Twos Complement NotationDate: 12/17/2001 at 09:42:16 From: Klaartje van Riemsdijk Subject: Sign byte I'm learning to make computer programs. I'm a fourth-generation programmer so I have not learned enough about bits and bytes. Could you answer the question how to make a 2-byte negative and at which place in the 16 bits the sign bit is placed and why there? Date: 12/17/2001 at 11:26:21 From: Doctor Tom Subject: Re: Sign byte Hello Klaartje, There are different ways to do it, depending on the architecture of the machine, but nowadays, almost every computer does it the same way, and it's called "two's complement." Here's the idea: We just pretend that we are looking at the bottom bits of an infinitely long number, so the number you write for decimal 5 is: 101 binary, or 00000000 00000101 in 16 bits or ...0000000000000000000000000000000000000101 in your imagination. All the positive numbers and zero have an infinite set of zeroes to the left. So to get -1, for example, just start with the infinite set of zeros (representing zero) and subtract 1. You have to keep carrying forever, but you eventually get: ...1111111111111111111111111111111111111111 or, in 16 bits, you represent -1 as: 11111111 11111111 You can do any negative number this way, to obtain: -2 = 11111111 11111110 -3 = 11111111 11111101 -4 = 11111111 11111100 and so on. The most negative number that can be represented in 16 bits is: -32768 = 10000000 00000000 The most positive number is: 32767 = 01111111 11111111 The highest order bit is 1 if the number is negative and zero if it's positive. Addition and subtraction are trivial - just do the addition all the way to the end. The hardware actually examines the carry bit out of the high-order position to see if overflow occurred, et cetera. There is a very easy way to find the representation of a negative number if you have the representation of the positive version. For example, if you know that 11 (decimal) is: 00000000 00001011 then to find -11, change all the bits: 11111111 11110100 and add 1: 11111111 11110101 It's easy to check that this is correct, since a positive and negative version of a number should add to zero. The positive and negative versions will have different bits in all the positions if we ignore the 1 that we added. Here's an example. Positive: 00101101 11011100 Negative: 11010010 00100011 +00000000 00000001 Add all three together. The sum of the first two is always: 11111111 11111111 and if we add 1, the carries go to infinity, making all zeroes. If you want details, look for a book on "computer arithmetic" and look up "two's complement notation" or "2s complement notation." - Doctor Tom, The Math Forum http://mathforum.org/dr.math/ |
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