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separated by a comma from the three figures which follow it—thus, 4,574 stands for 4 thousands 5 hundreds 7 tens 4 units, or four thousand five hundred and seventy-four.

If the thousands' figure be preceded by two others, they denote tens and hundreds of thousands, just as the two figures before the units' place stand for tens and hundreds of units. Thus, 574,574 stands for five hundred and seventy-four thousand five hundred and seventy-four.

The next group of three figures to the left of the thousands'

group, and separated from it by a comma, is called millions; the next before millions is called billions; the next trillions, and so on, according to the following table, which the pupil should learn by heart :

Numeration Table.
Units
Tens

of units.
Hundreds
Thousands
Tens

of thousands.
Hundreds
Millions
Tens

of millions.
Hundreds
Billions.
Tens

of billions.
Hundreds

&c. &c. From the preceding explanations we deduce the following practical rules :

To write in figures a number expressed in words.

Rule.-—Begin at the left hand, and write the figures which express the hundreds, tens, and units

} } } }

of the highest denomination; place a comma after them, and then the hundreds, tens, and units of the next denomination, and so on, taking care to supply with noughts the places of any denominations of figures which may be wanting.

To read or write in words a number expressed in figures.

Rule.-Divide the figures into groups of three, beginning at the right hand; read each group from the left hand, and add the proper denomination, according to the above table.

Examples. Write in figures seven hundred and fifty-three million four hundred and eight thousand and sixty-nine.

Here millions is the highest denomination, and the number of them being seven hundred and fiftythree, we write 753 (followed by a comma); the next denomination is thousands, and the number of them four hundred and eight, we therefore write 408 (followed by a comma), after the first group ; and the number is completed by writing 069 after the thousands, the o being used to supply the absence of the hundreds figure. Hence the required number is 753,408,069.

Write in words 430,005,080,017.

Beginning at the right hand, the first group is units, the second thousands, the third millions, the fourth billions ; hence the number is four hundred and thirty billion, five million, eighty thousand and seventeen.

Write in figures

L. (1) Fifteen thousand seven hundred and nineteen; 1 eighty-three thousand five hundred and sixty-four; ( forty-seven thousand three hundred

(8) nine

(9)

(2)

(6)

(6)

(8)

and twenty-nine ; ("fifty-six thousand three hundred and seventy ; (5) twenty-one thousand eight hundred and six; iB) nineteenthousand and forty-seven ;

thirty thousand four hundred and six; thousand and sixteen ; seventeen thousand and eight; (10) forty thousand and eighteen'; (11) sixty thousand and nine ; (12) forty-one thousand and ten; (18) ten thousand and one.

M. (1) Three hundred and forty-nine thousand four hundred and seventy-eight; eight hundred and seventeen thousand one hundred and forty ; (8) five hundred and sixty-three thousand seven hundred and eight; seven hundred and four thousand five hundred and sixty ; (5) three hundred and forty thousand two hundred and one ; two hundred and sixty thousand and seventy-four ; three hundred and twenty thousand and eleven ; five hundred and thirty thousand and ten; hundred thousand three hundred and six ; hundred thousand and eight"-four ; (11) three hundred thousand and ten.

N. (1) Three hundred and seventeen million five hundred and sixty-three thousand seven hundred and eighty ;

(2) two hundred and nine million four hundred and ten thousand five hundred and eight; (3) nine hundred and forty million seventeen thousand and sixty; (1) thirty-six million fifty thousand and eight;

(5) two hundred and one million five thousand and forty ; ten million one hundred thousand and one;

(7)

seven billion two hundred and sixty million eight thousand and forty ; (8) three hundred and seven billion forty million five thousand one hundred and sixty ; ) eleven : billion eleven thousand and eleven.

seven

(10) one

(6)

O. (1)

17/ 301008.

(18) ΙοΙΙΙΟ.

Read or write in words 5749.

(13)

IIO001.
(2)
63870.

(14)
5008o.

900019. (3) 403780.

70006o. .

(15)

10301.
(10)
580406.

(16)
301003

IIOOII. (5)

(11) 38074.

200002.

) (8)

(12) 50048.

304010. P. (!) 157386294

(11)

200000109.
310705600.
(3)

(13)
240106304.

40007019.
(1)
207310405.

900009900.
(5)
586004109.

(18)

130103301.

(16) 300708006.

310705006002.

(17) 50103006.

400060006600.

, (8)

(18)
20030056.

I1000040030.
IOIOIOI.

30006000043
()

(20) 800000010011.

(12) 13060005.

(10)

SECTION V.-THE ROMAN NOTATION. In the ancient Roman notation the number one was represented by a single strokel; ten, the second stage in the numeration, was denoted by two strokes X; a hundred, the third stage, by three strokes [; and a thousand, the fourth and last stage to which the Romans first proceeded by direct notation, by four strokes M.

In process of time the halves of these marks furnished characters for intermediate numbers. The half of X (ten) was V (five); the half of C (hundred) was L (fifty) and as the form of M (thousand) became changed into m, its half, namely n, was employed for five hundred.

The numerical strokes were finally displaced by such alphabetical characters as then most resembled them, hence I stood for one; V for five; X for ten; L for fifty ; C (the modern form of the old Roman letter L) for a hundred; D for five hundred; M for a thousand.

These seven letters, combined according to the following rules, serve to express all numbers what

ever.

1. If a letter be repeated, its value is also repeated—thus, II stands for two; III for three; XX for twenty ; CCC for three hundred.

2. If a letter be placed after another of greater value, its value is united to that of the greater—thus, XI stands for eleven; XIII for thirteen ; LX for sixty ; DCC for seven hundred.

3. If a letter be placed before another of greater value, its value is taken from that of the greaterthus, IX stands for nine; XL for forty; XC for ninety ; CD for four hundred; CM for nine hundred.

4. If a letter be placed between two other letters of greater value, its value is taken from the number expressed by the other two-thus, XIV stands for fourteen; XIX for nineteen; CXL for one hundred and forty.

5. A letter with a line drawn over it expresses as many thousands as the letter itself expresses units; thus V stands for five thousand ; X for ten thousand ; C for a hundred thousand.

Examples.—Read or write in Arabic figuresQ. (1) 1.

(6) M. (8) V.

(5) C.

(3) V. (4) L.

(9) C.
(10) M.

(11) X. (121 D

(2) X.

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