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ARITHMETIC.

SECTION 1.

Art. 1. ARITHMETIC is the science of numbers.

Any single thing, as a peach, a rose, a book, is called a unit, or one ; if another single thing is put with it, the collection is called two; if another still, it is called three ; if another, four; if another, five, &c.

The terms, one, two, three, four, &c., are the names of numbers. Hence,

2. NUMBER signifies a unit, or a collection of units.

Numbers are expressed by words, by letters, and by figures.

3. NOTATION is the art of expressing numbers by letters or figures. There are two methods of notation in use, the Roman and the Arabic.

I. ROMAN NOTATION. 4. The Roman Notation is the method of expressing numbers by letters ; and is so called because it was invented by the ancient Romans. It 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.

Quest.–1. What is Arithmetic? What is a single thing called ? If another is put with it, what is the collection called? If another, what? What are the terms one, two, three, &c. ? 2. What then is number? How are numbers expressed ? 3. What is Notation? How many methods of notation are in use? 4. What is the Roman notation? Why so called ? How many letters does it employ? What does the letter I, denote? V? X?L?C? D? M ?

5. To express the intervening numbers from to one a thousand, or any number larger than a thousand, we resort to repetitions and various combinations of these leto tors, as may be seen from the following

TABLE.

66

6 fifty.

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IX

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I denotes one.
II

two. III

three. IV

four. V

five. VI

six. VII

seven. VIII

eight.

nine. X

ten. XI

eleven. XII

twelve. XIII

thirteen. XIV

fourteen. XV

fifteen. XVI

sixteen. XVII

seventeen. XVIII eighteen. XIX

nineteen. XX

twenty. XXI

twenty-one. XXII

twenty-two. XXIII twenty-threc. XXIV

twenty-four. XXV

twenty-five. XXVI twenty-six. XXVII twenty-seven. XXVIII «

twenty-eight. XXIX twenty-nine. XXX

thirty.

XXXI denotes thirty-one.
XL

forty. XLI

forty-one. L LI

fifty-one. LX

sixty. LXI

sixty-one. LXX

seventy. LXXX eighty. XC

ninety. XCI

ninety-one. C

one hundred. CI

one hund. and one CIV

one hund. and four CX

one hund, and ten. CC

two hundred. CCV

two hund. and five. CCC

three hundred. CCCC four hundred, D

five hundred. DC

six hundred. DCC

seven hundred. DCCC eight hundred. DCCCC nine hundred. M

one thousand. MC

one thousand and

one hundred. MM

two thousand. MDCCCL“ one thousand eight

hundred and fifty.

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66

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QUEST.-5. What do the letters IV, denote? VI? VIII? IX? XI? XIV! XVI? XVIII? XIX? XXIV? XL? LXXX? XC? CIV? Expres seven by let ters on the slate or black-board. How express eleven? Thirteen Twouty-five? Nineteen? Forty-luur? Eighty-soven? Ninety-nine ?

OBs. 1. Every time a letter is repeated, its value is repeated. Thus, the letter 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 as many units as the less denotes ; if placed after the greater, the value of the greater is increased as many units as the less denotes. Thus, V denotes five; but IV denotes only four; and VI, bix. So X denotes ten; IX, nine; XI, eleven,

Noté. —The questions on the observations may be omitted, by beginners, till review, if deemed advisable by the teacher.

one,

II. ARABIC NOTATION. 6, The Arabic Notation is the method of expressing numbers by figures, and is so called because it is supposed to have been invented by the Arabs. It employs the following ten characters or figures, viz: 1 2 3 4 5 6 7 8 9 0 two, three, four, five,

six,

seven, eight, nine, naught. OBS. 1. The first nine are called significant figures, because each one always expresses a value, or denotes some number. They are also called digits, from the Latin word digitus, signifying a finger, because the ancients used to count on their fingers.

2. The last one is called naught, because when standing alone, it expresses nothing, or the absence of number. It is also called cipher or zero.

7. All numbers larger than 9, are expressed by different combinations of these ten figures. For example, to express ten, we use the 1 and 0, thus 10; to express eleven, we use two ls, thus 11 ; to express twelve, we use the 1 and 2, thus 12, &c.

QUEST.-Obs. What is the effect of repoating a letter ? If a letter of loss value is placed before another of greater valuo, what is the effect? If placed altor, what? 6. What is the Arabic notation? Why so called ? How many Agures does it employ? What are their names? Obs. What are the first nino callod? Why? What olse are they sometimes called? What is the last one called? Why? 7. How are numbers larger than nine expressed ? Express ton by figures. Eleven. Twolye. Fifteen.

The method of expressing numbers by figures from one to a thousand, may be seen from the following

TABLE.

1, one. 2, two. 3, three. 4, four. 5, five. 6, six. 7, seven. 8, eight. 9, nine. 10, ten. 11, eleven. 12, twelve. 13, thirteen. 14, fourteen. 15, fifteen. 16, sixteen. 17, seventeen. 18, eighteen. 19, nineteen. 20, twenty. 21, twenty-ona. 22, twen“.y-two. 23, twenty-three. 24, twenty-four. 25, twenty-five. 26, twenty-six. 27, twenty-seven. 28, twenty-eight. 29, twenty-nine. 30, thirty 31, thirty-one. 32, thirty-two. 33, thirty-three. 34, thirty-four. 35. thirty-five.

36, thirty-six. 37, thirty-seven. 38, thirty-eight. 39, thirty-nine. 40, forty. 41, forty-one. 42, forty-two. 43, forty-three. 44, forty-four. 45, forty-five. 46, forty-six. 47, forty-seven. 48, forty-eight. 49, forty-nine. 50, fifty. 51, fifty-one. 52, fifty-two. 53, fifty-three. 54, fifty-four. 55, fifty-five. 56, fifty-six. 57, fifty-seven. 58, fifty-eight. 59, fifty-nine. 60, sixty. 61, sixty-one. 62, sixty-two. 63, sixty-three. 64, sixty-four. 65, sixty-five. 66, sixty-six. 67, sixty-seven. 68, sixty-eight. 69, sixty-nine. 70, seventy.

71, seventy-one. 72, seventy-two. 73, seventy-three. 74, seventy-four. 75, seventy-five. 76, seventy-six. 77, seventy-seven 78, seventy-eight. 79, seventy-nine. 80, eighty. 81, eighty-one. 82, eighty-two. 83, eighty-three. 84, eighty-four. 85, eighty-five. 86, eighty-six. 87, eighty-seven 88, eighty-eight. 89, eighty-nine. 90, ninety. 91, ninety-one. 92, ninety-two. 93, ninety-three. 94, ninety-four. 95, ninety-five. 96, ninety-six. 97, ninety-seven 98, ninety-eight. 99, ninety-nine. 100, one hundred. 200, two hundred. 300, three hundred 400, four hundred. 900, nine hundred. 1000, one thousand,

QUEST.-How express fifteen? Twenty-five? Forty-seven? Thirty-six Seventy-three One hundred and one? One hundred and ten? One hundron and twonty! I'wo hundred and iftoen ?

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