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22. Is 52099 a prime number?
23. How many integral divisors has 57660 ?
24. What are the prime factors of 168432 ?

CHAPTER XXIII.

NUMERICAL APPROXIMATIONS.

The student will have already perceived, in circulating decimals, and the extraction of surd roots, that there are many arithmetical operations which never give an exact result. There are also others, which, by a tedious process, would furnish an exact answer, but in which we desire only an approximate value, and we would gladly know what part of our labor may be omitted without affecting the accuracy required. A few of the most important NUMERICAL APPROXIMATIONS will form the subject of the present chapter.

In ADDITION and SUBTRACTION of circulating decimals, it has been recommended to continue the repetends to five or six figures. If we wish to obtain the exact repetend, it will be necessary to change all the given repetends into others, containing as many figures as the least common multiple of the number of places in each repetend.

Add 17.5, 3.1, 419.0875, 1.98563, and 32.1278.

The numbers of repetend figures are 0, 1, 3, 2, and 4; the least common multiple of which is 12. The common repetend must there. fore consist of 12 figures, commencing at the lowest place of the given repetends, which is ten-thousandths. The numbers will be written as follows.

17.500000000000000 Adding the repetends, we find their sum

3.777777777777777 is 2814510279854. Dividing by the 419.087587587587587 rule for division by nines, we obtain a

1.985636363636363 quotient 2814510279856. The repetend 32.127812781278121 is written down, and the 2 carried to the

474.478814510279856 column of thousandths. Repetends, that thus commence and end at the same decimal places, are called similar and conterminuous.

999999

999 999

Subtract 1.742 from 2.937. 2.9379379

The common repetends have 6 figures. From 1.7424242 the right-hand figure of the remainder we subtract

1.1955136 1, because the repetend of the minuend is less than that of the subtrahend. The reason of this subtraction will become evident, if we change the repetends to fractions, and subtract 1.742 42 42 from 2,9 37 937 9

1. Add 17.69, 183, 25.75, 3.276, 194.43, and 649.287. 2. Add 3.219, 63,374, 285.12, 38.4, .037i, and 43.68. 3. Subtract 49.287i from 64. 4. Subtract 215.993i from 1842.2434. 5. Subtract 11.27 from 30.409. 6. Subtract 2856.036 from 3017.62591. 7. Subtract 43.763 from 288.1954.

8. Add 21.3, 28.72, 6.47, 19.345, 201.1593, and 419. 662434.

9. Add 7.83, 24.1, 79.142, 252.4163, and 17.3087. 10. Subtract 4.1956 from 21.28439113.

In MULTIPLICATION, if only a certain degree of accuracy is desired, the product may be obtained by writing the units’ figure of the multiplier under that figure of the multiplicand, whose place we would reserve in the product, and inverting the order of the remaining figures. In multiplying, we commence, for each partial product, with the figure of the multiplicand immediately above the multiplying figure, carrying the tens, which would arise from the multiplication of the two rejected figures at the right.

Required the product of 287.613952 by 15.98421, correct to the fourth decimal place. 287.613952

287.613952 12489.51

15.98421 2876.1395

2817613952 1438.0698

575 227904 258.8525

11504|55808 23.0091

2300911616 1.1505

2588525 568 575

14380697 60 29

287613952 4597.2818

4597.28180769792

The units' figure of the multiplier being placed under the 4th decimal of the multiplicand, and the whole multiplier reversed, the product of each figure by the one above it will be ten-thousandths. Therefore the right-hand figure of each partial product, will fall in the column of ten-thousandths. In the second product, multiplying 52 by 5 we obtain 260, which being nearer 300 than 200, we carry 3 to the product of 9 by 5.

The multiplication has also been performed in the usual way, the vertical line showing the figures that are rejected.

If the multiplicand does not contain enough decimal fig. ures to correspond with the inverted multiplier, the deficiency should be supplied by annexing zeroes. The same contraction may be applied to integers, if we wish only to obtain the thousands, millions, &c., of the product.

11. Required the product of 2869.174381 by 154.49216, true to three places of decimals.

12. Find the product of 176.2428 by 119.43, true to the second decimal place.

13. What is the integral part of the product of 49821.476 by 25.341 ?

14. What is the integral part of the product of 51763. 84926 by 2.4957 ?

15. Multiply 778148.3219 by 954.638, and reserve two decimal places.

16. Multiply 11817.93642 by 2581.36, and reserve two decimal places.

17. Multiply 4435.81977 by 6.9043, and reserve one decimal place.

18. What is the product of 7716.4295 by 19.87436, within .001?

19. Find the product of 63917.48219 by 587.618, within one ten-thousandth.

20. Find the product, true to the 4th decimal place, of 21.87964 by 2.38917.

In Division, a similar contraction may be made when the the divisor is large, which is also applicable in the extraction of roots.

The first quotient figure is of the same numerical value as

the figure of the dividend which stands immediately over the units of the divisor, at the first step of the division.

After the first remainder has been obtained, instead of bringing down the remaining figures of the dividend, we may cut off the right-hand figure of the divisor at each step, as in the following example. 342.15)28417.95255(83.057 342.15)28417.95255(83.057 27372 0

27372 01

[blocks in formation]

In the complete division the contraction is indicated by the vertical line. In each multiplication, the tens arising from the product of the quotient figure by the suppressed figure of the divisor, must always be carried as in contracted multiplication.

The right-hand figure of the quotient thus obtained, can. not always be relied upon. If greater accuracy is desired, the division may be extended further before commencing the contraction,

21. Divide 2704.1583 by 361.8901.
22. Divide 815.3796 by 21.55487.
23. Divide 14.289536 by 128.47.
24. Divide 2.81587 by 2643.
25. Divide 118.78 by 35.759.
26. What is the quotient of 6418 by 249.753 ?
27. What is the quotient of 297 by 36.8569.
28. Divide 2 by 375.814.
29. Divide 13 by .278113.
30. Divide 27 by 65.19428.

In DIVISION OF CIRCULATING DECIMALS, we may adopt the following rule.

Make the repetends of the divisor and dividend similar and conterminous, and from the result, considered as whole numbers, subtract the finite part of each. Perform the division with the remainders as with whole numbers, and the true quotients will be obtained.

Divide 36.9i by 5.273. 5.27327336.91919i

The example is here solved by 5 36

contracted decimal division. The 5273268) 36919155(7.001191

exact fractional quotient is 7701279 36912876

The effect of subtract19316

ing the finite parts of the divisor and 6279

dividend is the same as reducing the 5273

two numbers to improper fractions, 1006

and dividing the numerators. 527

5 273 278

or 7_ 23

479
474

5 5

31. Divide 27.5 by 3,341.
32. Divide 8.97 by 42.815.
33. Divide 39 by 63.4328.
34. Divide 4.02i by 65,378
35. Divide 17.58324 by 29.8.
36. Divide 41.319 by 28.72358
37. Divide 121.623 by 24.0184.
38. Divide 75.814 by 364.25.
39. Divide 4.3091 by 18.615843.
40. Divide 768.432 by 29,57961.
41. Divide 44.3808i by 27.85.

CONTINUED FRACTIONS, arise from the approximate valuation of fractions whose terms are large, and prime to each other. If, for example, we desire approximate values for the

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