It's easy to MULTIPLY signed numbers; you just look at the signs, and
if they are the same (+ times + or - times -), the product is
positive, while if they are different it is negative.
But addition and subtraction don't work that way. You have to know the
actual values in order to know the sign. In particular, the difference
between two negative numbers can be either positive or negative, or
even zero if they are the same.
Let's start with addition. If you add two positive numbers, the sum
will be positive: the first number puts you at some point to the right
of zero on the number line, and adding a positive number moves you
farther to the right.
-5 -4 -3 -2 -1 0 1 2 3 4 5
<--+---+---+---+---+---+---+---+---+---+---+-->
|>|>|
2 + 3 =5
Since you are always going to the right, the sum has to be positive.
If you add two negative numbers, you are always moving to the left,
since adding a negative number takes you to the left:
-5 -4 -3 -2 -1 0 1 2 3 4 5
<--+---+---+---+---+---+---+---+---+---+---+-->
|<|<|
-3 + -2 -2 + -3 = -5
But if the signs are different, you are going both left and right, and
the sign of the answer depends on which is "bigger." It's sort of a
tug-of-war; the stronger number wins. Let's add 2 + -3:
-5 -4 -3 -2 -1 0 1 2 3 4 5
<--+---+---+---+---+---+---+---+---+---+---+-->
|>| 2
|<| + -3
-1
Here I first went 2 units to the right, and then I went 3 units to the
left. It took 2 units to get me to 0, and that left one more unit to
the left, taking me to -1. Because 3 is bigger than 2, -3 "won" over
+2, and made the answer negative. The size of the sum is the
difference between the sizes of the two numbers.
The opposite thing happens if the first number is greater. Let's do
-3 + 2:
-5 -4 -3 -2 -1 0 1 2 3 4 5
<--+---+---+---+---+---+---+---+---+---+---+-->
|<| -3
|>| + 2
-1
Here, I went 3 units to the left, and adding 2 didn't take me far
enough to get back to zero, so the answer is negative. Its size again
is the difference between the sizes of the numbers.
Now, notice that what we've been doing is the same as subtraction.
Moving some amount to the left can be seen either as adding a negative
number, or subtracting a positive number. And moving to the right can
be thought of either as adding a positive number, or (and this is the
one that might seem odd) as subtracting a negative number. That's
because subtraction asks the question, what do we have to add to the
second number to get the first? For example, when we say
3 - 2 = 1
that's because
3 = 2 + 1
That is, 1 is the amount we have to add to 2 to get 3.
So what is 2 - -3? Well,
2 - -3 = 5
because
2 = -3 + 5
So as I said, subtracting a negative moves us to the right. We can
find the answer by changing the subtraction to an addition:
2 - -3 = 2 + 3 = 5
On the other hand,
-2 - -3 = -2 + 3 = 3 + -2 = 3 - 2 = 1
since after we change the addition to subtraction, the 3 takes us
farther to the right than the -2 took us to the left. Notice that I
did this without having to draw a number line; I just changed the
problem by changing the order so the 3 comes before the -2, and then
turning it back into a subtraction.
You should now be able to do any of your problems, even the one with
fractions. But you should be aware that those are not integers.
Because you have been introduced to signs as part of learning about
integers, you may think that "integer" means "signed number"; but
actually it means "whole numbers with signs." A signed fraction is a
rational number.
