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EXAMPLES. 1. Required to find whether the decimal equal to be finite or infinite ; and if infinite, of how many places the repetend will consist.

(2) (2) (2) AL=26=8=4=2=1; therefore, because the denominator vanishes in dividing, the decimal is finite, and consists of four places ; thus, 16)3.1995.

2. Required to find whether the decimal equal to 470% be finite or infinite ; and, if infinite, of how many places that repetend will consist.

(2) (2) (2) =42)112 = 56 = 28 = 14=7. Thus, 71999999; therefore, because the denominator, 112, did not vanish in di. viding by 2, the decimal is infinite ; and as six 9's were used, the circulate consists of six places, beginning at the fifth place, because four 2's were used in dividing.

3. Let I be the fraction proposed.
4. Let 287 be the fraction proposed.

SECTION XXXI.

ADDITION OF CIRCULATING DECIMALS.

EXAMPLE. 1. Let 3.5+7.651 +1.765+6.173+51.7 +3.7+27.631 and 1.003 be added together.

OPERATION,
Dissimilar. Similar and Conterminous.

3.5 = 3.5555555
7.651 = 7.6516516

Having made all the numbers similar

and conterminous by Sect. XXX., Case 1,765 = 1.7657657 III., we add the first six columns, as in 6.173 = 6.1737373 Simple Addition, and find the sum to 51.7 = 51.7777777 be 3591224 = 35091224 = 3.591227. 3.7 = 3.7000000 The repeating decimals .591227 we

write in their proper place, and carry 3 1.003 = 1.0030030

to the next column, and then proceed

? as in whole numbers. 103.2591227

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RULE. — Make the repetends similar and conterminous, and find their sum, as in common Addition. Divide this sum by as many 9's as there are places in the repetend, and the remainder is the repetend of the sum, which must be set under the figures added, with ciphers on the left when it has not so many places as the repetends. Carry the quotient of this division to the next column, and proceed with the rest as with finite decimals.

2. Add 27.56 +5,632 +6.7 + 16.356 +.71 and 6.1234 together.

Ans. 63.1690670868888. 3. Add 2.765+ 7.16674 +3.671 +.7 and .1728 together.

Ans. 14.55436. 4. Add 5.16345 +8.6381 +3.75 together.

Ans. 17.55919120847374090302. 5. Reduce the following numbers to decimals, and find their sum: 3, 4, and to

Ans. .587301.

Section XXXII.
SUBTRACTION OF CIRCULATING DECIMALS.

EXAMPLE. 1. From 87.1645 take 19.479167. OPERATION.

Having made the numbers similar 87.1645 = 87.164545 and conterminous, we subtract as in 19.479162 – 19.479 169 whole numbers, and find the remain

der of the circulate to be 5378, from 67.685377 which we subtract 1, and write the

remainder in its place, and proceed with the other part of the question as in whole numbers. The reason why I should be added to the repetend may be shown as

follows. The minuend may be considered 16% 58, and the subtrahend 78157 ; we then proceed with

these numbers as in Case II. of Subtraction of Vulgar gggg Fractions ; and the numerator 5377 will be the re. 8737 peating decimal. Q. E. D.

RULE. — Make the repetends similar and conterminous, and subtract as usual ; observing, that if the repetend of the subtrahend be greater than the repetend of the minuend, then the remainder on the right must be less by unity than it would be if the expressions were finite.

OPERATIO

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2. From 7. i take 5.02.

Ans. 2.08. 3. From 315.87 take 78.0378. Ans. 237.838072095497. 4. Subtract 7 from g.

Ans. .079365. 5. From 16.1347 take 11.0884.

Ans. 5.0462. 6. From 18.1678 take 3.27.

Ans. 14.8951. 7. From 3.123 take 0.7i.

Ans. 2.40595i. 8. From take it

Ans. .246753. 9. From take .

Ans. .i58730. 10. From , take for. Ans. .1764705882352941. 11. From 5.12345 take 2.3523456.

Ans. 2.7711055821666927777988883599994.

Section XXXIII.

MULTIPLICATION OF CIRCULATING DECIMALS.

1. Multiply .36 by .25. First Method.

In the first method, OPERATION.

we reduce the num.36= =*; .25 = tors.

bers to vulgar frac

tions, and then multiÀ X = = .0929 Answer.

ply and reduce them. 2. Multiply 582.347 by .08. Second Method.

In the second method, OPERATION.

we multiply as in whole 582.347 x .08 = 46.58778 Answer. numbers, but we add two

units to the product ; for 8 X 347=2776=27,16 = 279. Thus we see the repeating number is 778.

RULE. — Turn both the terms into their equivalent vulgar fractions, and find the product of those fractions as usual. Then change the vulgar fraction expressing the product into an equivalent decimal, and it will be the product required. But, if the multiplicand only has a repetend, multiply as in whole numbers, and add to the right-hand place of the product as many units as there are tens in the product of the lefthand place of the repetend. The product will then contain a repetend whose places are equal to those in the multiplicand.

3. Multiply 87.32586 by 4.37. Ans. 381.6140338. 4. Multiply 3.145 by 4.297.

Ans. 13.5 169533. 5. What is the value of 285714 of a guinea ? Ans. As. 6. What is the value of .461607142857 of a ton ?

Ans. 9cwt. Ogr. 261b. 7. What is the value of .284931506 of a year ?

Ans. 104da.

Section XXXIV.

DIVISION OF CIRCULATING DECIMALS.

1. Divide .54 by .15.

OPERATION. .54 = =

Having reduced the num

bers to vulgar fractions, we .15 = it too= = 75.

divide one by the other, and ( * 15 = 4 x =479.

change the quotient to a de270 = 339=3.506493 Ans. cimal. RULE. Change both the divisor and the dividend into their equivalent vulgar fractions, and find their quotient as usual. Change the vulgar fraction expressing the quotient into its equivalent decimal, and it will be the quotient required. 2. Divide 345.8 by ...

Ans. 518.83. 3. Divide 234.6 by .7.

Ans. 301.714285. 4. Divide .36 by .25. Ans. 1.4229249011857707509881.

Section XXXV.
MENTAL OPERATIONS IN FRACTIONS, &c.

If any number be divided into two equal parts, and into two unequal parts, the product of the two unequal parts together with the square of half the difference of the two unequal parts is equal to the square of one of the equal parts. Also,

The product of any two numbers is equal to the square of half their sum, less the square of half their difference. See Euclid's Elements, Book Second, Proposition Fifth.

Note. — A number is said to be squared when it is multiplied by itself; thus, the square of 5 is 5 x 5= 25.

From the above proposition we deduce the following rules. To multiply any number containing a half by itself.

Rule 1. — Multiply the whole number given in the question by the next larger whole number, and to the product add the square of the half

1. Multiply 54 by 5%.

OPERATION. 5 x 6= 30; } x {= ; 30+1=301 Ans. Note. - The whole number given is 5, and the next larger whole number is 6. 2. Multiply 74 by 74.

Ans. 561. 3. Multiply 31 by 31.

Ans. 121. 4. Multiply 91 by 97.

Ans. 901. 5. Multiply 114 by 11t.

Ans. 132] 6. Multiply 207 by 201.

Ans. 4201. 7. Multiply 301 by 307.

Ans. 9301 Note. — The same principle will hold good if we multiply any number by itself whose unit is a 5.

RULE 2. – Take the next least number that ends in a cipher, and multiply it by the next larger number ending in a cipher, and add to the product the square of 5=25, and the result will be the product. 8. Multiply 25 by 25.

Ans. 625. The next less number ending in a cipher is 20, and the next larger is 30; 30 x 20 = 600; 5 x 5=25; 600 +- 25 = 625 Ans. 9. Multiply 35 by 35.

Ans. 1225. 10. Multiply 85 by 85.

Ans. 7225. 11. Multiply 95 by 95.

Ans. 9025.

To find the product of two mixed numbers, whose fractional part is a half, and whose difference is a unit.

Rule 3. — Multiply the larger number without the fraction by itself, and from the product subtract the fractional part multiplied by itself, and the result will be the product. 12. Multiply 64 by 74.

Ans. 481.

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