MESSAGE
| DATE | 2014-09-23 |
| FROM | Ruben Safir
|
| SUBJECT | Subject: [NYLXS - HANGOUT] Negitive binary representations
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Anyone ever see this before?
http://www.allaboutcircuits.com/vol_4/chpt_2/3.html
Negative binary numbers
With addition being easily accomplished, we can perform the operation of
subtraction with the same technique simply by making one of the numbers
negative. For example, the subtraction problem of 7 - 5 is essentially
the same as the addition problem 7 + (-5). Since we already know how to
represent positive numbers in binary, all we need to know now is how to
represent their negative counterparts and we'll be able to subtract.
Usually we represent a negative decimal number by placing a minus sign
directly to the left of the most significant digit, just as in the
example above, with -5. However, the whole purpose of using binary
notation is for constructing on/off circuits that can represent bit
values in terms of voltage (2 alternative values: either "high" or
"low"). In this context, we don't have the luxury of a third symbol such
as a "minus" sign, since these circuits can only be on or off (two
possible states). One solution is to reserve a bit (circuit) that does
nothing but represent the mathematical sign:
. 1012 = 510 (positive)
.
. Extra bit, representing sign (0=positive, 1=negative)
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. 01012 = 510 (positive)
.
. Extra bit, representing sign (0=positive, 1=negative)
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. 11012 = -510 (negative)
As you can see, we have to be careful when we start using bits for any
purpose other than standard place-weighted values. Otherwise, 11012
could be misinterpreted as the number thirteen when in fact we mean to
represent negative five. To keep things straight here, we must first
decide how many bits are going to be needed to represent the largest
numbers we'll be dealing with, and then be sure not to exceed that bit
field length in our arithmetic operations. For the above example, I've
limited myself to the representation of numbers from negative seven
(11112) to positive seven (01112), and no more, by making the fourth bit
the "sign" bit. Only by first establishing these limits can I avoid
confusion of a negative number with a larger, positive number.
Representing negative five as 11012 is an example of the sign-magnitude
system of negative binary numeration. By using the leftmost bit as a
sign indicator and not a place-weighted value, I am sacrificing the
"pure" form of binary notation for something that gives me a practical
advantage: the representation of negative numbers. The leftmost bit is
read as the sign, either positive or negative, and the remaining bits
are interpreted according to the standard binary notation: left to
right, place weights in multiples of two.
As simple as the sign-magnitude approach is, it is not very practical
for arithmetic purposes. For instance, how do I add a negative five
(11012) to any other number, using the standard technique for binary
addition? I'd have to invent a new way of doing addition in order for it
to work, and if I do that, I might as well just do the job with longhand
subtraction; there's no arithmetical advantage to using negative numbers
to perform subtraction through addition if we have to do it with
sign-magnitude numeration, and that was our goal!
There's another method for representing negative numbers which works
with our familiar technique of longhand addition, and also happens to
make more sense from a place-weighted numeration point of view, called
complementation. With this strategy, we assign the leftmost bit to serve
a special purpose, just as we did with the sign-magnitude approach,
defining our number limits just as before. However, this time, the
leftmost bit is more than just a sign bit; rather, it possesses a
negative place-weight value. For example, a value of negative five would
be represented as such:
Extra bit, place weight = negative eight
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. 10112 = 510 (negative)
.
. (1 x -810) + (0 x 410) + (1 x 210) + (1 x 110) = -510
With the right three bits being able to represent a magnitude from zero
through seven, and the leftmost bit representing either zero or negative
eight, we can successfully represent any integer number from negative
seven (10012 = -810 + 110 = -710) to positive seven (01112 = 010 + 710 =
710).
Representing positive numbers in this scheme (with the fourth bit
designated as the negative weight) is no different from that of ordinary
binary notation. However, representing negative numbers is not quite as
straightforward:
zero 0000
positive one 0001 negative one 1111
positive two 0010 negative two 1110
positive three 0011 negative three 1101
positive four 0100 negative four 1100
positive five 0101 negative five 1011
positive six 0110 negative six 1010
positive seven 0111 negative seven 1001
. negative eight 1000
Note that the negative binary numbers in the right column, being the sum
of the right three bits' total plus the negative eight of the leftmost
bit, don't "count" in the same progression as the positive binary
numbers in the left column. Rather, the right three bits have to be set
at the proper value to equal the desired (negative) total when summed
with the negative eight place value of the leftmost bit.
Those right three bits are referred to as the two's complement of the
corresponding positive number. Consider the following comparison:
positive number two's complement
--------------- ----------------
001 111
010 110
011 101
100 100
101 011
110 010
111 001
In this case, with the negative weight bit being the fourth bit (place
value of negative eight), the two's complement for any positive number
will be whatever value is needed to add to negative eight to make that
positive value's negative equivalent. Thankfully, there's an easy way to
figure out the two's complement for any binary number: simply invert all
the bits of that number, changing all 1's to 0's and vice versa (to
arrive at what is called the one's complement) and then add one! For
example, to obtain the two's complement of five (1012), we would first
invert all the bits to obtain 0102 (the "one's complement"), then add
one to obtain 0112, or -510 in three-bit, two's complement form.
Interestingly enough, generating the two's complement of a binary number
works the same if you manipulate all the bits, including the leftmost
(sign) bit at the same time as the magnitude bits. Let's try this with
the former example, converting a positive five to a negative five, but
performing the complementation process on all four bits. We must be sure
to include the 0 (positive) sign bit on the original number, five
(01012). First, inverting all bits to obtain the one's complement:
10102. Then, adding one, we obtain the final answer: 10112, or -510
expressed in four-bit, two's complement form.
It is critically important to remember that the place of the
negative-weight bit must be already determined before any two's
complement conversions can be done. If our binary numeration field were
such that the eighth bit was designated as the negative-weight bit
(100000002), we'd have to determine the two's complement based on all
seven of the other bits. Here, the two's complement of five (00001012)
would be 11110112. A positive five in this system would be represented
as 000001012, and a negative five as 111110112.
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