III.

Positive and Negative Numbers.

56. Quantities Opposite in Kind.

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 considered is called a positive quantity; and a quantity that decreases the quantity considered is called a negative quantity.

57. Positive and Negative 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: 1

If we wish to add $2$ to $5$, we begin at $5$, count $2$ units forwards, and arrive at $7$, the sum required. If we wish to subtract $2$ from $5$, we begin at $5$, count $2$ units backwards, and arrive at $3$, the difference required. If we wish to subtract $5$ from $5$, we count $5$ units backwards, 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.

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: 2

If, now, we wish to subtract $7$ from $4$, we begin at $4$ in the positive series, count $7$ units in the negative direction (to the left), and arrive at $-3$ in the negative series; that is, $4-7=-3$.

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