What two methods of division? What is short division? Long division? When is short division employed? Give the rule for short division. The proof. Give the rule for long division. The proof. How do you divide by 10, 100, 1000, etc.? How, when there are ciphers at the right of the divisor?

What is Arithmetic? What is a number? represented? How many figures are there?

How are numbers

How does the value

value of a figure?

of figures increase? What is the simple Name the letters used in Roman notation. Give the rule for numeration. What is the local value of a figure?

What is Addition? What is the sum, or amount? What is the sign of addition? What does it show? What are like numbers? What is the sign of equality? Of what denomination is the sum, or amount? What is an operation? A solution? A conclusion? A rule? A proof?

What is Subtraction? What is the minuend? The subtranend? The remainder? Of what denomination is the remainder? What is the sign of subtraction? What does it show? Give the rule for subtraction.

What is Multiplication? What is the multiplicand? The multiplier? The product? The factors? Give the rule for multiplication. What is the method of proof? What is the square of a number? The cube?

COMPOUND NUMBERS.

49. With reference to their unit values, numbers are divided into two classes, Simple and Compound. 50. A Simple Number is one that denotes things of the same unit value.

51. A Compound Number is one composed of two or more numbers of different unit values.

NOTES.-1. When two numbers have the same unit, they are of the same kind, or denomination: thus, 3 dollars and 5 dollars

are of the same denomination, both dollars; and are simple numbers.

2. When two numbers have different units, they are of different denominations: thus, 3 dollars and 5 cents are of different denominations, dollars and cents; and taken together constitute a compound number.

3. The simplest class of Compound Numbers is Federal Money, because in it we pass from one denomination to another according to the scale of tens.

FEDERAL MONEY.

52. Federal Money is the legal currency of the United States.

The Denominations are mills, cents, dimes, and dollars.

Ten units of each denomination make one of the next higher: hence, the

TABLE.

10 mills, marked m, make 1 cent, marked ct.

53. A Coin is a piece of metal, stamped by authority of government, to be used as money.

The coins of the United States are made of gold, silver, nickel, and copper, and are as follows:

Gold: The double-eagle, eagle, half-eagle, three dollar piece, quarter-eagle, and dollar.

Silver: The dollar, half-dollar, quarter-dollar, dime, half-dime, and three-cent piece.

Nickel: The cent, two-cent, three-cent, and five. cent pieces

Copper:-The copper cent, though still in general use, is no longer coil.ed. The mill is not coined.

NOTATION AND NUMERATION.

54. Since ten units of one denomination make one unit of the next higher, Federal Money is written like simple numbers.

Mills occupy the first place on the right; cents, the second place; dimes, or tens of cents, the third place; dollars, the fourth; eagles, or tens of dollars, the fifth, etc.; illustrated in the margin, and read thus: 12 dollars, 36 cents, and 7 mills.

ILLUSTRATION.

12.367

Dollars are separated from cents by a period. The cents and dimes are read together as cents and tens of cents, and occupy two places at the right of the period. Mills occupy one place at the right of cents. Hence, the

Rule for Numeration.-Read the number at the left of the period as dollars; the first two figures at the right, as cents; and the third, if any, as mills.

55. Rule for Notation.-Write the dollars as in simple numbers, with a period at the right; in the next two places, write the cents, and in the third place, the mills.

NOTE-If the cents are less than ten, put a cipher next to the dollars; if there are no cents, put two ciphers in the place of cents.

EXAMPLES TO BE WRITTEN.

1. Five dollars and twenty-three cents.

2. Ten-dollars, nineteen cents, and five mills.
3. Fifteen dollars and twenty cents.

4. One hundred dollars, fifty cents, and four mills.
5. Nineteen dollars, nine cents, and two mills.

6. Eighty dollars and eight cents.

7. Eleven dollars, one cent, and one mill.

8. Six dollars, six cents, and seven mills.
9. Sixty-three dollars and six mills.

10. Six hundred dollars and ten cents.

11. Nine hundred and seven dollars and one cent. 12. One thousand and nine dollars, thirty cents, and five mills.

13. Twelve hundred and twelve dollars and sixty

cents.

14. Fifteen hundred and one dollars, nine mills. 15. Seventeen hundred dollars, one cent, and one mill.

16. One thousand five hundred and fifty dollars, and five mills.

17. Two thousand and seventy dollars, seventy cents, and seven mills.

18. Three thousand two hundred and one dollars, five cents, and five mills.

19. Four thousand dollars and seven mills.

20. Three hundred and three thousand and sixty dollars, ten cents, and five mills.

21. One million sixty thousand and thirty-five dollars and fifteen cents.

22. Twenty million three thousand and three dollars, thirty cents, and five mills.

23. Two hundred million dollars, sixty-nine cents, and three mills.

24. Nine hundred million twenty-three thousand and six dollars, seventy-five cents, and five mills.

25. Fourteen million nine thousand and three dollars, three cents, and three mills.

REDUCTION.

56. Reduction is changing a number from one denomination to another without altering its value.

It is of two kinds, Reduction Descending and Reduction Ascending.

Reduction Descending is changing a number from a higher denomination to a lower.

Reduction Ascending is changing a number from a lower denomination to a higher.

MENTAL EXERCISES.

EXAMPLE. In 5 cents there are how many mills?

SOLUTION. Since in 1 cent there are 10 mills, in five cents there are 5 times 10 mills, which is 50 mills.

CONCLUSION.-Therefore, in 5 cents there are 50 mills.

1. How many cents in 3 dimes? In 5 dimes? In 7? In 9?