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42

The product of numbers which are represented
by letters, is usually denoted by placing the
letters close together like a word, without any
mark between them. Thus, suppose the let-
ters a, b, and c, to represent any three numbers
whatever; then abc will denote the continued
product of those numbers, the same as axbxc.
By, or divided by; the sign of Division, signify-
ing
that the former of the two quantities between
which it is placed, is to be divided by the latter.
Thus, 82, denotes that 8 is to be divided by
2, and is read, 8 by 2, or, 8 divided by 2.

Division is also frequently denoted by placing
the dividend above, and the divisor below a
horizontal line. Thus, 2, denotes the same as
8÷2, viz., that 8 is to be divided by 2; and
6+8
denotes that the sum of 6 and 8 is to be
divided by the difference of 4 and 2.

4--2

The sign of Proportion. Thus, 2: 4 :: 7 : 14, denotes that the ratio of 2 to 4 is the same as the ratio of 7 to 14: read thus, as 2 is to 4, so is 7 to 14.

2

Denotes the second power or square of 4; and 43 denotes the third power or cube of 4, and so on for higher powers. Also, a“ or aa, denotes the second power of a; and a, or aaa, denotes the third power of a, &c.

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3

The radical sign, signifying that the quantity before which it is placed is to have some root of it extracted. Thus, v4 denotes the second or square root of 4; 8 denotes the third or cube root of 8; and 4/8 denotes the 4th root of 8, &c. Equal to; the sign of equality, signifying that the quantities, or sets of quantities, between which it is placed, are equal to each other. Thus, 100 cents 1 dollar, signifies 'that 100 cents are equal to 1 dollar: Also, 2+5=7, denotes that the sum of 2 and 5 is equal to 7, and is read thus, 2 plus 5 equal to 7.

A horizontal line drawn or placed over two or more quantities, signifies that all the quantities under it are to be considered jointly as one quantity; and the line is called a vinculum. Thus, 2+5x6, denotes that the sum of 2 and 5 is to be multiplied by 6. The same thing is also frequently denoted by including in a parenthesis the several quantities which are to be considered as one quantity. Thus, v(7+9) signifies the same as √7+9, viz. the square root of the sum of 7 and 9.

The reciprocal of any quantity, is that quantity inverted, or unity divided by it. Thus, the reciprocal of is, and the reciprocal of a or a 1

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is

Two points,standing beside each other, are used in this Work to separate the different denominations of compound numbers.

A single point is used to separate integers, or whole numbers, from decimal parts.

The signs and —, when annexed to the answers to mathematical questions, denote that. such answers are not exact; the former sign denoting that the answer to which it is annexed should be a small fraction greater, and the latter, that the answer should be a little less.*

Note. I have thought proper to explain the meaning and use of the foregoing characters in this part of the Book, though most of them are explained in the following Treatise, in the places where they are first used. It may be sufficient for the student, at first, to learn the use of only the signs + and and the sign of equality, =; and afterwards to make himself acquainted with the use of the other characters as soon as he shall have occasion for them.

*It is probable that some of the answers in this Work which have the sign+annexed to them are too great. In some cases it was difficult to determine whether the omission or addition of small fractions, in performing the numerical operations, made the answers or results too small, or too great; and all such answers, as well as those which were known to be too great, have the sign annexed to them.

ARITHMETIC.

ARITHMETIC is the Science of Numbers, and the Art of using them.

Arithmetic consists of two parts, Theoretical, and Practical. The Theoretical, considers the nature and quality of numbers, and demonstrates the reason of practical operations. The Practical, merely shows the method of working by numbers, so as to be most useful and expeditious for business. Theoretical Arithmetic is properly a Science, and Practical Arithmetic is an Art.

NUMBER is that which is used to express the relations of quantity.

UNITY, UNIT, or ONE, is the beginning of number, and signifies a single, or an individual thing, of any kind. One and one more, taken collectively, make a number called two: two and one more, make the number three: this increased by one, composes the number four: and thus, by the continual addition of unity or one, we may obtain the higher numbers, five, six, seven, eight, nine, ten, &c..

An Integer, or a Whole Number, is some precise quantity of units; as one, two, three, &c. Whole numbers are so called as distinguished from Fractions, which are broken numbers, or parts of numbers; as one-half, two-thirds, three-fourths, &c.

NOTATION AND NUMERATION

OF WHOLE NUMBERS.

Notation teaches how to write down in characters any number proposed in words.

Numeration teaches how to read, in proper words, any number expressed by characters.

The characters now generally used to denote numbers, are the ten Arabic numeral characters, commonly called Figures. These characters, and their names, which are the numbers they represent, are as follows, viz. 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven,8eight,9nine,0 nought,

*

cipher, or zero. The first nine of these characters are called significant figures, or digits, to distinguish them. from the cipher, 0, which of itself, is quite insignificant, or does not express any number. By these ten figures any numbers may be expressed.

The value of any figure when alone, is called its simple value, and is invariable. Thus, the figure 1, when alone, always denotes one; the figure 2, two; &c. Figures have also a local value; that is, a value which depends upon the place they stand in when two or more of them are combined or joined together, as in the following table.

NUMERATION TABLE.

Hundreds of millions.
Tens of millions."
Millions.

Hundreds of thousands.

Tens of thousands.
Thousands.
Hundreds.
Tens.

Units.

5 4 3 2 1

9 8 7 6 5 4 3 2 9876543

Note. The words at the head of this table, (viz. Units, Tens, Hundreds, &c.) show the local values of those figures over which they stand, and must be committed perfectly to memory.

ILLUSTRATION.

Here, each figure in the first place, at the right hand, denotes only its own simple value; but each figure in the second place, (counting from right to left,) denotes ten times its simple value; each figure in the third place denotes a hundred times its simple value, and so on;—the local value of a figure in any particular place being ten times its value in the next place to the right. Thus, the figure 9, in the first place, or column, at the right hand, signifies nine units, or simply nine; but in the second place it denotes nine tens, or ninety; in the third place it denotes

987654
98765
9876
987
98

9

* These ten characters were formerly all called by the general name of Ciphers; whence it came to pass that the art of Arithmetic was then often called Ciphering. The invention of these characters is usually ascribed to the Arabians: and it is said that they were first brought into Europe by the Moors, in the ninth century of the Christian era. The Roman method of Notation, by letters, had previously been in general use in Europe.

nine hundreds, and so on. The local values of all the other significant figures vary in the same manner, according to their distance from the place of units. Thus, in the number 9876, the 6, standing in the first or units' place, denotes six units, or six; the 7, in the second place, denotes seven tens, or seventy; the 8, in the third place, denotes eight hundreds; and the 9, in the fourth place, denotes nine thousands: and hence the expression 9876 is read thus, nine thousand, eight hundred and seventy-six.

In all expressions of whole numbers, the local values of the places of figures increase from right to left in the same tenfold ratio.*

Although the cipher, 0, does not of itself denote any number, yet, every cipher annexed to significant figures increases the local values of the latter in a tenfold ratio, by throwing them into higher places. Thus, 4 denotes only four; but 40 denotes forty; 400 denotes four hundreds; and 4000 denotes four thousands, &c.

The process of Numeration is more amply illustrated by the following table.

TABLE 2.†

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6 5 1, 2 3 7,

7 1 4, 9 7 0, 3 0 5, 0

&c.

Hund. of qunitill.
Tens of quintill.
Quintillions.

Hund. of quadrill.
Tens of quadrill.
Quadrillions.
Hund. of trill.
Tens of trill.
Trillions.

Hund. of bill.
Tens of bill.
Billions.

Hund. of mill.
Tens of mill.
Millions.

Hundreds of thou.
Tens of thousands.
Thousands.
Hundreds.
∞ Tens.
Units.

* Whole numbers are counted by tens and

combinations of tens; and

hence the reason why we make use of ten different characters to denote numbers. It may be well to observe that there is no reason in the nature of numbers, that they should be made to increase in a tenfold ratio: they might have been made to increase in 2, 3, 4, &c. fold, or any B

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