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Wednesday, September 7, 2011

Computer Architecture # 03 : Arithmetic: ONE’S COMPLEMENT ADDITION AND SUBTRACTION (5)

Although it is not heavily used in mainstream computing anymore, the one’s complement representation was used in early computers. 
One’s complement addition is handled somewhat differently from two’s complement addition: the carry out of the leftmost position is not discarded, but is added back into the least significant position of the integer portion as shown in  Figure 3-7. 
This is known as an end-around carry

We can better visualize the reason that the end-around carry is needed by examining the 3-bit one’s complement number circle in Figure 3-8. 
Notice that the number circle has two positions for 0. When we add two numbers, if we traverse through both 0 and +0, then we must compensate for the fact that 0 is visited twice. The end-around carry advances the result by one position for this situation.
Notice that the distance between 0 and +0 on the number circle is the distance between two integers, and is not the distance between two successive representable numbers. As an illustration of this point, consider adding (5.5)10 and (1.0)10 in one’s complement arithmetic, which is shown in Figure 3-9. (Note that we can also treat this as a subtraction problem, in which the subtrahend is negated by complementing all of the bits, before adding it to the minuend.) 
In order to add (+5.5)10 and (1.0)10 and obtain the correct result in one’s complement, we add the end-around carry into the one’s position as shown. This adds complexity to our number circle, because in the gap between +0 and 0, there are valid numbers that represent fractions that are less than 0, yet they appear on the number circle before 0 appears. If the number circle is reordered to avoid this anomaly, then addition must be handled in a more complex manner. The need to look for two different representations for zero, and the potential need to perform another addition for the end-around carry are two important reasons for preferring the two’s complement arithmetic to one’s complement arithmetic.

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