CHAPTER III
POSITIVE AND NEGATIVE NUMBERS.
61. Positive and Negative Quantities. If a person is en-
gaged 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 begin-
ning 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 2 3456789 10 11
0LIIIIIIIIII
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 oj
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numbers comes to an end.
34 POSITIVE AND NEGATIVE NUMBERS.
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 oppo-
sition 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
II11IIIII
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 num-
ber 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
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of a number is its value independent of its sign.
POSITIVE AND NEGATIVE NUMBERS. 35
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 -f- 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 who1e numbers, integra1 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 fractiona1 num-
bers, 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
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is often enclosed in a parenthesis, in order that the signs +
36 POSITIVE AND NEGATIVE NUMBERS.
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, + 4 + (— 8) 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
IIII11III
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, — 5.
70. If a and b represent any two numbers, we have
+ a + (+ b) = a + b. - a + (+ b) = - a + b.
+ a + (- b) = a - b. - a + (- b) = - a - b.
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 num*
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her to the result.
POSITIVE AND NEGATIVE NUMBERS. 37
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.
Exercise 9.
Perform mentally the indicated additions:
1.
2.
3.
4.
5.
6.
+8
-8
-7
-11
+ 11
-11
-7
+7
-7
+4
-4
-4
7.
8.
9.
10.
11.
12.
+5
-8
-9
+ 87
-37
-7
-7
-7
+ 24
-36
+40
-9
+9
-6
+ 36
-42
-20
13.
14.
15.
16.
17.
18.
+ 15
-15
-21
-20
-18
+ 17
+ 12
-12
+ 12
-30
+ 32
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-20