CHAPTER III. POSITIVE AND NEGATIVE NUMBERS. 61. Positive and Negative Quantities. If a person is engaged in trade, his capital will be increased by his gains, and diminished by his losses. Increase in temperature is measured by the number of degrees the mercury rises in a thermometer, and decrease in temperature by the number of degrees the mercury falls. In considering any quantity whatever, a quantity that increases the quantity under consideration is called a posi tive quantity; and a quantity that decreases the quantity under consideration is called a negative quantity. 62. The Natural Series of Numbers. If from a given point, marked 0, we draw a straight line to the right, and beginning from the zero point lay off units of length on this line, the successive repetitions of the unit will be expressed by the natural series of numbers, 1, 2, 3, 4, etc. Thus: 0 5 6 7 8 9 10 11 In this series if we wish to add 2 to 5, we begin at 5, count 2 units forwards, and arrive at 7. If we wish to subtract 2 from 5, we begin at 5, count 2 units backwards, and arrive at 3. If we wish to subtract 5 from 5, we count 5 units backwards from 5, and arrive at 0. If we wish to subtract 5 from 2, we cannot do it, because when we have counted backwards from 2 as far as 0, the natural series of numbers comes to an end. 63. Positive and Negative Numbers. In order to subtract a greater number from a smaller it is necessary to assume a new series of numbers, beginning at zero and extending to the left of zero. The series to the left of zero must proceed from zero by the repetitions of the unit, precisely like the natural series to the right of zero; and the opposition between the right-hand series and the left-hand series must be clearly marked. This opposition is indicated by calling every number in the right-hand series a positive number, and prefixing to it, when written, the sign +; and by calling every number in the left-hand series a negative number, and prefixing to it the sign. The two series of numbers may be called the algebraic series of numbers, and written thus: 4 -3' -2 -1 0 +1 +2 +3 +4 If, in this double series of numbers, we wish to subtract 4 from 2, we begin at 2 in the positive series, count 4 units in the negative direction (to the left), and arrive at 2 in the negative series; that is, 2 4 - 2. The result obtained by subtracting a greater number from a less, when both are positive, is always a negative number. In general, if a and b represent any two numbers of the positive series, the expression a-b will be a positive number when a is greater than b; will be zero when a is equal to b; will be a negative number when a is less than b. In counting from left to right in the algebraic series, numbers increase in magnitude; in counting from right to left, numbers decrease in magnitude. Thus, 3, 1, 0, +2, +4, are arranged in ascending order of magnitude. 64. The Absolute Value of a Number. The absolute value of a number is its value independent of its sign. 65. Every algebraic number, as + 4 or 4, 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. But the sign is never omitted. 66. Two algebraic numbers that have, one the sign +, and the other the sign, are said to have unlike signs. Two algebraic numbers that have the same absolute values, but unlike signs, cancel each other when combined. Thus, + 4 − 4 = 0; + a − a = 0. -- -. 67. 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. NOTE. In Arithmetic, if the things counted are whole units, the numbers that count them are called whole numbers, integral numbers, or integers, the adjective being transferred from the things counted to the numbers that count them. But if the things counted are only parts of units, the numbers that count them are called fractional numbers, or simply fractions, the adjective being transferred from the things counted to the numbers that count them. Likewise in Algebra, if the units counted are negative, the numbers that count them are called negative numbers, the adjective that defines the nature of the units counted being transferred to the numbers that count them. 68. Addition of Algebraic Numbers. An algebraic number 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 that denote the operations of addition and subtraction. signs Thus, + 4 + (-3) expresses the sum, and + 4 − (−3) expresses the difference, of the numbers + 4 and 3. 69. 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. -4 -3 -2 -1 0 +1 +2 +3 +4.. Thus, the sum of + 2 + (+3) is found by counting from +2 three units in the positive direction; that is, to the right and is, therefore, +5. The sum of+2+(-3) is found by counting from +2 three units in the negative direction; that is, to the left, and is, therefore, 1. The sum of 2+(+3) is found by counting from - 2 three units in the positive direction, and is, therefore, +1. The sum of 2+(-3) is found by counting from – 2 three units in the negative direction, and is, therefore, 70. If a and b represent any two numbers, we have 5. Therefore, from these four cases, we have the following Rule for Adding Two Algebraic Numbers: 1. If the numbers have like signs, find the sum of their absolute values, and prefix the common sign to the result. 2. If the numbers have unlike signs, find the difference of their absolute values, and prefix the sign of the greater number to the result. 71. The result is called the algebraic sum in distinction from the arithmetical sum; that is, the sum of the absolute values of the numbers. NOTE. If there are more than two numbers to be added, add two of the numbers, and then this sum to a third number, and so on; or find the sum of the positive numbers and the sum of the negative numbers, then the difference between the absolute values of these two sums, and prefix the sign of the greater sum to the result. |