BOOK TWO

PART I

NOTATION AND NUMERATION

DEFINITIONS

1. A unit is one.

A number is a unit or a collection of units of the same kind.

The unit of a number is one of the collection that makes the number.

The unit of 10 mills is 1 mill; of 10 cents is 1 cent; of 10 dimes is 1 dime; of 10 dollars is 1 dollar; of 10 eagles is 1 eagle; of 10 dozen is 1 dozen; of 10 pounds is 1 pound; of 10 boys is 1 boy; of 10 cords is 1 cord; of 10 halves is 1 half; of 10 tenths is 1 tenth; of 10 million is 1 million; of 10 is 1.

A unit is abstract when it is not applied to any particular kind of thing; as, 1, 1 ten, 1 hundred, 1 half, 1 eighth.

A unit is concrete when it is applied to some particular kind of thing; as, 1 dollar, 1 dime, 1 egg, 1 half dollar, 1 quarter dollar, 1 regiment, 1 company, 1 cord, 1 pound, 1 ton, 1 mile, 1 rod, 1 foot, 1 inch.

An abstract number is an abstract unit or a collection of abstract units of the same kind.

A concrete number is a concrete unit or a collection of concrete units of the same kind.

12

DEFINITIONS-NOTATION AND NUMERATION

Numbers like 1, 2, 3, 4, 5, 6, 20, 35, 70, 75, 76, etc., are called whole numbers, integers, or integral numbers.

Numeration is the art of reading numbers expressed by symbols.

Notation is the art of writing numbers by means of symbols. In arithmetic, the Arabic or decimal system of notation is used. In this system ten figures are used in writing numbers. These are:

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

The first nine of these figures are called significant figures because they signify or express value. They are also called digits. The last figure is called zero, cipher, or naught, and expresses no value. It is used with the other figures in expressing some numbers.

Another system of notation, called the Roman system, is sometimes used in numbering chapters of books, headings, and in a few other cases. It is explained on page 17.

NUMERATION

Study Recitation

1. Read the numbers written at the left.

3. Beginning with 1, what is the value of each number as compared with the number following it?

2.

1

10

100

1000

10000

100000

1000000

1111111

4. Beginning with 1,000,000, what is the value of each number as compared with the one before it?

5. In the sum of the several numbers, where is the significant figure in each of the numbers added to be found?

6. In the number 1,111,111, give the number represented by each figure.

In any whole number, the places occupied by the figures are called orders; the place at the right is of the first order, the second place is of the second order, the third place is of the third order, and so on to the left.

7. In 1,111,111, what is the value of the unit of the first order? Of each succeeding order? In 2,222,222, what is the value of the figure in each order?

8. The value of a unit in any order bears what relation to its value in the next higher order? To its value in the next lower order?

9. In whole numbers the first three orders constitute a period, and each three succeeding orders to the left constitute another period.

10. Beginning with the units, give the names of the orders in each period; as, units, tens, hundreds, thousands, etc. 11. Beginning at the right, give the names of the periods; as, units, thousands, millions, billions, trillions.

12. What is the name of the 1st period? The 3d? The 2d? The 5th? The 4th?