for the denominator."
II. "If the decimal be a pure circulate, place the circulating figures for the numerator, and as many 9's as there are places in the circle for the denominator."
III. "If there be cyphers prefixed to the repeating or circulating figures, annex a like number to the 9's in the denominator."
IV. "If the decimal be mixed, subtract the finite part from the whole decimal. The remainder is the numerator; and the denominator consists of as many 9's as there are places in the circle, together with as many cyphers as there are finite places before the circle."
Thus, \( \frac{23}{35} = \frac{23}{35} \)
From the whole decimal \( \frac{23562}{35} \)
We subtract the finite part \( \frac{235}{35} \)
and the remainder \( \frac{23327}{35} \) is the numerator.
The reason may be illustrated by dividing the decimal into two parts, whereof one is finite, and the other a pure repeater or circulate, with cyphers prefixed. The sum of the vulgar fractions corresponding to these will be the value of the decimal sought.
\( \frac{23562}{35} \) may be divided into \( \frac{235}{35} \) by rule I.
and \( \frac{23327}{35} \) by rules II. III.
In order to add these vulgar fractions, we reduce them to a common denominator; and, for that purpose, we multiply both terms of the former by 99, which gives \( \frac{23327}{35} \); then we add the numerators.
\( \frac{235}{35} \) or by method explained p. 295. col. 1. par. 3.
\[ \begin{array}{ccc} \text{Sum of numerators.} & & \\ 2115 & 23500 & 23205 \\ 2115 & 235 & 62 \\ 23265 & 23265 & 23327 \\ \end{array} \]
The value of circulating decimals is not altered, though one or more places be separated from the circle, and considered as a finite part, providing the circle be completed. For example, \( \frac{27}{35} \) may be written \( \frac{27}{35} \), which is reduced by the last of the foregoing rules to \( \frac{27}{35} \), or \( \frac{27}{35} \), which is also the value of \( \frac{27}{35} \).
And if two or more circles be joined, the value of the decimal is still the same. Thus, \( \frac{27}{35} = \frac{27}{35} \), which is reduced by dividing the terms by 101 to \( \frac{27}{35} \). All circulating decimals may be reduced to a similar form, having a like number both of finite and circulating places. For this purpose, we extend the finite part of each as far as the longest, and then extend all the circles to so many places as may be a multiple of the number of places in each.
Ex.: .34725, extended, .34725725725725, 1.4562, 14.592456245624,
Here the finite part of both is extended to two places, and the circle to 12 places, which is the least multiple for circles of 3 and 4 places.
Sect. ii. Addition and Subtraction of Intermediate Decimals.
To add repeating decimals, "Extend the repeating figures one place beyond the longest finite ones, and when you add the right-hand column, carry to the next by 9."
Ex.: .37524 or 37524 .28 .298 7 .8 88888 .328 .42 1 .643 643 .469/7 .7548 1 .73 73333 .38 .32 7
264046
To subtract repeating decimals, "Extend them as directed for addition, and borrow at the right-hand place, if necessary, by 9."
.93566 .646 .7358 .7382 .469 .84738 .53427 .62563 .68 .38 .08727 .11172
The reason of these rules will be obvious, if we recollect that repeating figures signify ninth-parts. If the right-hand figure of the sum or remainder be 0, the decimal obtained is finite; otherwise it is a repeater.
To add circulating decimals, "Extend them till they become similar (p. 319, col. 1, par. ult. &c.) and when you add the right-hand column, include the figure which would have been carried if the circle had been extended further."
Ex. 1st.] Extended. Ex. 2d.] Extended. .574 .574,574 .874,874874, 2.698 .269,869 .1463 .146333333, .428 .428 .158 .158585858, .37,983 .379,839 .32 .323,232323,
1.652,284, 1.503,026390,
Note 1. Repeaters mixed with circulates are extended and added as circulates.
Note 2. Sometimes it is necessary to inspect two or more columns for ascertaining the carriage; because the carriage from a lower column will sometimes raise the sum of the higher, so as to alter the carriage from it to a new circle. This occurs in Ex. 2.
Note 3. The sum of the circles must be considered as a similar circle. If it consist entirely of cyphers, the amount is terminate. If all the figures be the same, the amount is a repeater. If they can be divided into parts exactly alike, the amount is a circle of fewer places; but, for the most part, the circle of the sum is similar to the extended circles.
.3,868, .0842, 3 .368 .003094, .4,375, .0842 3 .57, .705, 4 .833492, .0842 3 .895 .70, 2 .62, .0842 3 .742 .705 2
To subtract circulating decimals. "Extend them till they become similar; and when you subtract the right-hand figure, consider whether it would have been borrowed if the circles had been extended further, and make allowance accordingly.
-5.72, .974, or -9.74974, .9,135, or -8,135135, -4.86, .96, .868686, .452907 or -4,529074,
-10,85, .106288, -3,606060, or -3,60
Sect. iii. Multiplication of Intermediate Decimals.
Case I. "When the multiplier is finite, and the multiplicand repeats, carry by 9 when you multiply the repeating figure: The right-hand figure of each line of the product is a repeater; and they must be extended and added accordingly."
Ex. 1.3494 -367 9446 809666 4048333 -0495246
If the sum of the right-hand column be an even number of 9's, the product is finite; otherwise, it is a repeater.
Case II. "When the multiplier is finite, and the multiplicand circulates, add to each product of the right-hand figure the carriage which would have been brought to it if the circle had been extended. Each line of the product is a circle similar to the multiplicand, and therefore they must be extended and added accordingly."
The product is commonly a circulate similar to the multiplicand; sometimes it circulates fewer places, repeats, or becomes finite; it never circulates more places.
Ex. .37,46 × .235 1. .674, × .78 2. .37, × .86 3. .625, × .42 4. .4793, × .48 5. .375, × 1.24 6. .2963, × .36
.08804,19,
Case III. "When the multiplier repeats or circulates, find the product as in finite multipliers, and place under it the products which would have arisen from the repeating or circulating figures, if extended."
Ex. 1st.] .958 × 8 2d.] .784 × .36, 8 36 7664 4794 7604 2352 7664 28224 7664 28224 29224
.8518
No. 28. The multiplication of intermediate decimals may be often facilitated, by reducing the multiplier to a vulgar fraction, and proceeding as directed p. 311. col. i. par. 6.
Thus,
4th.] \( \frac{3824}{7} = \frac{54}{9} \)
5th.] \( \frac{384 \times 28}{23} = \frac{88}{9} \)
Therefore, in order to multiply by \( \frac{3}{9} \), we take one third part of the multiplier; and, to multiply by \( \frac{2}{9} \), we take two thirds of the same. Thus,
6th.] \( \frac{784}{3} = \frac{2613}{3} \)
7th.] \( \frac{8761 \times 6}{3} = \frac{5406}{3} \)
As the denominator of the vulgar fractions always consists of 9's, or of 9's with cyphers annexed, we may use the contraction explained p. 298. col. 1. par. ult. &c.; and this will lead us exactly to the same operation which was explained p. 320. col. 2. par. ult. &c. on the principles of decimal arithmetic.
8th.] \( \frac{735 \times 326}{323} = \frac{237405}{323} \)
9th.] \( \frac{278 \times 365}{365} = \frac{101571}{365} \)
When the multiplier is a mixed repeater or circulate, we may proceed as in Ex. 5th and 8th; or we may divide the multiplier into two parts, of which the first is finite, and the second a pure repeater or circulate, with cyphers prefixed, and multiply separately by these, and add the products.
Thus, \( \frac{384 \times 28}{23} \) or by \( \frac{2}{9} = \frac{0768}{9} \) or thus, \( \frac{384}{9} \)
and by \( \frac{05}{9} = \frac{0213}{9} \)
\( \frac{09813}{9} \)
In the following examples, the multiplicand is a repeater; and therefore the multiplication by the numerator of the vulgar fraction is performed as directed p. 320. col. 2. par. 2.
Vol. II. Part I. In the following examples the multiplicand is a circulate, and therefore the multiplication by the numerator is performed as directed p. 320. col. 2. par. 4.
12th.] \( \frac{3}{8} \times \frac{5}{8} = \frac{15}{64} \)
9) 3,416 (.37,962,
27
71
63
99) 15010 (.15,16,
99
* 86
81
511
495
56
* 160
54
99
26
18
610
* 86
594
* 16
13th.] \( .12 \times 0.3 = \frac{3}{10} \)
99) 36,36 (.036730945821854912764,
666
723
306
936
453
576
813
216
183
846
543
486
903
126
273
756
633
396
* 036
In Ex. 13th, we have omitted the products of the divisor, and only marked down the remainders. These decimals are found, by adding the left-hand figure of the dividend to the remaining figures of the same. Thus, 363 is the first dividend, and 3 the left-hand figure, added to 63, the remaining figures gives 66 for the first remainder; and the second dividend, 666, is completed by annexing the circulating figure 6. The reason of which may be explained as follows. The highest place of each dividend shows, in this example, how many hundreds it contains; and as it must contain an equal number of ninety-nines, and also an equal number of units, it follows, that these units, added to the lower places, must show how far the dividend exceeds that number of ninety-nines. The figure of the quotient is generally the same as the first place of the dividend, sometimes one more. This happens in the last step of the foregoing example, and is discovered when the remainder found, as here directed, would amount to 99, or upwards; and the excess above 99 only, must in that case be taken to complete the next dividend.
14th.] \( .01 \times .01 = \frac{1}{99} \)
The number of places in the circle of the product is sometimes very great, though there be few places in the factors: but it never exceeds the product of the denominator of the multiplier, multiplied by the number of places in the circle of the multiplicand. Therefore, if the multiplier be 3 or 6, the product may circulate three times as many places as the multiplicand; if the multiplier be any other repeater, nine times as many; if the multiplier be a circulate of two places, ninety nine times as many; thus, in the last example, .01, a circulate of two places, multiplied by .01, a circulate of two places, produces a circulate of twice 99, or 198 places. And the reason of this limit may be inferred from the nature of the operation; for the greatest possible number of remainders, including 0, is equal to the divisor 99; and each remainder may afford two individuals, if both the circulating figures, 3 and 6, occur to be annexed to it. If the multiplier circulate three places, the circle of the product, for a like reason, may extend nine hundred and ninety-nine times as far as that of the multiplicand. But the number of places is often much less.
The multiplication of interminate decimals may be proven, by altering the order of the factors (p. 295. col. 2. par. 2.) or by reducing them both to vulgar fractions in their lowest terms, multiplying these as directed p. 310. col. 2. par. 3. and reducing the product to a decimal.
**Set. iv. Division of Interminate Decimals.**
**Case I.** "When the dividend only is interminate, proceed as in common arithmetic; but, when the figures of the dividend are exhausted, annex the repeating figure, or the circulating figures in their order, instead of cyphers, to the remainder."
Ex. The foregoing method is the only one which properly depends on the principles of decimal arithmetic; but it is generally shorter to proceed by the following rule.
"Reduce the divisor to a vulgar fraction, multiply the dividend by the denominator, and divide the product by the numerator."
Ex. 1st.] Divide .37845 by $g = \frac{1}{2}$
$9 \div 5)3.40605(.68121$
2d.] Divide .37845 by $g = \frac{1}{3}$
$3 \div 2)1.13539(.567683$
Note 1. Division by $g$ triples the dividend, and division by $g$ increases the dividend one-half.
Note 2. When the divisor circulates, the denominator of the vulgar fraction consists of $g$'s, and the multiplication is sooner performed by the contraction explained p. 295, col. 1, par. 1. It may be wrought in the same way, when the divisor repeats, and the denominator, of consequence, is $g$.
Note 3. If a repeating dividend be divided by a repeating or circulating divisor; or, if a circulating dividend be divided by a similar circulating dividend; or if the number of places in the circle of the divisor be a multiple of the number in the dividend; then the product of the dividend multiplied by the denominator of the divisor will be terminate, since like figures are subtracted from like in the contracted multiplication, and consequently no remainder left. The form of the quotient depends on the divisor, as explained at large, p. 316, col. 1, par. 1.—p. 318, col. 2, par. 3.
Note 4. In other cases, the original and multiplied dividend are similar, and the form of the quotient is the same as in the case of a finite divisor. See p. 322, col. 2, par. ult., &c.
Note 5. If the terms be similar, or extended till they become so, the quotient is the same as if they were finite, and the operation may be conducted accordingly; for the quotient of vulgar fractions that have the same denominator is equal to the quotient of their numerators.
Chap. XI. Of the Extraction of Roots.
The origin of powers by involution has already been explained under the article Algebra. There now remains therefore only to give the most expeditious methods of extracting the square and cube roots; the reasons of which will readily appear from what is said under that article. As for all powers above the cube, unless such as are multiples of either the square and cube, the extraction of their roots admits of no deviation from the algebraic canon which must be always constructed on purpose for them.
If the root of any power not exceeding the seventh power be a single digit, it may be obtained by inspection, from the following table of powers.
| S | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---| | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
S f 2 **Extraction of the Square Root**
**Rule I.** "Divide the given number into periods of two figures, beginning at the right hand in integers, and pointing toward the left. But in decimals, begin at the place of hundreds, and point toward the right. Every period will give one figure in the root."
**II.** "Find by the table of powers, or by trial, the nearest lesser root of the left hand period, place the figure so found in the quotient, subtract its square from the said period, and to the remainder bring down the next period for a dividend or resolvend."
**III.** "Double the quotient for the first part of the divisor; inquire how often this first part is contained in the whole resolvend, excluding the units place; and place the figure denoting the answer both in the quotient and on the right of the first part; and you have the divisor complete."
**IV.** "Multiply the divisor thus completed by the figure put in the quotient, subtract the product from the resolvend, and to the remainder bring down the following period for a new resolvend, and then proceed as before."
**Note 1st.** If the first part of the divisor, with unity supposed to be annexed to it, happen to be greater than the resolvend, in this case place 0 in the quotient, and also on the right of the partial divisor; to the resolvend bring down another period; and proceed to divide as before.
**Note 2.** If the product of the quotient-figure into the divisor happen to be greater than the resolvend, you must go back, and give a lesser figure to the quotient.
**Note 3.** If, after every period of the given number is brought down, there happen at last to be a remainder, you may continue the operation, by annexing periods or pairs of cyphers, till there be no remainder, or till the decimal part of the quotient repeat or circulate, or till you think proper to limit it.
**Ex. 1st.** Required the square root of 133225.
| Square number | 133225 | 365 root | |---------------|--------|----------| | | 9 | 365 |
x div. 66) 432 resolvend.
396 product.
1825
2190
1095
---
**Extraction of the Cube Root**
**Rule I.** "Divide the given number into periods of three figures, beginning at the right hand in integers, and pointing toward the left. But in decimals, begin at the place of thousands, and point toward the right. The number of periods shows the number of figures in the root."
**II.** "Find by the table of powers, or by trial, the nearest lesser root of the left hand period; place the figure so found in the quotient; subtract its cube from the said period; and to the remainder bring down the next period for a dividend or resolvend."
The divisor consists of three parts, which may be found as follows.
---
After getting half the decimal places, work by contracted division for the other half; and obtain them with the same accuracy as if the work had been at large.
III. "The first part of the divisor is found thus: " Multiply the square of the quot by 3, and to the product annex two cyphers; then enquire how often this first part of the divisor is contained in the resolvend, and place the figure denoting the answer in the quot."
IV. "Multiply the former quot by 3, and the product by the figure now put in the quot; to this last product annex a cypher; and you have the second part of the divisor. Again, square the figure now put in the quo for the third part of the divisor; place these three parts under one another, as in addition; and their sum will be the divisor complete."
V. "Multiply the divisor, thus completed, by the figure last put in the quot, subtract the product from the resolvend, and to the remainder bring down the following period for a new resolvend, and then proceed as before."
Note 1. If the first part of the divisor happen to be equal to or greater than the resolvend, in this case, place o in the quot, annex two cyphers to the said first part of the divisor, to the resolvend bring down another period, and proceed to divide as before.
Note 2. If the product of the quotient figure into the divisor happen to be greater than the resolvend, you must go back, and give a lesser figure to the quot.
Note 3. If, after every period of the given number is brought down, there happen at last to be a remainder, you may continue the operation by annexing periods of three cyphers till there be no remainder, or till you have as many decimal places in the root as you judge necessary.
Ex. ill. Required the cube root of 12812904.
Cube number 12812904 (234 root).
1st part 1200 2nd part 180 3rd part 9
1 divisor 1389 x 3 = 4167 product.
1st part 138700 2nd part 2760 3rd part 16
2 divisor 161476 x 4 = 645904 product.
PROOF.
Square 54756 234 234 936 702 468
Square 54756 Cube 12812904
2d.] Required the cube root of 28½.
28.750000 (3.06 root-27)
270000 5400 36
Div. 275436 x 6 = 1652616 prod.
97384 rem.
PROOF.
3.06 3.06 1836 918
Sq. 9.3636
If the cube root of a vulgar fraction be required, find the cube root of the given numerator for a new numerator, and the cube root of the given denominator for a new denominator. Thus, the cube root of \( \frac{8}{27} \) is \( \frac{2}{3} \), and the cube root of \( \frac{3}{8} \) is \( \frac{3}{2} \); and thus the cube root of \( \frac{1}{8} \) (\( \frac{1}{2} \)) is \( \frac{1}{2} \).
But if the root of either the numerator or denominator cannot be extracted without a remainder, reduce the vulgar fraction to a decimal, and then extract the root.