Each of the words, one, two, three, four, five, six, &c., denotes how many things are taken in the collection.

ΝΟΤΑΤΙΟΝ.

19. NOTATION is the method of expressing numbers either by letters or figures. The method by letters, is called Roman Notation; the method by figures is called Arabic Notation.

ROMAN NOTATION.

20. In the Roman Notation, seven capital letters are used, viz: I, stands for one; V, for five; X, for ten; L, for fifty; C, for one hundred; D, for five hundred; and M, for one thousand. All other numbers are expressed by combining the letters according to the following

NOTE. The principles of this Notation are these:

1. Every time a letter is repeated, the number which it denotes is also repeated.

2. If a letter denoting a less number is written on the right of one denoting a greater, their sum will be the number expressed.

3. If a letter denoting a less number is written on the left of one denoting a greater, their difference will be the number expressed.

EXAMPLES IN ROMAN NOTATION.

Express the following numbers by letters:

1. Eleven.

2. Fifteen.
3. Nineteen.
4. Twenty-nine.

5. Thirty-five.
6. Forty-seven.
7. Ninety-nine.

8. One hundred and sixty.

9. Four hundred and forty-one.

10. Five hundred and sixty-nine.

11. One thousand one hundred and six. 12. Two thousand and twenty-five.

13. Six hundred and ninety-nine.

14. One thousand nine hundred and twenty-five.
15. Two thousand six hundred and eighty.
16. Four thousand nine hundred and sixty-five.
17. Two thousand seven hundred and ninety-one.
18. One thousand nine hundred and sixteen.
19. Two thousand six hundred and forty-one.
20. One thousand three hundred and forty-two.

19. What is Notation? What is the method by letters called? What is the method by figures called?

20. How many letters are used in the Roman notation? Which are they? What does cach stand for?

NOTE.-What takes place when a letter is repeated? If a letter denoting a less number be placed on the right of one denoting a greater, how are they read? If the letter denoting the less number be written on the left, how are they read?

21. What is Arabic Notation? How many figures are used? What do they form? Name the figures. How many things does 1 express? How many things does 2 cxpress? How many units in 3? In 4? In 6? In 9? In 8? What does 0 express? What are the other figures called?

ARABIC NOTATION.

21. Arabic Notation is the method of expressing numbers by figures. Ten figures are used, and they form the alphabet of the Arabic Notation.

They are 0 called zero, cipher, or Naught.

1 expresses a single thing, or the unit of a number.

2

193488

5

6

7

two things
three things

four things

five things
six things
seven things
eight things

nine things

or two units. or three units. or four units. or five units. or six units.

or seven units. or eight units.

or nine units.

The cipher, 0, is used to denote the absence of a thing: Thus, to express that there are no apples in a basket, we write the number of apples is 0. The nine other figures are called significant figures, or Digits.

22. We have no single figure for the number ten. We therefore combine the figures already known.

writing 0 on the right hand of 1, thus:

This we do by

10, which is read ten.

This 10 is equal to ten of the units expressed by 1. however, but a single ten, and may be regarded as a unit, the value of which is ten times as great as the unit 1. called a unit of the second order.

It is,

It is

22. Have we a separate character for ten? How do we express ten? To how many units 1 is ten equal? May we consider it a single unit?

Of what order?

23. When two figures are written by the side of each other, the one on the right is in the place of units, and the other in the place of tens, or of units of the second order. Each unit of the second order is equal to ten units of the first order. When units simply are named, units of the first order are always meant.

Two tens, or two units of the second order, are written
Three tens, or three units of the second order, are written
Four tens, or four units of the second order, are written
Five tens, or five units of the second order, are written
Six tens, or six units of the second order, are written
Seven tens, or seven units of the second order, are written
Eight tens, or eight units of the second order, are written
Nine tens, or nine units of the second order, are written

23

20

30

40

50

60

70

80

90

These figures are read, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety.

The intermediate numbers between 10 and 20, between 20 and 30, &c., may be readily expressed by considering their tens and units. For example, the number twelve is made up of one ten and two units. It must therefore be written by setting 1 in the place of tens, and 2 in the place of units: thus,

Eighteen has 1 ten and 8 units, and is written

Twenty-five has 2 tens and 5 units, and is written
Thirty-seven has 3 tens and 7 units, and is written
Fifty-four has 5 tens and 4 units, and is written

12

18

25

37

54

Hence, any number greater than nine, and less than one hundred, may be expressed by two figures.

24. In order to express ten units of the second order, or one hundred, we form a new combination.

It is done thus,

100

by writing two ciphers on the right of 1. This number is read, one hundred.

23. When two figures are written by the side of each other, what is the place on the right called? The place on the left? When units simply are named, what units are meant? How many units of the second order in 20? In 30? In 40? In 50? In 60? In 70? In 80? In 90? Of what is the number 12 made up? Also 18, 25, 37, 54? What numbers may be exprsesed by two figures?

Now this one hundred expresses 10 units of the second order, or 100 units of the first order. The one hundred is but an individual hundred, and, in this light, may be regarded as a unit of the third order.

We can now express any number less than one thousand.

For example, in the number three hundred and seventy-five, there are 5 units, 7 tens, and 3 hundreds. Write, therefore, 5 units of the first order, 7 units of the second order, and 3 of the third; and read from the right, units, tens, hundreds.

In the number eight hundred and ninety-nine, there are 9 units of the first order, 9 of the second, and 8 of the third; and is read, units, tens, hundreds.

In the number four hundred and six, there are 6 units of the first order, 0 of the second, and 4 of the third.

Otens.

ohuns.

The right hand figure always expresses units of the first order; the second, units of the second order; and the third, units of the third order.

25. To express ten units of the third order, or one thousand, we form a new combination by writing three ciphers on the right of 1; thus,

1000

Now, this is but one single thousand, and may be regarded as a unit of the fourth order.

1,

10,

100,

Thus, we may form as many orders of units as we please: a single unit of the first order is expressed by a unit of the second order by 1 and 0; thus, a unit of the third order by 1 and two 0's; a unit of the fourth order by 1 and three 0's; a unit of the fifth order by 1 and four O's; and so on, for units of higher orders:

1000, 10000;

24. How do you write one hundred? To how many units of the second order is it equal? To how many of the first order? May it be considered a single unit? Of what order is it? How many units of the third order in 200? In 300? In 400? In 500 ? In 600? Of what is the number 375 composed? The number 899? The number 406 ? What numbers may be expressed by three figures? What order of units will each figure express?