11. Forty-seven quadrillions, sixty-nine billions, four hundred and sixty-five thousands, two hundred and seven.

12. Eight hundred quintillions, four hundred and twenty-nine millions, six thousand and nine.

13. Ninety-five sextillions, eighty-nine millions, eighty-nine thousands, three hundred and six.

14. Six quintillions, four hundred and fifty-one billions, sixtyfive millions, forty-seven thousands, and one hundred and four. 15. Write, in figures, nine hundred and ninety-nine billions, sixty-five millions, eight hundred and forty-one thousands, four hundred and eleven.

16. Four hundred and seventy nonillions, forty octillions, four millions, six thousands, two hundred and four.

17. Sixty-five sextillions, eight hundred quadrillions, seven hundred and fifty billions, seven hundred and fifty-one millions, nine hundred and seventy-five thousands, three hundred and

ten.

FORMATION AND NATURE OF NUMBERS.

21. The term, one, may refer to any single thing: it has no reference to kind or quality: it is called an abstract unit.

22. The term, one foot, refers to a single foot, and is called a concrete or denominate unit.

23. An abstract number is one whose unit is abstract: thus, three, four, six, &c., are abstract numbers.

24. A concrete or denominate number, is one whose unit is concrete or denominate thus, three feet, four dollars, five pounds, are denominate numbers.

25. A SIMPLE NUMBER is a single unit, or a single collection of units, either abstract or denominate.

21. Does the term, one, refer to the kind of thing to which it is applied? What is it called?

22. To what does one foot refer? What is it called?

23. What is an abstract number?

24. What is a concrete, or denominate number?

25. What is a simple number?

26. QUANTITY is anything which can be measured by a unit. 27. Numbers which have the same unit are of the same denomination: and numbers having different units are of different denominations. Thus, 4 yards and 6 yards are of the same denomination; but 4 yards and 6 feet are of different denominations.

28. If two or more denominate numbers, having different units, are connected together, forming a single expression, this is called, a compound denominate number. Thus, 3 yards 2 feet and 6 inches, is a compound denominate number.

29. We have seen (Art. 19) that when figures are written by the side of each other, thus,

678904,

the language implies that ten units of any place make one unit of the place next to the left.

30. When figures are written to express English Currency,

the language implies, that four units of the lowest denomination make one of the second; twelve of the second, one of the third; and twenty of the third, one of the fourth.

31. When figures are written to express Avoirdupois weight, thus:

T. cwt.
02. dr.
gr. lb.
27 17 2 24 11 10

the language implies, that 16 units of the lowest denomination

26. What is quantity?

27. When are numbers said to be of the same denomination? When of different denominations?

28. What is a compound denominate number?

29. When several figures are simply written by the side of each other, what does the language imply?

33. In the English Currency, how many units of the lowest denomination make one of the second? How many of the second one of the third ? How many of the third one of the fourth?

31. In the Avoirdupois weight, how many of the lowest make one of the second? How many of the second one of the third?

make one of the second; 16 of the second, one of the third 25 of the third, one of the fourth; 4 of the fourth, one of the fifth; and 20 of the fifth, one of the sixth.

All the other compound denominate numbers are formed on the same principle; and in all of them, we pass from a lower to the next higher denomination by considering how many units of the lower make one unit of the next higher.*

32. A SCALE expresses the relations between the different units of a number. There are two kinds of scales-uniform and varying. In the uniform scale, the number of units which make 1 of the next higher is 10. In English Currency, 4, 12, and 20, make up the varying scale; and 16, 16, 25, 4 and 20, in Avoirdupois weight.

The language of figures determines the unit of each place, and also, the law of change in passing from one place to another. This is called the decimal system, in which the units change according to the scale of tens.

If it be required to express a given number of units, of any * For the Tables of Denominate Numbers, see Appendix, page 383.

32. What is a scale? How many kinds of scales are there? What are they? What is the scale in the common system of numbers? What is the scale in English Currency? What in Avoirdupois weight?

33. If a row of O's be written, what does the language of figures determine? What is such a system called? How does the unit change? How do you express a given number of units of any order?

order, we first select from the arithmetical alphabet the figure which designates the number, and then write it in the place corresponding to the order. Thus, to express three millions, we write

3000000;

and similarly for all numbers.

UNITED STATES MONEY.

34. United States money affords an example of a system of denominate units, increasing according to the scale of tens: thus,

may be read 11 thousand 1 hundred and 11 mills; or, 1111 cents and 1 mill; or, 111 dimes, 1 cent, and 1 mill; or, 11 dollars, 1 dime, 1 cent, and 1 mill; or, 1 eagle, 1 dollar, 1 dime, 1 cent, and 1 mill. Thus, we may read the number with any one of its units as a base, or we may name them all; as 1 eagle, 1 dollar, 1 dime, 1 cent, 1 mill. Generally, in United States money, we read in the denominations of dollars cents and mills; and say, 11 dollars 11 cents and 1 mill.

United States money is denoted by the character, $. The figures expressing dollars are separated from those which denote cents and mills by a comma; thus,

$11,111

is read, 11 dollars 11 cents 1 mill; the figures on the left of the comma always denote dollars; the first two on the right denote cents, and the third, mills.

ALIQUOT PARTS.

One number is said to be an aliquot part of another, when

34. Are the numbers used in United States money abstract or denominate? According to what scale do the units change? How are dollars separated from cents and mills? What is an aliquot part? Name the aliquot parts of a dollar?

it is contained in that other an exact number of times.

Thus; 50 cents, 25 cents, &c., are aliquot parts of a dollar: so also, 2 months, 3 months, 4 months and 6 months are aliquot parts of a year. The parts of a dollar are sometimes expressed fractionally, as in the following

35. If we write the well-known signs of the English currency, and place 1 under each denomination, we shall have

Now, the signs £. s. d. and far. fix the value of the unit 1 in each denomination; and they also determine the relations between the different units. For example, this simple language expresses the following ideas:

1st. That the unit of the right hand place is 1 farthing-of the place next at the left, 1 penny-of the next place, 1 shilling of the next place, 1 pound; and

2d. That 4 units of the lowest denomination make one unit of the next higher; 12 of the second, one of the third; and 20 of the third, one of the fourth. Hence, 4, 12 and 20, make

up the scale.

36. If we take the denominate numbers of Avoirdupois weight, we have

Ton cut. qr. lb. 02. dr.

35. In English currency, is the scale uniform or varying? How does it vary?

36. Name the units of the scale in Avoirdupois weight.