Difference between Short and Long Division,

Rule for Long Division,

To divide by 10, 100, 1000, &c.,

When there are ciphers on the right of the divisor, -

Arithmetical Terms,

Multiplication of Fractions,

Division of Fractions,

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ARITHMETIC.

SECTION I.

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.X

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, ter; 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 iu 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? DY 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 letters, as may be seen from the following

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

OBS. 1. Every time a letter is repeated, its value is repted. 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, six. So X denotes ten; IX, nine; XI, eleven.

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

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 :

5 6 7 8

90

six, seven, eight, nine, naught.

2 3 4 one, two, three, four, five, 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 1s, thus 11; to express twelve, we use the 1 and 2, thus 12, &c.

QUEST. Obs. What is the effect of repeating a letter? If a letter of less value is placed beforo another of greater value, what is the effect? If placed after, what? 6. What is the Arabic notation? Why so called? How many figures does it employ? What are their names? Obs. What are the first nine called? Why? What else are they sometimes called? What is the last one called? Why? 7. How are numbers larger than nine expressed? Express ten by figures. Eleven. Twelve. Fifteen

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

QUEST.-How express fifteen? Twenty-five? Forty-seven? Thirty-six? Seventy-three? One hundred and one? One hundred and ten? One hundred and twenty? Two hundred and fifteen?