18. Numbers are written by putting each order in its own place, and if any order is not mentioned, naught must occupy its place.

Write the following numbers by means of figures:

1. Six thousand four hundred and twenty-one.

2. One thousand three hundred and five.

3. Two thousand and ninety-six.

4. Eight thousand and one.

5. Four thousand and nine hundred.

6. Sixty-one thousand three hundred and seven.
7. Twenty-three thousand seven hundred and five.
8. Two hundred and fifty-nine thousand.

9. Three hundred, forty-eight thousand and thirteen.

Read the following numbers, and name the number of units in each order:

19. Numbers containing more than three figures are separated into periods of three figures each, beginning at the right. The left-hand period may contain less than three figures.

The first period, counting from the right, is called the period of units, the second is called the period of thousands; the third is the period of millions, and so on, as shown in the following table:

Periods

trillions, billions, millions, thousands, units.

hundreds of

co tens of

units of

hundreds of
tens of
units of

hundreds of

- tens of

units of

co hundreds of

tens of
units of

hundreds of

- tens of

units of

Number .

3 1,

0, 2. 1 0,

5, The number written above is read, 231 trillions, 876 billions, 415 millions, 300 thousands, 210.

The table may be extended at pleasure; the units of the succeeding periods are quadrillions, quintillions, sextillions, etc.

Every period except the left-hand one must be complete, that is, it must contain three digits, but one or all these digits may all be ciphers.

Divide 14674268436173 into periods. Tell how many figures in each period, and read the number.

DEFINITION.

20. A Rule is a brief direction for performing work.

RULE FOR NOTATION.

Begin at the left and write the figures of each period in their proper order, filling all vacant places with ciphers.

RULE FOR

NUMERATION.

I. Begin at the right and separate the number into periods of three figures each, until you reach the left-hand period, which may have one, two, or three figures.

II. Begin at the left and read each period as if it stood alone, naming each period as you read its last figure.

Write, point off, and read the following numbers:

How many periods in the last number? Name the periods in this number. How many figures in each period? Give the rule for notation. Give the rule for numeration. Which period may have less than three figures?

CLASSIFICATION OF NUMBERS.

Count five. In this manner of counting, you mention the numbers without a thought of any object. Numbers used in this way are called abstract numbers.

Count the number of scholars in this class; the number of maps on the wall; the number of books on my desk; the number of panes of glass in the window. Numbers used in connection with the objects counted, indicating the number of objects, are called denominate or concrete.

DEFINITIONS.

21. An Abstract Number is one whose unit is not named; as three, five, seven.

22. A Denominate or Concrete Number is one whose unit is named; as three girls, five pounds seven pennies, eight pencils.

Name five abstract numbers.

Name five denominate numbers. Write four abstract numbers. Write five denominate numbers. What is an abstract number? What is a denominate number

23. An Integer or Integral Number is one which expresses one or more entire things.

24. The Unit of any number is one of the collection which constitutes the number.

25. Similar or Like Numbers are those that have the same kind of unit; as eight days and ten days, two yards seven feet, and six yards eleven feet.

26. Unlike Numbers are those that have different kinds of units; as eight horses and five cows, six pencils and four knives, two feet and three days.

What is a unit? Write a unit. What is a number? Write two numbers. What is arithmetic? What is notation? What is numeration? Name the three methods of notation. The Arabic notation employs how many figures? Write them. What are these figures called tak en separately? What is an abstract number? Give two examples. What is a concrete, or denominate number? Give three examples. What is an integer? What are like numbers? Give two examples. What are unlike numbers? Give two examples. What are the first four orders of units? Write four units of the third order. How many units of any order make one unit of the next higher order? What is the greatest number that can be expressed by one figure? What is the greatest number that can be expressed by two figures? What by three figures? When there are four figures in a number, of what orders is it composed? Give an example and name the order of each figure. What do units of the first order denote? What do units of the second order denote? What do units of the third order denote? What is the simple value of a figure? What is the local value of a figure? Give all the general principles of notation and numeration. Give the rule for notation. Give the rule for numeration. How are numbers expressed in the Roman notation? What are figures? Name all the orders to trillions, beginning with units. Name the first four periods. Why is the second order called tens? hundreds? Why fourth called thousands?

Why third called

How many are 1 and 1? How many are 1 and 1 and 1? How many are 1 and 1 and 1 and 1?