OPERATING WITH SIGNED NUMBERS
The number line can be used to demonstrate addition
of signed numbers. Two cases must be
considered; namely, adding numbers with like
signs and adding numbers with unlike signs.
ADDING WITH LIKE SIGNS
As an example of addition with like signs, suppose
that we use the number line (fig. 34) to add
2 + 3. Since these are signed numbers, we
indicate this addition as (+2) + (+3). This emphasizes
that, among the three + signs shown, two are
number signs and one is a sign of
Figure 34.Using the number line to add. operation.
Line a (fig. 34) above the number line shows
this addition. Find 2 on the number line. To
add 3 to it, go three units more in a positive
direction and get 5.
To add two negative numbers on the number line,
such as 2 and 3, find 2 on the number line
and then go three units more in the negative direction to get 5, as in b (fig.
34) above the number line.
Observation of the results of the foregoing operations
on the number line leads us to the following
conclusion, which may be stated as a law: To
add numbers with like signs, add the absolute
values and prefix the common sign.
ADDING WITH UNLIKE SIGNS
To add a positive and a negative number, such
as (4) + (+5), find +5 on the number line and
go four units in a negative direction, as in line
c above the number line in figure 34. Notice
that this addition could be performed in the
other direction. That is, we could start at 4
and move 5 units in the positive direction.
(See line d, fig. 34.)
The results of our operations with mixed signs
on the number line lead to the following conclusion,
which may be stated as a law: To add numbers
with unlike signs, find the difference between their absolute values and prefix
the. sign of the numerically greater number. The
following examples show the addition of the
numbers 3 and 5 with the four possible combinations of signs:
In the first example, 3 and 5 have like signs and
the common sign is understood to be positive. The sum of the absolute values is
8 and no sign is prefixed to this sum, thus
signifying that the sign of the 8 is
understood to be positive. In the second
example, the 3 and 5 again have like signs, but their common sign is negative.
The sum of the absolute values is 8, and this
time the common sign is prefixed to the sum. The
answer is thus 8.
In the third example, the 3 and 5 have unlike signs.
The difference between their absolute values
is 2, and the sign of the larger addend is negative.
Therefore, the answer is 2.
In the fourth example, the 3 and 5 again have unlike
signs. The difference of the absolute values
is still 2, but this time the sign of the larger
addend is positive. Therefore, the sign prefixed
to the 2 is positive (understood) and the
final answer is simply 2.
These four examples could be written in a different
form, emphasizing the distinction between the sign of a number and an
operational sign, as follows:
(+3) + (+5) = +8
(3) + (5) = 8
(+3) + (5) = 2
(3) + (+5) = +2
Practice problems. Add as indicated:
1. 10 + 5 = (10) + (+5) = ?
2. Add 9, 16, and 25
3. 7 1 
3 = (7) +
(1) + (3) = ?
4. Add 22 and 13
Answers:
1. 5
2. 0
3. 11
4. 35
SUBTRACTION
Subtraction is the inverse of addition. When subtraction
is performed, we "take away" the subtrahend.
This means that whatever the value of the
subtrahend, its effect is to be reversed when
subtraction is indicated. In addition, the sum
of 5 and 2 is 3. In subtraction, however, to
take away the effect of the 2, the quantity +2 must
be added. Thus the difference between +5 and
2 is +7.
Keeping this idea in mind, we may now proceed to examine the various
combinations of subtraction involving signed
numbers. Let us first consider the four
possibilities where the minuend is
numerically greater than the subtrahend, as in the following examples:
We may show how each of these results is obtained
by use of the number line, as shown in figure
35.
In the first example, we find +8 on the number line, then subtract 5 by
making a movement that reverses its sign.
Thus, we move to the left 5 units. The result
(difference) is +3. (See line a, fig. 3 5 .)
In the second example, we find +8 on the number
line, then subtract (5) by making a movement
that will reverse its sign. Thus we move to
the right 5 units. The result in this case is
+13. (See line b, fig. 35.)
In the third example, we find 8 on the number line, then subtract 5 by
making a movement that reverses its sign.
Thus we move to the left 5 units. The result
is 13. (See line c, fig. 35.)
In the fourth example, we find 8 on the number
line, then reverse the sign of 5 by moving 5
units to the right. The result is 3. (See
line d, fig. 35.)
Next, let us consider the four possibilities that
arise when the subtrahend is numerically greater
than the minuend, as in the following examples:
In the first example, we find +5 on the number line, then subtract 8 by
making a movement
Figure 35.Subtraction by use of the number line.
that reverses its sign. Thus we move to the left
8 units. The result is 3. (See line e, fig.
35.)
In the second example, we find +5 on the number
line, then subtract 8 by making a movement to the right that reverses its sign.
The result is 13. (See line f, fig. 35.)
In the third example, we find  5 on the number line, then reverse the sign
of 8 by a movement to the left. The result is 13. (See line g, fig.
35.)
In the fourth example, we find 5 on the number line, then reverse the sign
of 8 by a movement to the right. The result is 3. (See line h, fig.
35.)
Careful study of the preceding examples leads
to the following conclusion, which is stated
as a law for subtraction of signed numbers: In any subtraction problem, mentally
change the sign of the subtrahend and proceed
as in addition.
Practice problems. In problems 1 through 4, subtract
the lower number from the upper. In 5 through
8, subtract as indicated.
5. 1 (5) = ?
6. 6 (8) = ?
7. 14  7 (3) = ?
8. 9  2 = ?
Answers:
1. 27
2. 20
3. 4
4. 9
5. 6
6. 2
7. 10
8. 11
