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EXAMPLES.

210 be

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

(2) (2) (2)

??=,+= ?=?=3=1; therefore, because the de3 2)16

210

nominator vanishes in dividing, the decimal is finite, and consists of four places; thus, 16)3.0000.

.1875°

be

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

(2)

(2)

(2)

28001122)112=56=28=14=7. Thus, 7999999 therefore, because the denominator, 112, did not vanish in dividing 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 4. Let

be the fraction proposed.

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.

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3.5555555
7.6516516

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1.7657657

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=

Having made all the numbers similar and conterminous by Sect. XXX., Case III., we add the first six columns, as in 6.1737373 Simple Addition, and find the sum to = 3.591227.

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51.7 51.7777777 be 3591224 = 3591224 3.7 = 3.7000000 The repeating decimals .591227 we 27.63127.6316316 write in their proper place, and carry 3 1.003 1.0030030 to the next column, and then proceed

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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 27.565.632+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

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OPERATION.

87.1645 19.479167

= 87.164545
19.479167

=

67.685377

Having made the numbers similar and conterminous, we subtract as in whole numbers, and find the remainder of the circulate to be 5378, from 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 1 should be added to the repetend may be shown as follows. The minuend may be considered 165, and the subtrahend 78187; we then proceed with these numbers as in Case II. of Subtraction of Vulgar

OPERATION.

164548 78167

9

Fractions; and the numerator 5377 will be the re88333 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.

2. From 7.1 take 5.02.

Ans. 2.08.

3. From 315.87 take 78.0378. Ans. 237.838072095497.

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RULE.

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999

999.

In the first method, we reduce the numbers to vulgar fractions, and then multiply and reduce them.

In the second method, we multiply as in whole numbers, but we add two units to the product; for Thus we see the repeating

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. 4. Multiply 3.145 by 4.297.

Ans. 381.6140338.
Ans. 13.5169533.
Ans. 8s.

5. What is the value of .285714 of a guinea ?
6. What is the value of .461607142857 of a ton?

Ans. 9cwt. Oqr. 26lb.

7. What is the value of .284931506 of a year?

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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 .6.

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.

=

1. Multiply 5 by 51.

NOTE. ber is 6.

OPERATION.

5 × 6 = 30; × 1 = 1; 30+1 = 301 Ans.

-The whole number given is 5, and the next larger whole num

2. Multiply 7 by 74.
3. Multiply 31 by 31.
4. Multiply 94 by 91.
5. Multiply 11 by 114.
6. Multiply 201 by 204.
7. Multiply 301 by 301.

NOTE.

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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 × 20 = 600; 5 × 5=25; 600 +- 25 = 625 Ans.

9. Multiply 35 by 35.

10. Multiply 85 by 85. 11. Multiply 95 by 95.

Ans. 1225.

Ans. 7225.

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 6 by 74.

Ans. 481.

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