EXAMPLES FOR PRACTICE.

12

1. Change into equivalent decimals, 1, 3, 11, 1, 5, 15, and 1.

32

2. Reduce to decimals, 11, 137, 138, 118, 45, and, carrying interminants to seven decimal places.

49. It becomes important to observe what fractions will produce terminating decimals. Suppose a fraction in its lowest terms; then in reducing it to a decimal we multiply the numerator by 10, 100, 1000, &c. Now the numerator contains no factor common to the denominator, and by this multiplication we introduce the factors 2 and 5 as often as we please and no others. Unless, then, the denominator contains no other factor except twos and fives, this multiplication cannot render the numerator divisible by it. Hence the only fractions which will produce terminating decimals are those whose denominators contain only 2 and 5 as prime factors. All other fractions will produce circulating decimals, though in many cases the period is so long that it would be tedious to find it.

50. Decimals are most frequently used to make calculations on numbers that have been obtained by observations of some kind, by measuring, for instance, or weighing; and it is very seldom indeed that the accuracy of these observations can be relied on to within one five-thousandth part of the unit employed. Now if we cannot rely on the measurement beyond three decimal places, it is needless to carry the result derived from it any farther. In all operations with decimals, then, whether terminating or repeating, we may usually stop at the third or fourth place, and need very rarely go beyond the fifth or sixth. We may, however, attain any degree of exactness that may be required, by carrying the decimal far enough.

With respect to repeating decimals, if perfect accuracy be necessary, they must in most cases be reduced to vulgar fractions before they are added, subtracted, multiplied, or divided. In almost all the applications of decimals, however, an approach to accuracy is sufficient, and this is attained by carrying the decimal only to a moderate number of digits, and omitting the rest. If, in converting a vulgar fraction into a decimal, we stop after the third digit, for instance, adding unity to that digit, if the next be 5 or upwards, it does not differ from its exact value by more than one five-thousandth part of the unit employed. Thus, 172 differs from 172437 by 0004372, which is less than 0005. Similarly, 983 differs from 98276 by 0002317, which is also less than 0005.

51. To reduce any quantity or fraction of one denomination to the decimal of another denomination.

RULE XVIII.

Reduce the number of the lowest denomination to a decimal of the next higher denomination, prefix to this decimal the number of its denomination given in the question, if any, and reduce this also to a decimal of the next higher order, and so on till all the numbers of the given denominations are exhausted, and the decimal of the required denomination has been obtained: the last result will be the answer.

EXAMPLES.

52.

Or, reduce the given quantities to the lowest denominations when there are more than one, and also the integer to which it is referred, to the same denomination; then divide the given quantity by the integer thus reduce.

Ex. I. (Ex. 7 above). The given quantity 700 feet, being all of one denominator requires no further reduction. The integer mile, reduced to the same denomination, is 6080 feet; then 700 divided by 6080 gives o‘115.

Ex. 2. (Ex. 8 above). 8 inches and 3 quarters are 35 quarters, and 1 foot reduced to the same denomination is 48 quarters; then 35 divided by 48 gives o'729.

EXAMPLES FOR PRACTICE.

1. Express as decimals of an hour 17m; 29m; 42m; 25m; 48m; and 58m.

4.

Add together 2.095 hours, '07 days, '05 weeks, and express the same as the decimal of 365.25 days.

5. A nautical mile is 6082.66 feet, and an imperial mile is 5280 feet; express each of these miles as decimals of the other. Also find how near the results are to the decimal values of and 3.

6. A sidereal day is 23h 56m 4509; express this as a decimal of a common day—that is, of 24h and give the result to nine decimal places.

7. Express as decimals of a day the following quantities:-12d 14h 13m 128; 29d 17h 11m 45; 15d 17h 48m 54s; and 119d 5h 19m 158.

8. Express as decimals of a degree the following quantities:—8° 11′ 15′′; 19° 40′ 45′′; 104° 16′ 7′′; and 82° 19′ 30′′.

9.

10.

II.

If 90 degrees correspond to 100 French grades, how many degrees are there in the sum of 4145 degrees and 41°45 grades.

A mètre is 39 37079 English inches, a kilomètre is 1000 mètres; express as decimals of each other a kilomètre and an English mile.

If the length of a degree of latitude is 365000 feet, and a mètre one ten-millionth of 90 degrees: find its length in feet.

12. Express in figures thirty-four and two thousandths, and by it divide 28255662. What alteration must be made in the quotient if the decimal part in the dividend be moved eight places to the left?

13. The sidereal year being 365d 6h 9m 95.6, and the tropical year 365d 5h 48m 4987: reduce their difference to the decimal of a tropical year.

14.

Supposing the velocity of electricity be 288,000 miles per second, and the earth's circumference to be 25,000 miles: calculate to seven places of decimals the time of transmission of an electric telegraph to the antipodes.

15. A French mètre is 39'37 inches nearly: show that a foot is equal to 304 mètres nearly.

To find the value in a lower denomination of any decimal of a higher denomination.

RULE XIX.

Note the number of parts which the unit or integer of the given quantity contains of the next inferior denomination, and multiply the given decimal by this number; the product is the given quantity expressed in that denomination.

2°. If this product has a decimal part, multiply this decimal by the number of parts which the unit of the present denomination contains of the next inferior denomination to that just before employed; this product is the quantity which the given decimal contains of the next denomination.

3°. Proceed (if there still be decimals), in like manner, to the lowest denomination in which the decimal is required to be expressed.

Ans. (in the lowest denomination required) 699-430 feet.

Ex. 2. Find the number of seconds in 0-7 of a minute.

7יס

The next inforior denomination to that of minutes is seconds) x 60

of which the number in a minute is......................

Ex. 3. Find the number of inches and eighths in 0:48 of a foot,

The next inforior denomination to that of feet is inches, of
which the number in a foot is

The next proposed inferior denomination to inches is eighths,
of which the number in an inch is

Ans. 420 seconds.

0'48

X 12

5.76 inches.

Ex. 4. What is the value of ·625 of a cwt. ?

The next inferior denomination to that of a cwt. is qrs., of
which the number in a cwt. is

The next inferior denomination to qrs. is lbs., of which the
number in a quarter is

X 8

6:08 eighths.

.625

X 4

2500 qrs.

X 28

Find the length of a tropical year which contains 365-242218 days.

53. Logarithms are numbers arranged in Tables for the purpose of facilitating arithmetical computations. They are adapted to the natural numbers, 1, 2, 3 . . . in such a manner that by means of them

the operation of Multiplication is changed into that of Addition;

No proof can here be offered that numbers must exist possessing the properties under which we call them logarithms; neither can any account be here given of the methods of computing such logarithms. The reader will accept the statement that if such numbers exist, bearing the properties aforesaid, they are called logarithms. He must also accept the

54. Take any whole numbers, as 18, 813, 6489; the first consists of two, the second of three, and the third of four figures or digits. Again, in the mixed number 739 815, the whole number or integral part (739) consists of three digits.

55. By multiplying a number by itself, one, two, three, &c., times successively, we obtain the second, third, fourth, &c., powers of that number; hence, a power of a number is the number arising from successive multiplication by itself. Thus, 3 x 39 is the square or second power of 3; and 5 × 5 × 5 = 125, the cube or third power of 5; and so on.

These operations are denoted by means of Indices, or small figures placed on the right of the numbers, a little above the line; thus, 22 = 2 X 2 = 4, 33 = 3 × 3 × 3 = 27, and 25 = 2 X 2 X 2 X 2 X 2 = 32, where the Index or exponent denotes the number of factors employed.

56. When there are a series of numbers, such that each is found from the previous one by the addition or subtraction of the same number, they are said to be in arithmetical progression. 1, 3, 5, 7, 9, 11, &c., are in

arithmetical progression.

57. Again, the numbers 3, 6, 12, 24, &c., are in geometrical progression, for each number is formed from the one immediately preceding by multiplying by 2. If we take the following series of powers, 31, 32, 33, 3, 3, &c., we find that the exponents proceed in arithmetical progression, and the quantities themselves in geometrical progression.

58. DEF.-Logarithms are a series of numbers in arithmetical progression answering to another series in geometrical progression, so taken that o in the former corresponds with 1 in the latter.

I

Thus, o, 1, 2, 3, 4, 5, 6, &c., are the logarithms or arithmetical series, and 1, 2, 4, 8, 16, 32, 64, &c., are the numbers or geometrical series, answering thereto-the latter being called the natural number.

Or, 0, 1, 2, 3, 4, 5, the logarithms,

and 1, 5, 25, 125, 425, 5125, the corresponding numbers.

Or, o, I, 2, 3,

4, 5, the logarithms,

and 1, 10, 100, 1000, 10000, 100000, the corresponding numbers. In which it will be seen, that by altering the common ratio of the geometrical series, the same arithmetical series may be made to serve as

tables which are published, recording logarithms for the several numbers to which they profess to belong, though he cannot at present verify the computation of these several logarithms; and he will be informed how he may use these tables to effect with comparative ease many calculations which would otherwise be most laborious.

The truth is, though it requires for its demonstration higher algebra than this work presupposes the reader to be acquainted with, that not only has every number a logarithm, but it has an infinite variety of logarithms, constructed, as the term is, on different scales or bases. The base of any system of logarithms is defined by the fact that in that system unity is its logarithm.

Any number might be used as a base; but in fact there are only two numbers which are ever really used.

The one is an unterminating decimal, 2.7182818

denoted generally by the letter

e. This is the base of what is called the natural or Naperian system; and the advantage of it consists in the ease with which logs. are computed, to this base; but which we cannot here explain.

The other is 10, which is the base in ordinary use, and with this base log. 101. Logarithms to this base are the only ones which will now be considered in their practical use.