NOTES.-1. Though the letters used in the above table have been employed the Roman numerals for many centuries, the marks or characters used origi nally in this notation are as follows:

2. The system of Roman Notation is not well adapted to the purposes of numerical calculation; it is principally confined to the numbering of chapters and sections of books, public documents, etc.

11. Five hundred fifty-five.

12. Seven hundred ninety-eight.

13. One thousand three.

14. Twenty thousand eight hundred forty-five.

THE ARABIC NOTATION

48. Employs ten characters or figures to express numbers.

Thus,
Figures,
Names and
values.

· 0 1 2 3 4 5 6 7 8 9 naught one, two, three, four, five, six, seven, eight, nine.

cipher.

or

49. The cipher, or first character, is called naught, because it has no value of its own. It is otherwise termed nothing, and zero. The other nine characters are called significant figures, because each has a value of its own. They are also called digits, a word derived from the Latin term digitus, which signifies finger.

50. The ten Arabic characters are the Alphabet of Arithmetic. Used independently, they can express only the nine numbers that correspond to the names of the nine digits. But when combined according to certain principles, they serve to express all numbers. 51. The notation of all numbers by the ten figures is accomplished by the formation of a series of units of different values, to which the digits may be successively applied. First ten simple units are considered together, and treated as a single superior unit; then, a collection of ten of these new units is taken as a still higher unit; and so on, indefinitely. A regular series of units, in ascending orders, is thus formed, as shown in the following

52. The various orders of units, when expressed by figures, are distinguished from each other by their location, or the place they occupy in a horizontal row of figures. Units of the first order are written at the right hand; units of the second order occupy the second place; units of the third order the third place; and so on, counting from right to left, as shown on the following page :

53. In this notation we observe

1st. That a figure written in the place of any order, expresses as many units of that order as is denoted by the name of the figure used. Thus, 436 expresses 4 units of the 3d order, 3 units of the 2d order, and 6 units of the 1st order.

2d. The cipher, having no value of its own, is used to fill the places of vacant orders, and thus preserve the relative positions of the significant figures. Thus, in 50, the cipher shows the absence of simple units, and at the same time gives to the figure 5 the local value of the second order of units.

54. Since the number expressed by any figure depends upon the place it occupies, it follows that figures have two values, Simple and Local.

55. The Simple Value of a figure is its value when taken alone; thus, 4, 7, 2.

56. The Local Value of a figure is its value when used with another figure or figures in the same number. Thus, in 325, the local value of the 3 is 300, of the 2 is 20, and of the 5 is 5 units. NOTE. When a figure occupies units' place, its simple and local values are

the same.

57. The leading principles upon which the Arabic notation is founded are embraced in the following

GENERAL LAWS.

I. All numbers are expressed by applying the ten figures to dif ferent orders of units.

II. The different orders of units increase from right to left, and decrease from left to right, in a tenfold ratio.

III. Every removal of a figure one place to the left, increases its local value tenfold; and every removal of a figure one place to the right, diminishes its local value tenfold.

58. In numerating, or expressing numbers verbally, the various orders of units have the following names:

59. This method of numerating, or naming, groups the successive orders into periods of three figures each, there being three orders of thousands, three orders of millions, and so on in all higher orders. These periods are commonly separated by commas, as in the following table, which gives the names of the orders periods to the twenty-seventh place.

and

tens

∞ units
hundreds
→ tens

} of septillions.

of sextillions.

∞ hundreds

of quadrillions.

tens

of trillions.

→ units

hundreds
→ units
or tens

of billions.

hundreds
∞ tens

of millions.

O units

O units

9 8,7 6 5,4 3 2,10 9,8 7 6,5 5 6,7 8 9,0 1 2,3 4 5
ninth eighth seventh sixth fifth fourth

third

second first period. period. period. period. period. period. period. period. period.

NOTE. This is the French method of numerating, and is the one in general use in this country. The English numerate by periods of six figures each.

60. The names of the periods are derived from the Latin numerals. The twenty-two given on the following page extend the numeration table to the sixty-sixth place or order, inclusive.

61. From this analysis of the principles of Notation and Numera-ion, we derive the following rules:

RULE FOR NOTATION.

I. Beginning at the left hand, write the figures belonging to the highest period.

II. Write the hundreds, tens, and units of each successive period in their order, placing a cipher wherever an order of units is omitted.

RULE FOR NUMERATION.

I. Separate the number into periods of three figures each, commencing at the right hand.

II. Beginning at the left hand, read each period separately, and give the name to each period, except the last, or period of units. NOTE.-Omit and in reading the orders of units and periods of a number.

EXAMPLES FOR PRACTICE.

Write and read the following numbers:

1 One unit of the 3d order, two of the 2d, five of the 1st. Ans. 125; read, one hundred twenty-five.

2. Two units of the 5th order, four of the 4th, five of the 2d, six of the 1st. Ans. 24056; read, twenty-four thousand fifty-six. 3. Seven units of the 4th order, five of the 3d, three of the 2d, eight of the 1st.