|
Section
04 Part 02 – Signed and Unsigned |
|
“A positive attitude may not solve all your problems, but it will annoy enough people to make it worth the effort.” ~Herm Albright, quoted in Reader's Digest, June 1995 |
Introduction
In mathematics,
you can have negative numbers (-1, -2, -3, etc). This section looks at signed and unsigned,
and how they can be used in programming to “fake” (or rather pretend) that
negative numbers exist.
Unsigned
Right
now, you’ll most probably agree with me, that for a byte of data; 00 is the lowest
number, and FF Is the highest number.
For a word of data; 0000 is the lowest
number, and FFFF is the highest number.
And for a long-word of data; 00000000 is the
lowest number, and FFFFFFFF is the highest number.
I
hope we’re cool with that so far...
Now,
this is what we refer to as “unsigned”.
Unsigned numbers are always positive from 00, right up to FF.
Signed
There
will be times however, where you’ll need negative numbers to get some sort of
calculation done. This is where “signed”
comes in.
When
a byte of data is treated as signed; 00 to 7F is treated as positive, while 80
to FF is treated as negative:
|
Unsigned |
|||
|
Lowest
number |
Highest number |
||
|
00 01 02 03 04 05 06 07 ... |
... F8 F9 FA FB FC FD FE FF |
||
|
|
|||
|
Signed |
|||
|
Lowest
number |
Highest number |
||
|
80 81
82 83 ... |
... FC FD FE FF 00 01 02 03 04 ... |
... 7C 7D 7E 7F |
|
As
you can see for “signed” the numbers 80 to FF are red for negative:
|
... |
||
|
03 |
= |
+3 |
|
02 |
= |
+2 |
|
01 |
= |
+1 |
|
00 |
= |
0 |
|
FF |
= |
-1 |
|
FE |
= |
-2 |
|
FD |
= |
-3 |
|
... |
||
This
is a process known as "Two's Complement", it is the idea of using the MSB (most
significant bit) to signify the value is negative:
|
80 = 1000 0000 |
All
numbers from 80 to FF will have that bit 1 (set),
while the positive numbers from 00 to 7F will have it 0 (clear).
For
word and long-word, the same applies:
|
8000 = 1000 0000 0000
0000 |
|
80000000 = 1000 0000 0000
0000 0000 0000 0000 0000 |
It’s
important not to confuse byte, word and long-word at this stage, for good
reasons.
The
byte FC is -4, but if you were to read FC as a word; 00FC, then the number is no longer
negative, it is read as positive for word because it is between 0000 and 7FFF. To have it as -4 for word, you need to extend it as FFFC. The same goes for
long-word, the number needs to be extended.
FFFFFFFC is -4 for long-word.
Conversion
It
should be made very clear however,
that simply changing the MSB will not convert the polarity of a number from
positive to negative, or negative to positive directly. For example, the number 04 in binary 0000 0100. Setting the MSB will give us 1000 0100 in hex 84, while 84 is
a negative number, it is not -4, it is in fact -7C.
Negative
numbers start from FF (which is -1), and proceed
downwards, FE (-2), FD (-3), FC (-4), and so on...
There
is of course a simpler method to convert a number from positive to negative (or
negative to positive), which can be done by performing a "logical compliment" onto the number (you
may remember the "NOT" instruction), and then adding 1 to the number afterwards,
for example, the positive number 04 (0000 0100), perform "logical compliment" by swapping the bits:
|
0000 0100 |||| |||| VVVV VVVV 1111 1011 |
This
gives us FB (1111 1011), then adding 1 to it will
give us FC (-4), and that is the correct conversion, it also
works for converting negative numbers into positive. It is a strong belief that the CPU does this
internally for the "NEG" instruction (which will be explained shortly
in the next part).
Further
info
Your
next question may probably be “how do I treat a number as unsigned or signed?”. It’s a good
question, the answer is simple, but not easy to explain, so bear with me
please.
The
68k has what is known as a “Conditional Code Register” (CCR), and every time an
instruction is processed, information is saved to the CCR, this information can
include:
As
you can see, information for both signed and unsigned are
stored away in the CCR. So to sum it up,
the number is treated as both signed and unsigned in most cases.
There
are instructions that we’ll get to later on, that will read the CCR and perform
an action depending on the signed or unsigned results. So you’ll have complete control over how the
number is treated.
Important:
The above information
about the CCR is very minor; more information on the CCR will be explained
later on in time. It is explained here
briefly so you can understand when a number is treated as signed or unsigned,
and nothing more. So don’t worry if you
don’t quite understand the CCR just now, you’re not expected to.