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ARTICLE 1. ARITHMETIC is the science of numbers.
The operations of arithmetic are performed by the aid of five distinct rules, viz.: Numeration, Addition, Subtraction,Multiplication, and Division. These are usually called the FUNDAMENTAL RULES of arithmetic, because all other niles are founded upon them.
What is Arithmetic ? How many distinct rules has it for its operations ? Repeat their names. What are these usually called ? Why are they so called ?
2. NUMERATION explains the method of reading written numbers.
Notation is the writing down of numbers.
Various methods of notation and numeration were used by the ancients. We shall content ourselves with mentioning two, the common or Arabic method, and the Roman method.
In the common method ten characters are employed. These characters when written are,
1, 2, 3, 4, 5, 6, 7, 8, 9,0
When printed, they become,
1, 2, 3, 4, 5, 6, 7, 8, 9, 0.
They have the following names:
1 is called One, or a Unit,
O is called Naught, Cipher, or Zero. Each of these characters, except the zero, is called a digit* ; and the first nine, when taken together, are called the nine digits.
Any digit is called a significant figure. What is numeration? How is the common method sometimes called ? In this method how many characters are employed ? What are the names of these char acters ? What are called digits ? What is a significant figure ?
3. The significant figures have unchanging values ; that is, they always represent units or ones ; but the units which they represent differ in value.
When a significant figure stands disconnected from other figures, the value of its unit is called its simple value. When such figure stands in connection with other figures, the value of its unit will depend upon the place which it occupies, and is therefore called its local value.
Thus, in the number 3456, which consists of four sig. nificant figures standing in connection with each other, each figure expresses units; but units of different values. The right-hand figure, 6, expresses six units, whose value is their simple value; that is, each unit is a single one. The second figure, 5, expresses five units; but each unit is ter times greater than each unit of the first figure; therefore the 5 may be read 5 tens, equal to fifty units of simple value. The units expressed by the third figure, 4, are ten times greater than the units expressed by the second figure, and one hundred times greater than those expressed by the first figure; the third figure is therefore read 4 hundreds. The last figure, 3, expresses units ten times greater than the units in 4, and one thousand times greater than the units in 6, and is read 3 thousands.
* From the Latin, digitus, a finger; because the ancients used to do they reckon on their fingers. Originally 10 was also called a digit.
Hence this property:
When figures are connected in a line as in the number 3456, the units which they express are said to be of different orders. Thus, 6 occupies the first place, and its units are of the first order, that is, they have their simple value. The 5 occupies the second place, and its units are of the second order, or tens. The 4 occupies the third place, and its units are of the third order, or hundreds The 3 occupies the fourth place, and its units are of the fourth order, or thousands. Hence the above number is three thousand four hundred and fifty-six.
To numerate and read the numbers in the following table, proceed thus: Begin with the upper line 3. The first place only being occupied, you numerate Units. Then read, three units, or simply three. In the second line two places are occupied—then numerate Units, Tens-read fifty-four. In the third line three places are occupied ; then numerate Units, Tens, Hundreds—read two hundred and sixty-seven, and so proceed.