number it is the unit which the number measures.

Thus, in seven dollars

it is a dollar, in twelve bushels it is a bushel, in forty-three dozen of eggs it is a dozen of eggs, in six tens it is one ten, &c.

7. ARITHMETIC is the SCIENCE OF NUMBERS and ART of

NUMERICAL COMPUTATION.

(a.) It treats of numbers with reference to their nature and use, thei properties and relations; explains the various methods of representing them; and includes the theory of all numerical operations, as well as the practical methods of performing them.

8. The operations of which numbers are susceptible are four in number; viz., Addition, Subtraction, Multiplication, and Division.

9. Among the characters used to indicate numerical operations or relations are the following:

-:

(a.) The sign of equality, called equal, or equal to, signifies that the quantities between which it is placed are equal to each other.

(b.) The sign of addition, called plus or and, signifies that the quantities between which it is placed are to be added together.

Thus, 7+4=11, is read, seven plus four equal eleven; or, seven and four equal eleven; and means that seven added to four equal eleven. (c.) -The sign of subtraction, called minus or less, signifies that the number following it is to be subtracted.

Illustration. 9—63, is read, nine minus six, or nine less six, equal three, and means that nine diminished by six equal three.

(d.) X The sign of multiplication, called times or multiplied by, signifies that the numbers between which it is placed are to be multiplied together.

Illustration. 7X5= 35, is read, seven times five equal thirty-five; or, seven multiplied by five equal thirty-five.

(e.) The sign of division, called divided by, means that the number before it is to be divided by that following it.

Illustration. 123: 4, is read, twelve divided by three equal four, and means that twelve divided by three gives four for a quotient.

(f) Division may also be expressed by writing the num

ber to be divided above the number by which it is to be divided, with a line between them.

Illustration.

12

3

= = 4, means the same as 12÷3=4; i. e., that the

quotient of twelve divided by three equals four.

(g.) Such expressions as 12, 16, 7, are usually called fractions, and read thus: twelve thirds, sixteen fourths, seven eighths; though they may, with equal correctness, be read as twelve divided by three, sixteen divided by four, seven divided by eight. When read as fractions, the number above the line is called the numerator, and the number below it the denominator. See Section X.

(h.) The more common use of fractions is to express the value of one or more such parts as are obtained by dividing a unit into a given number of equal parts, or, which is the same thing, to express the value of one or more equal parts of such kind that a given number of them will equal a unit. Thus, 3 (read three fourths) is used to express the value of three such parts as would be obtained by dividing a unit into four equal parts; or, in other words, the value of three equal parts of such kind that it would take four of them to equal a unit.

(i.) The numerical value of a fraction is the same, whether we consider that it expresses a division to be performed, or a certain number of equal parts, and, in either case, it is obvious that a fraction must equal unity whenever its numerator equals its denominator. Thus, 1 = ===&, &c.

==

SECTION II.

NOTATION AND NUMERATION.

2. Definition of Terms.

NOTATION and NUMERATION treat of the various methods of representing and expressing numbers.

(a.) The distinction usually made between notation and numeration is, that the former treats of the methods of representing numbers by written characters, while the latter treats of the methods of reading them, or of expressing them in words.

3. Methods of representing Numbers, and Origin of the Decimal System.

Numbers may be represented by material objects or visible marks, by words, and by figures.

(a.) As our ideas of number are derived primarily from material objects, so the most natural and obvious method of communicating them to others is by exhibiting as many such objects as there are units in the number considered.

(b.) It is probable that in the earlier stages of society numbers were represented only in this way, the fingers being, as a general thing, made use of as counters. Thus, three fingers would be shown as a symbol for the number three, five fingers for the number five, and the fingers of both hands for the number ten.

(c.) Such a method would naturally lead a people using it to represent large numbers by exhibiting the fingers of both hands as many times as there are tens in the numbers considered, and by thus leading them to reckon by tens, would lay the foundation for a system of numbers similar to the one in general use, which is known as the DECIMAL*

SYSTEM.

4. Nature of the Decimal System of Numbers.

(a.) The fundamental idea of the Decimal System is, that ten single things may be regarded as forming a single collection or group; ten of these groups as forming a larger group; and so on, ten groups of one size forming a new group of a larger size, each capable of being regarded and dealt with as a single thing or unit. This idea renders it easy to represent the largest numbers, by having names for each of the first ten numbers, and for each group formed by combining ten of the smaller

ones.

(b.) In conformity with it, we might count thus: one, two, three, four, five, six, seven, eight, nine, ten, one and ten, two and ten, three and ten, four and ten, five and ten, six and ten, seven and ten, eight and ten, nine and ten, two tens, two tens and one, two tens and two, &c., to nine tens and eight, nine tens and nine, ten tens or one hundred, one hundred and one, &c., to nine hundreds nine tens and eight, nine hundreds nine tens and nine, ten hundreds or one thousand, &c.

(c.) Adopting this method, and forming compound words by dropping the conjunction, we should count from ten thus: one-ten, two-ten, three-ten, four-ten, five-ten, six-ten, seven-ten, eight-ten, nine-ten, two-tens, twotens one, two-tens two, &c.

*The word decimal is derived from the Latin word decem, which signifles ten.

(d) Changing the word ten to teen, and dropping the hyphen in counting from ten to two-tens, we should have oneteen, twoteen, threeteen, fourteen, fiveteen, sixteen, seventeen, eighteen, and nineteen.

(e.) By now changing five to fif, three to thir, and substituting for one-teen and two-teen the words eleven and twelve, signifying respectively one left and two left, (i. e., ten and one left, ten and two left,) we should have the familiar names, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, and nineteen.

(f) Substituting the syllable ty for tens in the words two-tens, threetens, &c., would give the words twoty, threety, fourty, fivety, sixty, seventy, eighty, and ninety; and again changing the three to thir, four to for, and the five to fif, and substituting twen, derived from twain, for two, would give the familiar names twenty, thirty, forty, fifly, sixty, seventy, eighty, and ninety.

(g.) These changes would enable us to count from two-tens or twenty thus: twenty-one, twenty-two, &c., to twenty-nine, thirty, thirty-one, &c., to ɔne hundred, one hundred and one, &c.

(h.) This method of expressing numbers is the one now in general use.

5. Arabic and Roman Methods of Notation.

(a.) Numbers are usually represented to the eye by characters called figures, though sometimes by letters of the alphabet. (b.) The method by figures is called the Arabic Method, because it was introduced into Europe by the Arabs.

NOTE. The Arabs probably obtained it from the Persians, who had obtained it from the Hindoos. Its origin has never been satisfactorily determined.

(c.) The method by letters is called the Roman Method, cause it was used by the ancient Romans.

7. The Place of a Figure determines its Denomination.

Each of these figures represents as many units as its name indicates; but the size or denomination of those units is determined by the place or position of the figure with reference tc a period or dot, called the decimal point.

8. Names and Position of the Decimal Places.

(a.) The figure immediately at the left of the point represents ones, or simple units; the second figure at the left represents tens, (i. e., units of the denomination or value of ten ones, or ten simple units;) the third figure represents hundreds; the fourth represents thousands, and so on; the figure in any place always representing ten times the value it would represent if it stood one place farther towards the right.

(b.) Hence each place has its peculiar name, the first, second, third, and fourth places being called, respectively, the units' place, the tens' place, the hundreds' place, and the thousands' place. Moreover, the position of these places is marked by the figures occupying them. Hence each figure performs a double office, viz., it marks a place, and indicates as many of the denomination of that place as its name indicates. (c.) The following will illustrate this:

(d.) In the above example each zero marks a place, and shows that there are none of the denomination of the place it occupies expressed in that place.

(e.) As another illustration, take the expression 2503. Here each figure marks a place, and denotes as many of the denomination of that place as its name implies; i. e., the 3 marks the units' place, and shows that there are 3 units; the O marks the tens' place, and shows that there are no tens; the 5 marks the hundreds' place, and shows that there are 5 hundreds; and the 2 marks the thousands' place, and shows that there are 2 thousands. The number is read two thousand five hundred and three.