The numbers from one to nine, inclusive, compose the first order of units. Ten is considered the first unit of the second order; and as in the first, so also in the second order, the units are reckoned from one to nine. Thus the number formed by

the combination of ten with ten is named twenty (two tens).

Between ten and twenty there are nine numbers, of which the names of two, viz., eleven and twelve, are irregular. The names of the other seven are formed by combining the names of the simple numbers from three to nine, inclusive, with ten. Thus the number formed by

the combination of ten with one is named eleven (ten and one left). twelve (ten and two left).

eleven with one

The names of the remaining seven are thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen.

The names of the nine numbers between twenty and thirty are formed by annexing to the word twenty the names of the simple units from one to nine inclusive. The names are twentyone, twenty-two, twenty-three, twenty-four, twenty-five, twenty-six, twenty-seven, twenty-eight, twenty-nine.

The names of the numbers intermediate between thirty and forty, forty and fifty,. . . . ninety and ninety-nine, are similarly formed by annexing in succession the words one, two, ... to the word expressing the number of tens.

The name of the number formed by the combination of ninety-nine with one, that is, of nine tens with one ten, is hundred.

.....

.....

As tens are made units of the second order, and reckoned from one to nine, so hundreds are made units of the third order, and reckoned from one to nine. The names are one hundred, two hundred, three hundred, . . . . . nine hundred. The names of the ninety-nine numbers between one hundred and two hundred, two hundred and three hundred, . . . . . nine hundred and nine hundred and ninety-nine, are formed by annexing to the hundreds the names of the consecutive numbers from one to ninety-nine. Thus, one hundred and one, one hundred and two, . . . one hundred and ten, one hundred and eleven,... one hundred and ninety-nine, . . . nine hundred and ninety-nine.

The number formed by the combination of nine hundred and ninety-nine with one, or of nine hundred with one hundred, is named thousand.

Thousands are made units of the fourth order, and reckoned from one to nine, thus: one thousand, two thousand, . . . nine thousand.

...

As the assemblage of ten hundred or one thousand has been made an unit of the fourth order, so that of ten thousand is made an unit of the fifth order, and the assemblage of ten tens of thousands an unit of the sixth order.

A single word, however, is not employed to signify either ten thousand or ten times ten thousand, but the names of the consecutive thousands from ten thousand to nine hundred and ninety-nine thousand are formed by placing successively the names in the series beginning with eleven and ending with nine hundred and ninety-nine before the word thousand; thus, eleven thousand, . . . . . ninety-nine thousand, one hundred thousand, one hundred and one thousand.

The names of the nine hundred and ninety-nine numbers between one thousand and two thousand, two thousand and three thousand, . . . . . nine hundred and ninety-nine thousand, and ten hundred thousand, are formed by writing the names from one to nine hundred and ninety-nine after each of the successive thousands, thus :-One thousand and one, one thousand and two, . . . one thousand and ninety-nine, one thousand one hundred, one thousand one hundred and one, . . . one thousand nine hundred and ninety-nine, thousand nine hundred and ninety-nine, .

.....

ninety-nine

nine hundred and

ninety-nine thousand nine hundred and ninety-nine, ten hundred thousand, or one thousand thousand.

The number ten hundred thousand is named million.

The number one million is made an unit of the seventh order.

The number ten hundred thousand millions, or one million millions, is, by British writers*, named billion; one million billions, trillion; one million trillions, quadrillion, &c.

The number ten billions is made an unit of the fourteenth order.
one hundred billions

and so on.

fifteenth

3. This nomenclature is simple and comprehensive, but not convenient for the effecting of combinations among different numbers.

An abridged notation is founded on the following considerations:-One, ten, hundred, thousand, . . . . are the names of the successive orders of units, and the terms one, two, three, ...nine, express the repetitions of each sort of unit in any given number. Nine distinct symbols or figures are employed to represent the nine simple units, and the convention is made that every figure placed to the left of another shall be held to express units of the order immediately superior to those of that other figure.

The symbols employed are 1 (one), 2 (two), 3 (three), 4 (four), 5 (five), 6 (six), 7 (seven), 8 (eight), 9 (nine). Besides these it is necessary that there be a figure without value to occupy the place of the units of any order which may be wanting. The character adopted is 0 (nothing, or zero).

In contradistinction to this character, 1, 2, 3, . . . . 9, are called significant figures. These have two values: the one belonging to the figure itself, the other derived from the place

* By foreign writers, the number ten hundred millions is named billion; ten hundred billions, trillion; ten hundred trillions. quadrillion, &c.

'which it occupies. That is termed the absolute, and this the relative, value of the figure.

4. As illustrations of the use of these characters, let it be required 1. to express in figures the number seven hundred and sixty-nine.

This number is composed of nine (9) units, six (6) tens, and seven (7) hundreds; and is, therefore, in accordance with the conventions already made, to be written 769.

2. Let it be required to express in figures the number five thousand eight hundred and forty-three.

This number, which is composed of 3 units, 4 tens, 8 hundreds, and 5 thousands, when translated into figures, stands thus, 5843.

3. Let the number four hundred and thirty be taken as a third example.

Four hundred and thirty, being composed of 0 units, 3 tens, and 4 hundreds, is written, 430.

5. The ten figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, (sometimes also called cyphers), are sufficient to express every whole number. For any such number is composed of ones, or simple units, tens, or units of the second order, hundreds, or units of the third order, &c. The order of each sort of unit is denoted by its place; the number of the units of each order is expressed by one of the figures 1, 2, 3, . . . . . 9; and the places of deficient units of any order are occupied by the character 0.

.....

If the names of the successive orders of units are arranged

as follows:

Order of units, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1,

Names.

hundred thousand billions.
trillions.

ten thousand billions.
thousand billions.
hundred billions.
ten billions.

billions.

hundred thousand millions.
ten thousand millions.
thousand millions.
hundred millions.
ten millions.

hundred thousands.
ten thousands.
thousands.
chundreds.

→ millions.

tens.

- units.

it is found that there are six orders of units, or six places of figures between simple units and units of millions; six places between units of millions and units of billions, &c. Whence, if a number is divided into periods of six figures, beginning at the lowest figure, the first period is composed of hundred thousands, ten thousands, thousands, hundreds, tens, and units of simple units; the second of hundred thousands, ten thousands, thousands, hundreds, tens and units of millions; the third of hundred thousands, ten thousands, thousands, hundreds, tens and units of billions, &c.

6. It is required to express the following numbers in figures

:

1st. Twenty-five. 2nd. Fifty-two.

3rd. Two hundred and fifty.

4th. Two hundred and fifty-five. 5th. Two thousand five hundred. 6th. Two thousand and fifty.

7th. Two thousand and fifty-five.

8th. Two thousand five hundred and fifty-five.

9th. One thousand eight hundred and fifty-three.

10th. Seventeen thousand.

11th. Seventeen thousand and seventeen.

12th. Seventeen thousand and seventy.

13th, Fifty-eight thousand seven hundred and forty-six. 14th. Fifty thousand seven hundred and forty.

15th. Fifty-eight thousand and forty-six.

16th, One hundred and ninety thousand nine hundred.

17th. One hundred and ninety thousand and nine.

18th. Eight hundred and six thousand three hundred and seven.

19th. Three millions seven hundred and fifty-six thousand four hundred

and eighty-nine.

20th. Twelve millions seven thousand and sixteen.

21st. Five billions four hundred and seventy-two thousand six hundred and eighty-seven millions nine hundred and thirty-four thousand five hundred and twenty-one.

22nd. Eighty-seven trillions five hundred and eighty-four thousand billions three thousand and seventy millions sixty thousand and sixteen.

6 a. It is required to express the following numbers in words:

1st. 56; 65; 50; 60.

2nd. 100; 101; 110; 111.

3rd. 432; 423; 342; 324; 243; 234.

4th. 18; 180; 1800; 1860; 1086.