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• Our final challenge is figuring out how to represent signed integers, for example, what

• should be our representation for the number -2000?

• In decimal notation, the convention is to precede the number with a "+" or "-" to indicate

• whether it's positive or negative, usually omitting the "+" to simplify the notation

• for positive numbers. We could adopt a similar notation -- called

• "signed magnitude" -- in binary, by allocating a separate bit at the front of the binary

• string to indicate the sign, say "0" for positive numbers and "1" for negative numbers.

• So the signed-magnitude representation for -2000 would be an initial "1" to indicate

• a negative number, followed by the representation for 2000 (as described on the previous two

• slides). However there are some complications in using

• a signed-magnitude representation. There are two possible binary representations

• for zero: "+0" and "-0". This makes the encoding slightly inefficient

• but, more importantly, the circuitry for doing addition of signed-magnitude numbers is different

• than the circuitry for doing subtraction. Of course, we're used to thatin elementary

• school we learned one technique for addition and another for subtraction.

• To keep the circuitry simple, most modern digital systems use the two's complement binary

• representation for signed numbers. In this representation, the high-order bit

• of an N-bit two's complement number has a negative weight, as shown in the figure.

• Thus all negative numbers have a 1 in the high-order bit and, in that sense, the high-order

• bit is serving as the "sign bit" – if it's 1, the represented number is negative.

• The most negative N-bit number has a 1-bit in the high-order position, representing the

• value -2^(N-1). The most positive N-bit number has a 0 in

• the negative-weight high-order bit and 1's for all the positive-weight bits, representing

• the value 2^(N-1)-1. This gives us the range of possible values

• for example, in an 8-bit two's complement representation, the most negative number is

• -2^7 = -128 and the most positive number is 2^7 – 1 = 127.

• If all N bits are 1, think of that as the sum of the most negative number with the most

• positive number, i.e., -2^(N-1) + 2^(N-1)-1, which equals -1.

• And, of course, if all N bits are 0, that's the unique representation of 0.

• Let's see what happens when we add the N-bit values for -1 and 1, keeping an N-bit answer.

• In the rightmost column, 1 plus 1 is 0, carry the 1.

• In the second column, the carry of 1 plus 1 plus 0 is 0, carry the 1.

• And so onthe result is all zero's, the representation for 0… perfect!

• Notice that we just used ordinary binary addition, even when one or both of the operands are

• negative. Two's complement is perfect for N-bit arithmetic!

• To compute B - A, we can just use addition and compute B + (-A).

• So now we just need to figure out the two's complement representation for –A, given

• the two's complement representation for A. Well, we know that A + (-A) = 0 and using

• the example above, we can rewrite 0 as 1 + (-1).

• Reorganizing terms, we see that –A equals 1 plus the quantity (-1) – A.

• As we saw above, the two's complement representation for -1 is all 1-bits, so we can write that

• subtraction as all 1's minus the individual bits of A: A_0, A_1, … up to A_N-1.

• If a particular bit A_i is 0, then 1-A_i = 1 and if A_i is 1, then 1-A_i = 0.

• So in each column, the result is the bitwise complement of A_i, which we'll write using

• the C-language bitwise complement operator tilde.

• So we see that –A equals the bitwise complement of A plus 1.

• Ta-dah! To practice your skill with two's complement,

• try your hand at the following exercises. All you need to remember is how to do binary

• addition and two's complement negation (which is "bitwise complement and add 1").

Our final challenge is figuring out how to represent signed integers, for example, what

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B1 complement representation negative signed binary positive

1.2.6 Signed Integers: 2's complement

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林宜悉 posted on 2020/03/29
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