13. In d, read units' period; thousands' period; millions' period. 14. In d, read billions' period; trillions' period; quadrillions'. 15. In e, read quadrillions' period; trillions' period; billions' period; etc.

16. In ƒ, read each period in succession, beginning at the left. Ex. 13. Units' period, 429, four hundred twenty-nine. 4 in hundreds' order is read four hundred; 2, in tens' order, twenty; 9, in units' order, nine.

Thousands' period, 016, sixteen. O is not read; 1 in tens' order is read with 6, the figure in units' order, sixteen.

Millions' period, 306, three hundred six. 3 in hundreds' order is read three hundred; 0 is not read; 6, in units' order, six.

The process as a whole

17. Read 37000016328.

Ans. 37 billion, 16 thousand, 328. Pointing off into periods of three figures each, 37,000,016,328; numerating to learn the name of the highest period, units, thousands, millions, billions; reading and naming the periods, 37 billion, 16 thousand, 328.

NOTE. - The plural form of the period's name should be used in numerating, but the singular form, in reading. The word "and" should not be used.

NOTATION - ARABIC

In writing an integer there are two steps:

The left-hand period is written with one, two, or three figures. The other periods are then written in succession with three figures each.

Writing the left-hand period

The Arabic notation employs two devices: The number in each order is represented by one of the symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

The name of the order is represented by relative position.

Write as a left-hand period:

Three hundred eight,

3 hundred 0 ten 8, 308

32. Seven; seventy; seven hundred; nine hundred sixty-eight. 33. Twenty; sixteen; thirty-six; forty-five; twelve; five.

34. Ninety; four hundred; five hundred sixty; one hundred one; nineteen; seventy-five.

35. Nine hundred ninety-nine; two; sixteen; eighty; fortyfive; thirteen.

Ex. 32. 7; 70; 700; . .

Writing the other periods

The same devices are employed as before, but each period must contain three figures. See examples 7 and 8.

Write as a full period:

36. Zero; eight; eighteen; ninety; fifty-six; forty.

37. Thirty; one; six; seventy-five; one hundred; three.

38. Five hundred six; nine; seventeen; sixty-eight; four; seven; seventy; twenty.

39. Sixteen; thirty-six; forty-five; five; fourteen; ninety-nine. Ex. 36. 000; 008; 018;. . . .

The process as a whole

40. Write thirty-six trillion, one hundred seventy-six million, six.

Ans. 36,000,176,000,006. We think 36 trillion, and write 36,; we think no billion, and write 000, (the work now appears 36,000,); we think one hundred seventy-six million, and write 176, (36,000,176,); we think no thousand, and write 000, (36,000,176,000,); we think six, and write 006. (36,000,176,000,006.)

Write:

41. Two billion, seventeen thousand, one hundred twenty-six. 42. 307 quadrillion, 20 billion, four hundred seventy-seven. 43. 300 sextillion, 4 trillion, 30 million, 98 thousand, sixty. 44. 65 quadrillion, 700 billion, 99 million, 999 thousand, 999. 45. Nineteen million, 75 thousand, seven hundred twenty-four. 46. Five million, 7 hundred 20 thousand, 6 hundred thirty. 47. 30 quintillion, 300 trillion, 475 million, 4 thousand, 16. 48. 555 sextillion, 505 million, five hundred thousand, 500. 49. 826 quadrillion, 469 billion, 8 million, 95 thousand, 18.

Numeration table

50. Memorize the following table both by orders and by periods.

368, 496, 743, 978, 432, 389, 986, 725.

NUMBERS NAMED-TO A THOUSAND

In naming numbers to a thousand, it is thought best to consider individuals as collected into groups of ten, ten of these groups as collected into a higher group, and ten of these higher groups as collected into a still higher group. The individuals are called units; the groups, orders. Thus:

Ten units are considered a group of ten; ten tens, a group of a hundred; ten hundreds, a group of a thousand.

The individuals forming a group of ten are given independent names, and these names are repeated with the names of the orders to form larger numbers.

Thus:

....

NOTE. One ten, one ten one, one ten two, two ten, two ten one, have been abbreviated to ten, eleven, twelve,. twenty, twenty-one,

....

From the above, it will be seen that numbers to a thousand may be expressed by only ten different symbols, if the number in each order is expressed by a figure, and if the names of orders are expressed by relative position. Thus:

900, 901, 902, 903, 904, 905, 906, 907, 908,. 997, 998, 999.

The Arabic notation affords these devices, and therefore harmonizes with the plan of naming numbers. The oversight of these devices by the Romans and most other nations of antiquity, accounts for their laborious methods of computation.

NUMBERS NAMED - ABOVE A THOUSAND

When the number of individuals is larger than a thousand, it is thought best to consider the individuals as collected into groups of a thousand, a thousand of these groups as collected into a higher group, a thousand of these higher groups as collected into a still higher group, and so on. The individuals are called units; the groups, periods. Thus:

A thousand units are considered a group of a thousand; a thousand thousands, a group of a million; a thousand millions, a group of a billion; and

so on.

The individuals forming a group of a thousand are named as on the preceding page, and these names are repeated with the names of the periods to form larger numbers. Thus:

1 thou. 999, . 1 mil. 999 thou. 999,

1 thousand, 1 thou. 1, . . . . million, 1 mil. 1,

.. 999 thou. 999, 999 mil. 999 thou. 999,

From the above, it will be seen that numbers larger than a thousand may be expressed by only ten different symbols, if the number in each period except the left-hand is written with three figures, and if the names of periods are expressed by relative position. Thus :

1,000; 1,001; 1,002; 1,003; 1,000,000; 1,000,001; 1,000,002;

1,999;

1,999,999

999,999;

999,999,999;

NOTE.-The people of the United States and of France follow the foregoing plan indefinitely; the English, only to a million.

When the number of individuals is larger than a million, the English and most European nations think it best to consider the individuals as collected into groups of a million, a million of these groups as collected into a higher group, a million of these higher groups as collected into a still higher group, and so on. The individuals are called units; the groups, periods. Thus :

A million units are considered a group of a million; a million millions, a group of a billion; a million billions, a group of a trillion; and so on.