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14. Multiplication and Division by 10, 100, 1000, etc.
(a.) A consideration of the foregoing principles shows that removing the figures of any number one place to the left, or the decimal point one place to the right, multiplies the number by 10; while removing the figures one place to the right, or the point one place to the left, divides the number by 10. A similar change of two places would multiply or divide by 100; a change of three places would multiply or divide by 1000, etc.
ILLUSTRATIONS. — Removing the point in 43.28 one place to the right, gives 432.8 = 10 times 43.28; while removing the point one place to the left, gives 4.328 = .1 of 43.28.
Removing the point in 2.36 three places to the right, leaves one place to be filled with a zero, or gives 2360 = 1000 times 2.36; while removing it three places to the left, leaves two places to be filled with zeroes, or gives .00236 = .001 of 2.36.
(b.) Hence, 1st, To multiply a number by 10, 100, 1000, or any other power* of ten, remove the decimal point as many places to the right as there are zeroes used in writing the given multiplier.
2d. To divide a number by 10, 100, 1000, or any other power of ten, remove the decimal point as many places to the left as there are zeroes used in writing the given divisor.
3d. When by such change any places are left vacant between the number and the point, they should be filled by zeroes. ILLUSTRATIONS.—67 x 100 = 6700.
67 100 = .67 6.7 x 100 = 670.
6.7 • 100 = .067 .67 X 100 = 67.
.67 • 100 = .0067 Note.— These operations are very important. If thoroughly mastered, they will remove in advance most of the difficulties connected with decimal fractions. 1. 36 X 10.
11. 6597 X 1000. 2. 423 X 100.
12. .29 • 100. 3. 62.7 x 10.
13. 67.3 = 10. 4. 539.6 X 100.
14. 42.79 X 10. 5. 247 • 10.
15. .0006 = 10. 6. 3697 • 100.
16. 75000 X 100. 7. 4.28 • 1000.
18. .0039 X 100. 8. 3 • 100.
17. 57.897 X 10. 9. .04 X 100.
19. 87 X 1000000. 10. .7279 X 10.
20. 87 • 1000000.
* For definition of the word power, see 68.
15. Roman Method of Notation. a.) The Roman method of notation represents numbers by letters of the alphabet.
(b.) The letters used are I, V, X, L, C, D, and M.
(c.) The letter I stands for one; V for five; X for ten ; L for fifty; C for one hundred; D for five hundred; and M for one thousand.
(a.) If a letter is repeated, it indicates that the number for which it stands is to be repeated.
ILLUSTRATIONS.—1= 1, II = 2, III = 3, X= 10, XX = 20, XXX = 30. Letters are rarely repeated in this way more than three times.
(e.) If a letter representing a number stands before a letter representing a larger number, the former is to be subtracted from the latter; but if the letter representing the larger number be placed first, the values of the two are to be added together.
ILLUSTRATIONS.-IV = 1 from 5 = 4, XL = 10 from 50 = 40, etc.; while
XVI. = 16.
XVII. = 17.
*XXXVI. = 36.
CCXLIX. = 249.
MDCCCXXXIII. = 1833.
MDCCCLVII. = 1857. (f.) A dash placed over a number makes it represent thousands instead of ones.
ILLUSTRATIONS. — V = 5000; XI = 11 thousands ; MDCCXLIIDCXXI = 1742621.
(a.) The money used in the United States is called UNITED STATES MONEY, or FEDERAL MONEY.
10 mills = 1 cent.
10 dimes = 1 dollar. 10 cents = 1 dime.
10 dollars =1 eagle. (b.) The coins of the United States are — the cent; the THREE-CENT PIECE; the HALF-DIME, worth 5 cents; the DIME, worth 10 cents; the QUARTER-DOLLAR, worth 25 cents; the HALF-DOLLAR, worth 50 cents; the DOLLAR, worth 100 cents; the THREE-DOLLAR PIECE; the EAGLE, worth 10 dollars; the DOUBLE-EAGLE, Worth 20 dollars; the HALF-EAGLE, worth 5 dollars; the QUARTER-EAGLE, worth 21 dollars; and the FIFTY-DOLLAR PIECE.
(c.) These coins are made of gold, silver, copper, and nickel. Those worth more than a dollar are made of gold, and those worth less than a dollar are made of silver, with the exception of the cent, which is either made of copper, or of copper and nickel. The dollar is coined both of gold and of silver, though most frequently of gold.
(d.) The character, $, placed at the left of figures, shows that they represent dollars, or values in United States money.
(e.) Values in Federal money are usually expressed in dollars and cents; or, in dollars, cents, and mills. The dollar is the unit; and, as dimes, cents, and mills are, respectively, tenths, hundredths, and thousandths of a dollar, the figures representing them are written as decimal fractions.
IllustRATIONS.- $4.27 = 4 dollars and 27 cents; or 427 cents = 4 dollars, 2 dimes, and 7 cents.
$93.068 = 93 dollars, 6 cents, and 8 mills; or, 93068 mills. $50.203 = 50 dollars, 20 cents, 3 mills = 50203 mills, etc.
17. English or Sterling Money. (a.) This money is used in England.
(b.) The coin which represents a pound is called a SOVEREIGN. Among the English coins are also the GUINEA, worth 21 shillings, and the crown, worth 5 shillings. The SOVEREIGN usually passes, in the United States, for $4.84, though it varies from $4.83 to $4.86.
(c.) We can form an approximate estimate of values in English money, by regarding the pound as about 5 dollars, the shilling as about 25 cents, and the penny as about 2 cents. An income of £200 per year is very nearly equivalent to an income of $1000 per year; a man who owes' £500 owes not far from $2500, etc. etc.
(a.) The term SHILLING, as used in New England, New York, and some other parts of the Union, does not refer to the English shilling. A New York shilling is 123 cents, while a New England shilling is 163 cents. The NINEPENÇE of New England equals the shilling of New York. These terms, thus used, are very inconvenient; and will, it is hoped, soon be entirely superseded by the decimal currency of the United States.
18. Avoirdupois Weight.
(a.) Iron, flour, sugar, wool, coal, and almost all articles, except gold, silver, and jewels, are weighed by AVOIRDUPOIS Weight.
(b.) Formerly, the quarter was reckoned at 28 pounds, the hundredweight at 112 pounds, and the ton at 2240 pounds; and they are thus reckoned, at the present time, in Great Britain, and at the United States' custom-house. Merchants, however, most frequently buy and sell by the SHORT Ton, or the ton of 2000 pounds.
19. Troy Weight.
(a.) Troy-Weight is used in weighing gold, silver, and precious stones, and also in philosophical experiments. It is made the standard of weights.
20. Apothecaries' Weight. (a.) This weight is used only in compounding and mixing medicines, Avoirdupois Weight being used in buying and selling them. FULL TABLE.
20 gr. = 17. 3 scruples = 1 dram.
37 = 13. 8 drams=1 ounce.
83 = 13. 12 ounces = 1 pound.
123 = lib.