changing the place to which we refer the point. It is the more cbjeo tionable, as it tends to convey the false idea that the zero is essential to the multiplication. As a matter of fact, annexing any figure whatever to a number will, if the decimal point is omitted, multiply it by 10, for it will change the place to which the decimal point is referred; but if the annexed figure is other than zero, its value will be added to the product of the number by 10. Thus, annexing 7 to 43 gives 437, which

equals 10 times 43, plus 7. The principle involved is this: that every change which is made in the position of figures with reference to the decimal point, whether it is made by changing the position of the point, or by writing other figures between the given figures and the point, alters the value they represent, by multiplying or dividing them by 10 or some power of ten. A figure can only alter the value expressed by other figures when it is written between them and the point.

(e.) How will you express in figures the results of the following indicated operations?

(f.) Let the student now tell by inspection, without changing the place of the point or re-writing the numbers, the result of the above indicated operations.

*The student should remember that when the decimal point is not marked, it is always understood to belong at the right of the given figures.

Thus eighty-seven multiplied by ten equals eight hundred and seventy ; eighty-seven hundredths multiplied by ten equals eighty-seven tenths, or eight and seven tenths, &c. He should learn to do this without the slightest hesitation.

(a.) The value of a number may be expressed in terms of any other decimal denomination as well as in units, by making the requisite change in the place of the point. Thus :

847

34.7 tens, = 8.47 hundreds, = .847 of a thousand, = .0847 of a ten-thousand, &c.

8478470 tenths, =84700 hundredths, =847000 thousandths, &c. 642.06 = 6.4206 hundreds, : 64206 hundredths, &c.

(b.) Express the value of each of the following in tenths, then in tens; in hundredths, then in hundreds; in millionths,

26. French Method of Numeration.

The foregoing method of numeration is called the French method. It divides the figures expressing a number into periods of three figures each, making a unit of one period equal to one thousand units of the next lower period.

Thus one million equals one thousand thousands; one billion equals one thousand millions; one trillion equals one thousand billions, &c.

27. English Method of Numeration.

(a.) There is another method, called the English method, which divides the figures expressing a number into periods of six figures each, making a unit of one period equal to one million units of the next lower period.

(b.) By this method one billion equals one million millions; one trillion equals one million billions, &c. This is illustrated in the following example:

(c.) In this method, as in the French, we read the figures in each period as though they stood alone, calling afterwards the name of the period.

(d.) The above number would be read, 9 quadrillions, 752730 trillions, 665897 billions, 253060 millions, 934578.

28. Comparison of French and English Methods.

(a.) It will be readily seen that while the English periods bear the same name as the French, and while one thousand and one million represent the same number in the two systems, one billion, one trillion, or a unit of any higher denomination is much greater in the English system than in the French. Thus, an English billion equals a French trillion; an English trillion equals a French quintillion.

(b.) The French method is the one generally used in this country and on the continent of Europe, and being much more convenient than the English, has been adopted in part in England, and is likely to come into general use there.

29. Numbers to be read according to the English Method.

30. The Roman Method.

(a.) The Roman method of notation represents numbers by letters of the alphabet.

(b.) It is now chiefly used in numbering the sections or chapters of a book, the pages of a preface or introduction, the year of the Christian era, or when it is necessary to distinguish one class of numbers from another.

(c.) The letters used are the following, viz.:
:-

I, which stands for One.

V, which stands for Five.

X, which stands for Ten.
L, which stands for Fifty.

C, which stands for One Hundred.
D, which stands for Five Hundred.
M, which stands for One Thousand.

(d.) Other numbers are represented by repetitions and combinations of these letters.

(e.) When a letter is repeated, it indicates that the number it represents is to be repeated.

Thus: II. two; III. = three; XX,= twenty; XXX.= thirty, &c.

(f.) If a letter expressing one number be placed before a letter expressing a larger number, the former is to be subtracted from the latter; but if the letter expressing the larger value be placed first, the values of the two are to be added together.

=

=

Thus: IV. four; IX. nine; XL. forty, &c.; while VI. six; XI. eleven; LX. = sixty, &c.

=

(g.) In the following columns, the letters stand for the numbers written against them:

I.. One.

Two.

II..
III.. Three.

IV.. Four.
V.. Five.

VI.. Six.

VII.. Seven.

VIII. . Eight.

IX.. Nine.

X.. Ten.

XI.. Eleven.

XII.. Twelve.

(h.) A dash placed over a letter makes it express thousands

instead of ones. Thus, V. 5000; VI.

(i.) Read the following numbers:

XCVIII.

DCCXLII.

MDCCLXXVI.

XXDCCCXCIX.

CCXLVIIICCCL.

DCLIDLXXI.

DCCCXCIXCCCXXXIII.