magnitude.
Every algebraic number, as or , consists of a sign or and the absolute value of the number. The sign shows whether the number belongs to the positive or
negative series of numbers; the absolute value shows the place the number has in the positive or negative series.
When no sign stands before a number, the sign is always understood. Thus means the same as , means the same as . But the sign is never omitted.
Two algebraic numbers which have, one the sign , and the other the sign , are said to have unlike signs.
Two algebraic numbers which have the same absolute values, but unlike signs, always cancel each other when combined. Thus ; .
60. Double Meanings of the Signs $+$ and $-$.
The use of the signs and to indicate addition and subtraction must be carefully distinguished from the use of the signs and to indicate in which series, the positive or the negative, a given number belongs. In the first sense they are signs of operations, and are common to Arithmetic and Algebra; in the second sense they are signs of opposition, and are employed in Algebra alone.
In Arithmetic, if the things counted are whole units, the numbers which count them are called whole numbers, integral numbers, or integers, where the adjective is transferred from the things counted to the numbers which count them. But if the things counted are only parts of units, the numbers which count them are called fractional numbers, or simply fractions, where again the adjective is transferred from the things counted to the numbers which count them.
Likewise in Algebra, if the units counted are negative, the numbers which count them are called negative numbers, where the adjective which defines the nature of the units counted is transferred to the numbers that count them.
A whole number means a number of whole units, a fractional number means a number of parts of units, and a negative number means a number of negative units.
61. Addition and Subtraction of Algebraic Numbers.
An algebraic number which is to be added or subtracted is often enclosed in a parenthesis, in order that the signs and , which are used to distinguish positive and negative numbers, may not be confounded with the and signs that denote the operations of addition and subtraction. Thus expresses the sum, and expresses the difference, of the numbers and .
62. Addition.
In order to add two algebraic numbers, we begin at the place in the series which the first number occupies, and count, in the direction indicated by the sign of the second number, as many units as there are in the absolute value of the second number. 3