B1 Intermediate 1 Folder Collection
After playing the video, you can click or select the word to look it up in the dictionary.
Loading...
Report Subtitle Errors
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 that – in 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 on – the 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").
    You must  Log in  to get the function.
Tip: Click on the article or the word in the subtitle to get translation quickly!

Loading…

1.2.6 Signed Integers: 2's complement

1 Folder Collection
林宜悉 published on March 30, 2020
More Recommended Videos
  1. 1. Search word

    Select word on the caption to look it up in the dictionary!

  2. 2. Repeat single sentence

    Repeat the same sentence to enhance listening ability

  3. 3. Shortcut

    Shortcut!

  4. 4. Close caption

    Close the English caption

  5. 5. Embed

    Embed the video to your blog

  6. 6. Unfold

    Hide right panel

  1. Listening Quiz

    Listening Quiz!

  1. Click to open your notebook

  1. UrbanDictionary 俚語字典整合查詢。一般字典查詢不到你滿意的解譯,不妨使用「俚語字典」,或許會讓你有滿意的答案喔