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Sunday, April 3, 2011

Computer Architecture # 02 : Data Representation : AN EARLY LOOK AT COMPUTER ARITHMETIC(7)

We will explore computer arithmetic in detail in Chapter 3, but for the moment, we need to learn how to perform simple binary addition because it is used in representing signed binary numbers. Binary addition is performed similar to the way we perform decimal addition by hand, as illustrated in Figure 2-5. Two binary numbers A and B are added from right to left, creating a sum and a carry in each bit position. Since the rightmost bits of A and B can each assume one of two values, four cases must be considered: 0 + 0, 0 + 1, 1 + 0, and 1 + 1, with a carry of 0, as shown in the figure.

The carry into the rightmost bit position defaults to 0. For the remaining bit positions, the carry into the position can be 0 or 1, so that a total of eight input combinations must be considered as shown in the figure. 

Figure 2-5    Example of binary addition.

Notice that the largest number we can represent using the eight-bit format shown in Figure 2-5 is (11111111)2 = (255)10 and that the smallest number that can be represented is (00000000)2 = (0)10. The bit patterns 11111111 and 00000000 and all of the intermediate bit patterns represent numbers on the closed interval from 0 to 255, which are all positive numbers. Up to this point we have considered only unsigned numbers, but we need to represent signed numbers as well, in which (approximately) one half of the bit patterns is assigned to positive numbers and the other half is assigned to negative numbers. Four common representations for base 2 signed numbers are discussed in the next section.

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