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ARITHMETIC.

(1.) The marks used in Arithmetic are

1, 2, 3, 4, 5, 6, 7, 8, 9; which stand for one, two, three, four, five, six, seven, eight, nine. *

These marks are called figures; and by help of these figures, and another mark, 0, to stand for nought, or nothing, any number

may be written down. The mark 0 or nought, may also be called a figure; so that there are ten different marks or figures used in Arithmetic.

(2.) You must remember that a figure is only one of these marks: when you see two or three or more of them written side by side, you see a number: thus, 24 is a number, of two figures; it stands for twenty-four; also 37 is a number of two figures; it stands for thirty-seven: and so on. Twenty-four means two tens and four; thirty-seven means three tens and seven. And in like mạnner forty-eight means four tens and eight; and this number, written in figures, is 48.

You see then, that, in a number of two figures, the first, or left-hand figure, tells us how many tens there are in the number; and the second, or right-hand figure, tells us how many ones, or units there are in the number, besides the tens: one, you are to remember, is also called unit, or unity.

(3.) From what has now been said, you see that the word number does not mean the same thing as the word figure: there are only ten different figures, or single marks, but by joining two or more of these together, we may write down as many numbers as we please. The single figures themselves are also called numbers, as well as sets of two, three, or more figures : thus, 5, 7, 6, &c., are numbers of one figure each ; 57, 75, 76, &c., are numbers of two figures; and 576, 756, &c., are numbers of three figures. The number 57 is fiftyseven, the number 75 is seventy-five, the number 76 is seventy

* Arithmetic may be defined as the science which teaches how to perform computations by numbers. It would perhaps be of but little use to a beginner to give a formal definition of Arithmetic in the text.

B

six, and so on. The number 576 is five hundred and seventysix, the number 756 is seven hundred and fifty six; and in any number of three figures, the first figure on the left hand tells us how many hundreds there are in the number; the next figure tells us how many tens there are, besides the hundreds; and the last figure tells us how many units there are, besides the hundreds and tens: 'there may be no units after the hundreds and tens; if so, a nought or 0, is put for the last figure : thus, five hundred and seventy, would be written in this way, 570; and seven hundred and fifty, would be written 750. Should there be no tens after the hundreds, then, in the same way, a 0 is put in the place of tens: thus, 506, means five hundred, no tens, and six units; that is, five hundred and six; also, 605 means six hundred and five; and 600 means six hundred, without any tens or units besides. You see, therefore, that if you write down a single figure, you mean so many units; but if you put a nought to the right of it, you mean ten times as many, and if two noughts, one hundred times as many.

(4.) When a number has four figures, as, for instance, the number 3562, the first figure on the left hand tells us how many thousands there are in the number; the next figure, how many hundreds besides ; the third figure, how many tens; and the fourth, or last figure, bow many units besides ; so that the number just written is three thousand five hundred and sixty-two: if, instead of five hundreds, there bad been no hundreds, the number would have been written 3062; that is, three thousand and sixty-two.

In like manner, 3502 is three thousand five hundred and tro; also, 3002, is three thousand and two; and 3000, is three thousand only.

(5.) From the explanations which have now been given, you see that when a figure stands by itself, that is, without any figures beside it, it stands for a certain number of units : thus, 6 stands for six units, or six ones; and that when a figure does not stand by itself, but at the right-hand end of a row of figures, it still stands for units: thus, the 6 in the number 346 is still six units; but the 6 in the number 562 stands for ten times 6, or sixty; the 6 in the number 4637 stands for ten times sixty, or six hundred; the 6 in the number 6253 stands for ten times six hundred, or six thousand. It is on this account that we say that the last figure of any number occupies the place of units; the next figure,

on the left, the place of tens; the next, the place of hundreds; the next, the place of thousands; and so on, as in the following Table, which is called the

Numeration Table. 3 7

6 8

3

00 millions.

o tens.

units.

a hundreds.

co tens of millions.

&c. s hundreds of thousands of millions.

co thousands.

tens of thousands of millions.

2 thousands of millions.

a hundreds of millions.

* hundreds of thousands.

s tens of thousands.

SO on.

(6.) The number written above, which is a number of twelve figures, is therefore three hundred and seventy-five thousand two hundred and sixty-eight millions, four hundred and thirty-six thousand, two hundred and ninety-seven. If we were to put another figure before the first figure 3 above, we should have a number of thirteen figures, and the new figure would be in the place of billions ; another new figure put before this would be in the place of tens of billions, and

You should learn this table, by repeating the words, units, tens, hundreds, thousands, &c., till you can say them in order without looking at the book; and you must notice, that whichever of these you pronounce, the next you pronounce is always ten times it: thus, ten is ten times unit; a hundred is ten times ten; a thousand is ten times a hundred, and so on.

(7.) Until you have learned the Numeration Table, you will not know the meaning of a large number written in figures. Suppose the number 465287 were shown to you; you could tell what it means only by knowing the Numeration Table. You would point to the 7 and say units, to the 8 and say tens, to the 2 and say hundreds, to the 5 and say thousands, to the 6 and say tens of thousands, and to the 4 and

say hundreds of thousands ; and you would thus know the number to be four hundred and sixty-five thousand, two hundred and eighty-seven. And in this way you are now required to write in words the following numbers : *.

Exercises in Numeration. 1. 2763 35162

45280 2. 56106

82030

910257 3. 173004 6789523

3486025 4. 1142060 1110111

4362800 5. 64370253

99874062

35006200 6. 73892531

875062035

107926500 7. 7539336210 326972573971 415862314203 8. 730254062810 173004202604. 502130065080

(8.) Write the following numbers in figures :1. Á mile contains one thousand seven hundred and sixty

yards. 2. It has been found that there are more than five hundred

and forty-six thousand persons in the world who are

deaf and dumb. 3. The

expense of building London Bridge was two millions

of pounds. 4. The expense of the Britannia Tube, over the Menai

Straits, was six hundred and twenty-one thousand

eight hundred and sixty-five pounds. 5. The quantity of gold collected at California, in the year

1850, is estimated at three hundred and twelve thou

sand five hundred ounces. 6. The money received by the London and North-western

Railway for passengers and goods, during the first half of the year 1851, was one million one hundred and

seventy-seven thousand five hundred and eighty pounds. 7. The number of visitors to the Great Exhibition, on

Tuesday, the 7th of October, 1851, was one hundred and nine thousand nine hundred and fifteen. [This

was the greatest number on any one day.] 8. The money received by the Great Exhibition amounted

altogether to five hundred and five thousand one hun

dred and seven pounds. 9. In the month of September, 1851, there were one million

one hundred and fifty-five thousand two hundred and

forty visits paid to the Exhibition. * The answers to all the questions and exercises are placed at the end.

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