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NOTES.–1. The subject of circulating decimals was first developed by Dr. Wallis, Professor of Geometry at Oxford, born in 1616.

2. Circulates which begin at the same place are usually called similar, and those which end at the same place, conterminous. It is more precise to include both of these in similar and give another name to those which begin at the same place. Surely circulates are not entirely similar unless they begin and end alike.

3. There being no word employed to denote a similarity of origin, the term coöriginous, expressing a co-origin, is suggested. Its appropriateness may be seen by comparing it with conterminous, a co-termination.

4. In reading a mixed circulate, read the decimal and then name the repeating part; thus .206 is read, “the mixed circulate 206 thousandths, in which 06 repeats.”

REDUCTION OF CIRCULATES. 316. The Reduction of Circulates is conveniently treated under four cases.

CASE 1. 317. To reduce a common fraction to a circulate. 1. Reduce to a circulate.

OPERATION. SOLUTION.—Annexing ciphers to the 7 and divid

22)7.000 ing by 22, as in Art. 271, we find 22 equals the circu

.31818+ late .318. Hence the following

=.3i8 Rule.- Annex ciphers to the numerator and divide by the denominator, until the terms begin to repeat, and then place the period over the first and the last terms of the repeating

part.

Reduce the following common fractions to circulates :
2. 11
Ans. .90. 8. 33.

Ans. .96. 3. j. Ans..39. 9. 21.

Ans. .954. 4. 43. Ans. .9772.

Ans. .952380. 5. 11. Ans. .846153. 11. 3. Ans. .9642857i. 6. 14 Ans. .9285714. 12. 33. Ans. .99695121. 7. 34. Ans. .9714285. 13. 114. Ans. .99759615384. Prove, by actual division, the following principles : 1. That i=.i.

4. That 900g=.000i. 2. That g'=.ii.

5. That goo=.0000i. 3. That odg=.00i.

6. That 79 99g=.00000i.

ABBREVIATED METHOD OF REDUCTION. 318. An Abbreviated Method may be employed when the circulate consists of many figures.

1. Reduce / to a circulate. SOLUTION.-13.14; now ? is 2 times 1, hence & equals 2x.14or .284; substituting the value of ', we have .14284; now =4 times , hence, =4x.14284, or .57144; substituting this value, we have = .14285714%, which we see begins to repeat; hence, į=.142857.

NotE.-The solution so clearly indicates the method, that no rule need be given for it.

Reduce the following fractions to circulates: 2. 4. Ans. .02439. 7. 3. Ans. .0188679245283. 3. . Ans. .047619. 8. . Ans. .0126582278481.

Ans. .076923. 9. . Ans. .0588235294117647. 5. g. Ans. .025640. 10. 1. Ans..052631578947368421. 6. 7. Ans. .01369863. I

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CASE II. 319. To reduce a pure circulate to a common frac. tion.

NOTE.—There are three distinct methods of explaining this case, two of which are given here and the other under Geometrical Progression.

1. Reduce .648 to a common fraction. SOLUTION 1st.–Since .ooi equals ts, as OPERATION. shown in Art. 317, .648, which is 648 times ..voi=g), .ooi, equals 648 times gig, which is 648, .648=$18= 4, Ans. and this, reduced to its lowest terms, is SOLUTION 20.—Let F represent the

OPERATION. common fraction, then we will have

F=.648648 etc. F=.648648 etc; multiplying by 1000

1000 F= 648.648648 etc. to make a whole number of the repeat 999 F= 648 ing part, we have 1000 times the frac

F=$$$= 34, Ans. tion equals 648.648 etc.; subtracting once the fraction from 1000 times the fraction, we have 999 times the fraction equals 648; hence the fraction equals 595=*.

Rule.- Take the repetend for the numerator of a fraction, and as many 9's as there are places in the repetend for the denominator, and reduce the fraction to its lowest terms.

Reduce the following circulates to common fractions:
2. .054.
Ans. z r. | 7. .888.

Ans. s. 3. .324. Ans. 34. 8. .980i.

Ans. 1981: 4. .370. Ans. 19. 9. .860139.

Ans. 14. 5. .296.

Ans. 787. 10. .986013. Ans. 141 6. .962. Ans. 29. 11. .923076. Ans. 15.

CASE III. 320. To reduce a mixed circulate to a common fraction.

1. Reduce .318 to a common fraction.

OPERATION.

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SOLUTION 1st.-.318=ij of 3.18, which by the preceding case equals I of 3, or b of 34, which equals 35., and this reduced to its lowest terms equals zz.

=+=*='s. SOLUTION 20.—Let F represent the OPERATION common fraction, then we shall have

F=.3181818 etc. F=.31818 etc.; multiplying by 10 10 F=3.181818 etc. to make a whole number of the non

1000 F=318.1818 etc. repeating part, we have 10 times the

990 F= 315 fraction equals 3.1818 etc.; multiply

F=}}=1, Ans. ing this by 100 to make a whole number of the repeating part, we have 1000 F=318.1818 etc.; subtracting 10 F from 1000 F, we have 990 F=315; hence F=;to=7.

Rule 1.— Write beneath the repetend as many 9's as there are places in the repetend, annex this to the finite part, and divide the result by 1 with as many ciphers annexed as there are places in the finite part.

Rule II.-Subtract the finite part from the whole circulate, and write under the remainder as many 9's as there are figures in the repetend, with as many ciphers annexed as there are places in the finite part, and reduce the resulting fraction to its lowest terms. 2. Reduce .772 to a common fraction.

OPERATION. SOLUTION.–Subtract 7, the finite part, from .772,

.772 and we have .765; dividing by two 's with one cipher annexed, we have , which, reduced to its lowest terms, equals 12.

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Reduce the following circulates to common fractions : 3. .954. - Ans. 21. 12. 4.006.

Ans. 4150. 4. .527. Ans. 13. 13. 5.003.

Ans. 5150 6. .105. Ans. 15. 14. .04573170.

Ans. 3 6. .945. Ans. 35. 15. .82142857.

Ans. 21 7. 4.8i. Ans. 432. 16. .910714285.

Ans. 5 8. .7954. Ans. 35. 17. .96604938271. Ans. 311 9. .6590. Ans. 22. 18. 3.41; 2.0 Ans. 3]; 218. 10. 28 5714.* Ans. 284. 19. .040}; 2.0301: Ans. 3 ; 2132 11. .4857142. Ans. 37. 20. oži}; 0104. Ans. hangi dan zare

CASE IV. 321. To reduce dissimilar repetends to similar ones.

322. To solve this case we need to remember the following principles :

PRINCIPLES. 1. Any terminate decimal may be considered interminate, its repetend being ciphers. Thus, .45=.450 or .45000, etc.

2. A simple repetend may be made compound by repeating the repeating figure. Thus, .3=.33=.3333, etc.

3. A compound repetend may be enlarged by moving the right hand dot towards the right over an exact number of periods. Thus, .245=.24545, etc.

4. Both dots of a repetend may be moved the same number of places to the right without changing its value. Thus, .5378=.53783, or .537837, etc., for each expression developed will give the same result.

5. Dissimilar repetends may be made coöriginous by moving both dots of the repetend to the right until they all begin at the same place.

6. Dissimilar repetends may be made conterminous by moving the right hand dots of each repetend over an exact number of periods of each repetend until they end at the same place.

1. Make .45, .4362 and .813694 similar. SOLUTION.–To make these repe

OPERATION. tends similar they must be made to

45=.45454545454545 begin and end at the same place. To do this we first move the left hand

.4362=..43623623623623 dots so that the repetends begin at .813694=.81369436943694 the same place (Prin. 5), and then move the right hand dots over an exact number of periods so that they will end at the same place. Now the number of places in the periods are respectively 2, 3, and 4; hence the number of places in the new periods must be a common multiple of 2, 3, and 4, which is 12; we therefore move the right hand dot so that each repetend shall contain 12 places.

Rule.-I. Expand the repetends, and move the left hand dots toward the right so that they all begin at the same place.

II. Move the right hand dots so that the number of terms in each period shall be the least common multiple of the number of terms in the given periods.

2. Make 25.3, .375, and .473 similar.
3. Make 4.632, .325, and 43.32 similar.
4. Make 6.324, 3.34, .6532, and i1.oi similar.
5 Make .327, .435, 3.7642, and 6.789 similar.
6. Make 46.326, 46.326, and 46.326 similar.
7 Make io.i, 20.12, 3401.0i, and .o7 similar.
8. Make .64, 4.32, 44.53, and 3.25 similar.
9. Make .0.101, 2.0.103, 345.3, and .00043 similar.

ADDITION OF CIRCULATES. 323. Addition of Circulates is the process of finding the sum of two or more circulates.

1. Find the sum of 3.24, .685, and 4.32.

SOLUTION.-Since only similar fractional OPERATION. units can be added, the repetends must first be made similar. Having done this, we add as

3.24=3.2424242 in finite decimals, observing to add 1 to the .685= .6858585 right hand column, since this would be neces 4.32=4.3243243 sary if the repetends were expanded, and we have for the sum 8.2526071.

8.232607i

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