EXAMPLES.

Ex. 1. Let it be required to divide 256434 by 346.

2422

1423

1384

394

Looking at the leading figure of the divisor, and also at that 346)256434(741 quotient. of the dividend, with the view of seeing whether the latter contains the former, which it does not, 3 being greater than 2; we therefore commence with the number 25, formed by the first two figures of the dividend, and seeing that 3 is contained in 25 8 times, we should put 8 for the first quotient figure; but bearing in mind that when the whole divisor is multiplied by this 8 we 346 must attend to the carryings; we perceive that 8 is too great, we therefore try 7, and find 7 times 346 to be 2422, a number less 48 than 2564 above it, so that we can subtract; the remainder is 142, which, when the next figure of the dividend is brought down, becomes 1423. We now take this as a dividend, and looking at the leading figures in this new dividend and the divisor, we see that the latter will go 4 times, we therefore put 4 for the second quotient figure, and multiplying and subtracting we get 39 for the second remainder, and, by bringing down another figure we get 394 for a new dividend; the divisor goes into this once, so that the quotient is 741, and the final remainder 48; this remainder must be annexed with the divisor underneath to the quotient figures, so that the complete quotient is 941, which is the 346th part of 256434. Ex. 2. Divide 108419716214 by 5783.

It must be noticed that if any dividend formed by a remainder and a figure brought down should be less than the divisor, that the divisor will go no times in that dividend; so that a o will be the corresponding quotient figure; and that, then, a second figure must be brought down as in the operation annexed. The steps marked are inserted merely to show the principle. In practice we simply put down the two noughts in the quotient, and go at once to 32214 for the divisor.

28. Whenever the divisor can be separated into two factors, the division may be effected by the following rule:

RULE XI.

1o. Divide by one factor, setting down the quotient and remainder.

2°. Divide the quotient by the other factor, setting down the quotient and remainder; the second quotient thus obtained is the required quotient.

3°. The proper remainder is found by multiplying the second remainder by the first divisor, and to the product adding the first remainder.

This rule may be extended to the case of the divisor being divisible into any number of factors, as follows, always setting down as remainder the product of the partial remainder by all the previous divisors increased by the previous remainder.

Ex. 1. Divide 569736869 by 15.

EXAMPLES.

Here the remainder 2 in the first quotient is 2 units of the upper line; but the remainder 4 in the second line consists of 4 15 units of the second line; and as each unit in the second line is three times as great as each unit in the upper line, the remainder 4 is equal to 3 X 4 units of the upper line, i.e., is equal to 12 ordinary units, hence the whole remainder is 2 + 12, or is 14. Ex. 2. Divide 8327965 by 72 and 99.

To deduce the remainders which would have been left had the divisions been performed by 72 and 99 in the usual way, we may observe that the first partial remainder 4 must be units; but the second dividend being so many collections of 9 units each, the second remainder must be regarded as so many collections of 9 units each; hence the true remainders in these examples are respectively

1 × 9+4=13, and 9 × 9 + 4 = 85.

4 | 2671998

Ex. 3. Divide 2671998 by 192.

Ans.-Quotient = 13916
Remainder = 126

13916....5 × 6 × 4+6=126

29. Division may also be abridged where the divisor is terminated by a cypher or cyphers; we proceed as follows:

RULE XII.

1°. Cut off the cyphers from the divisor, and as many figures from the right hand of the dividend as there are cyphers so cut off at the right hand end of the divisor, then proceed with the remaining figures in the usual manner (Rule X or XI), and if there are anything remaining after the division annex those figures which are cut off from the dividend; otherwise, the figures cut off will be the remainder.

In the first of these examples you mark off with a turned comma the cypher or o in the divisor, and the first figure 6 to the right in the dividend; this is equivalent to dividing both divisor and dividend by 10. You next divide the remaining figures 370419, to the left in the dividend, by the divisor 2, according to Rule IX; thus is obtained the quotient 185209, and remainder 1; to this remainder you annex the figure 6, which was cut off, and you have the complete remainder 16. The quotient may now be correctly represented thus, 18520918.

In the second example you follow the same rule; that is, you cut off two cyphers in the divisor and two figures in the dividend, and obtain the quotient in the usual way, which is 4378, and remainder 31; to this 31 annex the two figures cut off from the dividend, and you have the complete remainder 3101.

30. Verification of Division.-(1.) Multiply the quotient by the divisor, or the divisor by the quotient, and to the product add the remainder, if there be one. The result ought to be the same as the dividend; because we are only adding the divisor the same number of times, as it was subtracted in the operation of division.

(2.) Subtract the remainder, if any, from the dividend, and divide the difference so obtained by the quotient. The result should be equal to the divisor, if the working be correct.

1. Express in figures, ten thousand and four.

2. 29483 +7648 + 32479 + 586 + 298364 + 98765 + 897 + 789 + 5678 + 99.

3. From 6794006897 take 3985160534.

4. Multiply 94785830 by 78060.

5. Divide 5688208152 by 594.

6. Express in figures, one hundred million, one hundred thousand and one hundred.

7. Add together 90473, 9456, 268, 59, 45694, 5437, 87668497, 2837, 9865, 3652, 999, and 8888.

8. Find the difference between 100000000000 and 87649786.

9. Multiply 326904678 by 3060900.

11. Express in figures, one hundred and three million, eighty thousand, two hundred

and seven.

12.

Add together 69074, 6745, 723, 29, 931648, 9005, 76245, 54267, 47096, and 7777. 13. From 78600070000 take 6974208506.

14. Multiply 167409678 by 768900.

15. Divide 60000007006490088805 by 98706543.

16. Express in words and in figures how much greater the value of one 5 is than the other

in the number 658457.

17. Multiply 129847 by 468. If, in the process, you shift all the figures resulting from the multiplication of the multiplicand by 4 two places farther to the left and then add, of what two numbers will the result be the product?

18. What number subtracted from 850967 will leave 3946?

The 365th part of a number is 101001, what is the number?

19. The digits in the units' and millions' places of a number are 4 and 6 respectively, what will be the digits in the same places when 99999 is added to the number. What number must be added to sixty-nine thousand four hundred and twenty-seven to produce three hundred and twenty-five millions, seven thousand and twenty-one ? Find the sum, difference, and product of 12345678 and 288144412.

20.

21.

Find the sum, difference, and product of 1234567 and 4321089.

22. 15996 tons of coal are exported in 43 ships: how many tons does each ship on the average carry?

23. How many years of 365 days each in 46355 days?

24. How often can you subtract 6 from 47112?

25. How many ships, each carrying 673 men, can transport an army of 22882 men ? 26. By what number must you divide 7460020 in order that the quotient may be 52907 and the remainder 133?

27. 2036809 divided by a certain number gives a quotient 2031 with a remainder of 1747 : find the dividing number?

28. A ream of paper contains 20 quires of 24 sheets each; on each page there is room for 34 lines of writing: how many may be written in the ream?

29. What is the number of holes in a sheet of perforated zinc, containing 1519 square inches, if there be 85 in the square inch.

30. What will remain after subtracting 213 as often as possible from 83216?

31. The product of two numbers is 1270374 and half of one of them is 3129: what is the other number?

32. Find the sum, difference, product, and quotient of 1653125 and 13225. 33. Find the sum, difference, product, and quotient of 9765625 and 78125.

DECIMAL FRACTIONS.

31. ARITHMETICAL operations become lengthy and troublesome if they involve many vulgar fractions of different denominations; it becomes necessary, therefore, to devise a method of expressing fractions in such a manner that they may be easily reduced to the same denomination. To effect this all fractions are reduced to others having for denominators 10, 100, 1000, &c. Such fractions are called decimal fractions.

32. Decimals occur so frequently in all computations relating to Nautical Astronomy, that it becomes absolutely necessary to have a knowledge of their application and their relation to Vulgar Fractions.

33. In the Notation of Integers or common numbers, the actual value of each figure depends upon its position with respect to the place of units, its value in any one position being one-tenth of what it would be if it stood one place further to the left; thus the number 1111 denotes one thousand, one

hundred, one ten, and one unit, or 1000 + 100 + 10 + 1, where the second unit beginning with the right hand one is ten times the first, the third is ten times the second, the fourth ten times the third, and so on; or beginning with the first on the left, the second is the tenth part of the first, the third the tenth part of the second, and so on, till we come down to the last unit, which is merely one; or in other words, the figures decrease in a tenfold ratio from left to right.

34. Now we may evidently extend this principle still further, and on the same plan may represent one-tenth of one, one-tenth of this, or one-hundredth of one, one-thousandth of one, and so on, by simply putting some mark of separation between the integers and these fractions. The mark actually used is a dot or full stop, and is called the decimal point, thus 1.* The unit (or 1) next the dot, on the left, is ; the unit one place from this on the left is 10; the next is 100; the next 1000, and so on. In like manner, the unit next the decimal point, on the right, is to, the next ʊ, the next voʊʊ, and so on. In other words, any figure one place to the right of the unit's place will be one-tenth of what it would be if it were in the unit's place, and will thus really denote a decimal fraction; any figure two places to the right of the unit's place will be one-hundreth of what its value would be if it were in the unit's place, and so on for any number of figures, as in the following table, which may be regarded as an extension of the numeration table.

35. This being agreed upon, it follows that a decimal may either be considered as the sum of as many fractions as it contains digits, or as a single fraction; thus:

36. Hence, a decimal is always equivalent to the vulgar fraction whose numerator is the decimal considered as integral, that is, the number itself, when the decimal point is suppressed, and whose denominator is 1 followed by as many cyphers as there are decimal places in it.

37. We generally speak of any figure in a decimal as being in such a place of decimals; thus, for instance, in 3.14159, we should say that the 5 is in the fourth place of decimals, the 9 in the fifth place, and so on, reckoning from left to right.

38. The figures 1, 2, 3, 4, 5, 6, 7, 8, 9, in a decimal are sometimes called significant figures or digits; thus, in such a decimal as '0002345, we should

The decimal point should be put at the top of the line of figures, thus-57, because 5.7 with a stop at the bottom is used in most works to mean 5 X 7 = 35.