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ARITHMETIC.

DEFINITIONS.

ARTICLE 1. Quantity is anything that can be measured.
A Unit is a single thing, or one.
A Number is a unit or a collection of units.

An Abstract Number is a number, whose units have no reference to any particular thing or quantity ; as two, five, seven.

A Concrete Number is a number, whose units have reference to some particular thing or quantity; as two books, five feet.

The Unit of a Number is one of the same kind as the number; thus, the unit of six is one, and the unit of six pounds is one pound.

Arithmetic is the science of numbers, and the art of computing by them.

A Rule is a prescribed mode for performing an operation.

The Introductory Processes of arithmetic are Notation, Numeration, Addition, Subtraction, Multiplication, and Division.

The last four are called the fundamental rules, because upon them depend all other arithmetical processes.

NOTATION.

2. Notation is the art of expressing numbers by figures or other symbols.

There are two methods of notation in common use; the Roman and the Arabic.

QUESTIONS. — Art. 1. What is quantity ? A unit? A number? An abstract number? A concrete number? Arithmetic ? A rule? Which are the introductory processes? What are the last four called ? —2. What is notation? How many kinds of notation in common use? What are they?

3. The Roman Notation, or that originated by the ancient Romans, employs in expressing numbers seven capital letters, viz. : I, V, X, L, C,

D

M. one, five, ten, fifty, one hundred, five hundred, one thousand.

All the other numbers are expressed by the use of these letters, either in repetitions or combinations.

1. By a repetition of a letter, the value denoted by the letter is repeated ; as, XX represents twenty ; CCC, three hundred.

2. By writing a letter denoting a less value before a letter denoting a greater, the difference of their values is represented ; as, IV represents four ; XL, forty.

3. By writing a letter denoting a less value after a letter denoting a greater, the sum is represented ; as, VI represents six ; XV, fifteen.

4. A dash (-) placed over a letter makes the value denoted a thousand-fold; as, V represents five thousand; IV, four thousand.

TABLE.

I
II
III
IV
V
VI
VII
VIII
IX
X
XX
XXX
XL
L
LX
LXX

one.
two.
three.
four.
five.
six.
seven.
eight.
nine.
ten.
twenty.
thirty

LXXX
XC
С
CC
CCC
CCCC
D
DC
DCC
DCCC
DCCCC
M
MD
MM
X
M

eighty. ninety. one hundred. two hundred. three hundred. four hundred. five hundred. six hundred. seven hundred. eight hundred. nine hundred. one thousand. fifteen hundred. two thousand. ten thousand. one million.

forty.

fifty. sixty. seventy.

3. Why is the Roman notation so called? By what are numbers expressed in the Roman nutation? What effect has the repetition of a letter? The effect of writing a letter expressing a less value before a letter denoting a greater? Of writing a letter after another denoting a greater value? How many fold is the value denoted by a letter made by placing a dash over it? Repeat the table.

The Roman notation is now but little used, except in number. ing sections, chapters, and other divisions of books.

EXERCISES IN ROMAN NOTATION.

Write the following numbers in letters :1. Ninety-six.

Ans. XCVI. 2. Eighty-seven. 3. One hundred and ten. 4. One hundred and sixty-nine. 5. Two hundred and seventy-five. 6. Five hundred and forty-two. 7. One thousand three hundred and nineteen. 8. One thousand eight hundred and fifty-eight.

4. The Arabic Notation, or that made known through the Arabs, employs in expressing numbers ten characters or figures, viz. : –

1, 2, 3, 4, 5, 6, 7, 8, 9, 0. one, two, three, four, five, six, seven, eight, nine, cipher. The first nine are called digits, from digitus, the Latin signifying a finger, because of the use formerly made of the fingers in reckoning. The cipher is called naught, or zero, from its expressing the absence of a number, or nothing, when standing alone.

5. The particular position a figure occupies with regard to other figures is called its PLACE ; as in 32 (thirty-two), counting from the right, the 2 occupies the first place, and the 3 the second place.

The digits have been denominated significant figures, because each of itself always represents so many units, or ones, as its name indicates. But the size or value of the units represented by a figure differs according to the place occupied by it.

Thus, in 366 (three hundred and sixty-six), each of the figures, without regard to its place, represents units, or ones; but the 6 occupying the first place represents 6 single units; the 6

3. What use is now made of Roman notation ? —4. How many characters are employed in the Arabic notation? What are the first nine called, and why?

The cipher? What does it represent when standing alone ? 5. What is meant by the place of a figure? What have the digits been denominated? Why? How does the size or value of units represented by figures differ?

occupying the second place represents 6 tens, or 6 units each ten times the size or value of a unit of the first place; and the 3 occupying the third place represents 3 hundreds, or 3 units each one hundred times the size or value of a unit of the first place.

6. The cipher, when connected with other figures, occupies a place that otherwise would be vacant; as in 10 (ten), where it occupies the vacant place of units; and in 304 (three hundred and four), where it occupies the vacant place of tens.

7. The Simple Value of a unit is the value expressed by a figure standing alone ; or, in a collection, when standing in the right-hand place.

Thus 6 alone, or in 26, expresses a simple value of six single units, or ones.

The Local Value of a unit is the value expressed by a figure when it is used in combination with another figure or figures, and depends upon the place the figure occupies.

The local values expressed by figures will be made plain by the following

TABLE.

Hund. of Thousands.
Tens of Thousands.
Millions.
Thousands.
Hundreds.

Tens.
co Units.

The figures in this table are read thus :

9 98 9 8 7 9 8 7 6 9 8 7 6 5

Nine.
Ninety-eight.
Nine hundred eighty-seven.
Nine thousand eight hundred seventy-six.
Ninety-eight thousand seven hundred sixty-five.
Nine hundred eighty-seven thousand six hundred

fifty-four.
Nine millions eight hundred seventy-six thousand

five hundred forty-three.

[blocks in formation]

6. What does a cipher occupy when written in connection with other fig. ures ? —7. What is the simple value of a unit? The local value of a unit ? The design of the table ?

In the table, any figure in the right-hand or units' place expresses the local value of so many units; but the same in the second place expresses the local value of so many tens, each of the value of ten ones; in the third place, the local value of so many hundreds, each of the value of ten tens; in the fourth place; the local value of so many thousands, each of the value of ten hundreds; and, in general,

The value expressed by any figure is always made tenfold by each removal of it one place to the left hand.

NUMERATION.

8. Numeration is the art of reading numbers when expressed by figures.

9. There are two methods of numeration in common use : the French and the English.

10. The French Method is that in general use on the continent of Europe and in the United States. It separates figures into groups, called periods, of three places each, and gives a distinct name to each period.

FRENCH NUMERATION TABLE.

Units.

7, 8 6 7, 123,

78, 478,

1 2 7, 8 9 4,

6 3 8.

Period of
Sextil-
lions.

Period of
Quintil-
Lions.

Period of
Quadril-
lions.

Period of
Trillions.

Period of
Billions,

Period of
Millions.

Period of
Thousands.

Period ot

Unito.

7. What value is expressed by a figure standing in the right-hand or units’ place? In the second place? In the third ? How do figures increase from the right towards the left? —8. What is numeration ?-9. What are the two methods of numeration in common use ? - 10. Where is the French method more generally used ? Repeat the French Numeration Table. Name the different periods in the table.

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