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4. The art of expressing numbers by letters or figures, is called NOTATION. There are two methods of notation in use,
the Roman and the Arabic. 5. The Roman method employs seven capital letters ; viz:I, V, X, L,C,D, M. When standing alone, the letter I denotes one ; V, five; X, ten ; L, fifty ; C, one hundred ; D, five hundred ; M, one thousand. To express the intervening numbers from one to a thousand, or any number larger than a thousand, we resort to repetitions and various combinations of these letters. The method of doing this will be easily learned from the following
TABLE. I denotes one.
XXX denote thirty. II two.
forty. IIL three.
sixty. V five.
LXX seventy VI six.
LXXX eighty. VII seven.
ninety. VIII eight.
one hundred. IX nine.
one hundred and one X ten.
one hundred and ten XI eleven.
two hundred. XII twelve.
three hundred. XIII thirteen.
CCCC four hundred. XIV fourteen.
five hundred. XV fifteen.
six hundred. XVI sixteen.
DCC seven hundred. XVII seventeen.
eight hundred. XVIII eighteen.
DCCCC - nine hundred. XIX nineteen.
one thousand. XX twenty.
two thousand. XXI twenty-one.
MDCCCXLV, one thousand eight XXII twenty-two, &c.
hundred and forty-five.
Quest.-4. What is notation? How many methods are there in use? What are they? 5. What does the Roman method employ? What does each of these letters denote when standing alone? How are the intervening numbers from one to a thousand expressed? How denote Two ? Four! Six? Eight? Nine? Fourteen? Sixteen? Nineteen? Twentyfour ? Twenty-eight? What does XL denote ? LX ? XC? CX ?
N.B. Questions on this table should be varied, and be continued by the teacher till the class becomes perfectly familiar with it.
Obs. 1. The learner will perceive from the table above, that every time a letter is repeated, its value is repeated. Thus, I, standing alone, denotes one ; II, two ones, or two, &c. So X denotes ten; XX, twenty, &c.
2. When two letters of different value are joined together, if the less is placed before the greater, the value of the greater is diminished; if placed after the greater, the value of the greater is increased. Thus V denotes five; but IV denotes only four; and VI, six. So X denotes ten; IX, nine; XI, eleven, &c.
3. A line or bar (-) placed over a letter, increases its value a thousand times. Thus V denotes five, ï denotes five thousand ; X, ten; X, ten thousand, &c.
4. This method of expressing numbers was invented by the Romans; hence it is called the Roman Notation. It is now seldom used, except to denote chapters, sections, and other divisions of books and discourses.
6. The common method of expressing numbers is by the Arabic Notation. The Arabic method employs the following characters or figures, viz :
1 2 3 4 5 6 7 8 9 0 one, two, three, four, five, six, seven, eight, nine, zero.
The first nine are called significant figures, because each one always has a value, or denotes some number. They are also called digits, from the Latin word digitus, which signifies a finger.
The last one is called a cipher, or nought, because when standing alone it has no value, or signifies nothing.
Obs. It must not be inferred, however ,that the cipher is useless ; for when placed on the right of any of the significant figures, it increases th value. It may therefore be regarded as an auxiliary digit, whose office, it will be seen hereafter, is as important as that of any other figure in the system.
Note.—The pupil must be able to distinguish and to write these characters, before he can make any progress in Arithmetic.
7. It will be seen that nine is the greatest number that
QUEST.-Obs. What is the effect of repeating a letter? If a letter is placed before another of greater value, what is the effect? If placed after, what? When a letter has a line placed over it, how is its value affected? Why is this method of notation called Roman? To what use is it chiefly applied ? 6. How are numbers commonly expressed? How many characters does this method employ? What are their names ? What are the first nine called? Why? What else are they called ? What is the last one called? Why? Obs. Is the cipher useless? What may it be regarded?
can be expressed by any single figure. All numbers larger than nine are expressed by combining together two or more of the ten characters just explained. To express ten, for example, we combine the 1 and 0, thus 10; eleven is expressed by two ls, thus 11 ; twelve, thus 12; two tens, or twenty, thus 20 ; one hundred, thus 100, &c.
The numbers from one to a thousand are expressed in the following manner : 1, one.
81, eighty-one, &c. 2, two,
90, ninety. 3, three.
91, ninety-one, &c. 4, four.
100, one hundred. 5, five.
101, one hundred and one. 6, six.
102, one hundred and two. 7, seven.
103, one hundred and three. 8, eight.
110, one hundred and ten. 9, nine.
ill, one hundred and eleven. 10, ten.
112, one hundred and twelve. 11, eleven.
120, one hundred and twenty. 12, twelve.
130, one hundred and thirty. 13, thirteen.
140, one hundred and forty. 14, fourteen.
150, one hundred and fifty. 15, fifteen.
160, one hundred and sixty. 16, sixteen.
170, one hundred and seventy. 17, seventeen.
180, one hundred and eighty. 18, eighteen.
190, one hundred and ninety. 19, nineteen.
200, two hundred. 20, twenty.
300, three hundred. 21, twenty-one, &c.
400, four hundred. 30, thirty.
500, five hundred. 31, thirty-one, &c.
600, six hundred. 40, forty.
700, seven hundred. 41, forty-one, &c.
800, eight hundred. 50, fifty.
900, nine hundred. 51, fifty-one, &c.
990, nine hundred and ninety. 60, sixty.
991, nine hundred and ninety-one 61, sixty-one, &c.
992, nine hundred and ninety-two 70, seventy
998, nine hundred & ninety-eight 71, seventy-one, &c.
999, nine hundred & ninety-nine. 80, eighty.
1000, one thousand.
QUEST.-7. What is the greatest number that can be expressed by one figure? How are larger numbers expressed ? How express ten? Eleven? Twelve ? Twenty? What is the greatest number that can be expressed by two figures ? How express a hundred? One hundred and ten? One hundred and forty-five ? Five hundred and sixty-eight? What is the greatest number that can be expressed by three figures ? How express a thousand ?
Note. Questions on the foregoing table should be continued till the class becomes familiar with the mode of expressing any number from 1 to 1000. They may be answered orally; but the best way is to let the pupil write the figures denoting the number upon the blackboard, and at the same time pronounce the answer audibly.
Obs. 1. The names thirteen, fourteen, fifteen, &c., are obviously derived from three and ten, four and ten, five and ten, &c., which by contraction become thirteen, fourteen, fifteen, &c., and are therefore significant of the numbers which they denote. The names eleven and twelve, are generally regarded as primitive words; at all events, there is no perceptible analogy between them and the numbers which they represent. Had the terms oneteen and twoteen been adopted in their stead, the names would then have been significant of the numbers one and ten, two and ten; and their etymology would have been similar to that of the succeeding terms.
2. The names twenty, thirty, forty, &c., were formed from two tens, three ţens, four tens, &c., which were contracted into twenty, thirty,
forty, &c. 3. The terms twenty-one, twenty-two, twenty-three, &c., are compounded of twenty and one, twenty and two, &c. All the other numbers as far as ninety-nine are formed in a similar manner.
4. The terms hundred and thousand are primitive words, and bear no analogy to the numbers which they denote. The numbers between a hundred and a thousand are expressed by a repetition of the numbers below a hundred. Thus we say one hundred and one, one hundred and two, one hundred and three, &c.
8. It will be perceived from the foregoing table, that the figures standing in different places have different values. Thus the digits, 1, 2, 3, &c., standing alone, or in the right hand place, respectively denote units or ones. But when they stand in the second place, they express tens: thus the 1 in 10, 12, 15, &c., expresses ten, or ten ones; that is, its value is ten times as much as when it stands in the first or right hand place, and it is called a unit of the second order. So the other digits, 2, 3, 4, &c., standing
QUEST.-Obs. From what is the term thirteen formed? Fourteen? Six. teen? Eighteen? What is said of the names eleven and twelve? How are the terms twenty, thirty, &c., formed? What is said of the terms hundred, and thousand ? How are the numbers between a hundred and a thousand expressed? 8. Does the same figure always express the same value? What does each of the digits, 1, 2, 3, &c., denote, when standing in the right hand place? What does the figure 1 denote when it stands in the second place? What is its value then? What do the other figures denote when standing in the second place? What do they denote when in the third place!
in the second place, denote two tens, three tens, four tens, &c.
When standing in the third place, they express hundreds : thus the 1 in 100, 102, 123, &c., denotes a hundred, or ten tens ; that is, its value is ten times as much as when it stands in the second place, and it is called a unit of the third order. In like manner, 2, 3, 4, &c., standing in the third place, denote two hundred, three hundred, four hundred, &c.
When a digit occupies the fourth place, it expresses thousands : thus the 1 in 1000, 1845, &c., denotes a thousand, or ten hundreds ; that is, its value is ten times as much as when it stands in the third place, and it is called a unit of the fourth order.
Thus it will be seen that ten units make one ten, ten tens make one hundred, and ten hundreds make one thousand; that is, ten in an inferior order are equal to one in the next superior order. Hence we may infer, universally, that
9. Numbers increase from right to left in a tenfold ratio ; that is, each removal of a figure one place towards the left, increases its value ten times.
10. The different values which the same figures have, are called simple and local values.
The simple value of a figure is the value which it expresses when it stands alone, or in the right hand place. Hence the simple value of a figure is the number which its name denotes. (Art. 6.)
The local value of a figure is the increased value which
Quest.-What is a figure called when it occupies the third place ? What is its value then? What is it called when in the fourth place? What is its value? What do the other figures denote when standing in the fourth place? How many units are required to make one ten? How many tens make a hundred? How many hundreds make a thousand ? Generally, how many of an inferior order are required to make one of the next superior order? 9. What is the general law by which numbers increase? What is the effect upon the value of a figure to remove it one place towards the left? 10. What are the different values of the same figure called? What is the simple value of a figure? What the local value ? Upon what does the local value of a figure depend? Obs. Why is this system of notation called Arabic? What else is it sometimes called ? Why?