5.
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We then presented the methods for converting numbers in one base to the other. A number can be converted to base 10 simply by expanding the positional notation. Base 10 can be converted to another base by successive division. Since 8 and 16 are powers of 2, there is a simple method for mapping between these bases. All we need do is group the binary representation into adjacent groups of three bits for base 8 and four bits for base 16.
Finally, we looked at the mechanics of binary addition and subtraction, using our intuition about how these operations work for base 10. For the addition operation, when the column sum exceeds the quantity that can be represented by a single digit, the "overflow" amount is carried over to the next higher-order column. Similarly for subtraction, when the column difference is negative, we must borrow an amount equal to the underlying base from the adjacent higher-order column. This guarantees that the column difference yields a nonnegative result. Both carries and borrows can cascade from the rightmost columns toward the left columns.
In this appendix, we concentrated on the representation of nonnegative numbers in a variety of alternative number bases. We will study number representations in more detail in Chapter 5. In particular, we will examine alternative methods for representing negative numbers in binary, as well as the detailed circuitry for implementing binary addition and subtraction.
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This file last updated on 07/16/96 at 05:10:10.
Appendix Review
In this appendix, we have examined the methods for representing numbers in positional notation. We use positional notation to represent numbers in a variety of different bases, including base 10(decimal), base 2 (binary), base 8 (octal), and base 16 (hexadecimal). Of course, it is also possible to represent numbers in bases other than these four.We then presented the methods for converting numbers in one base to the other. A number can be converted to base 10 simply by expanding the positional notation. Base 10 can be converted to another base by successive division. Since 8 and 16 are powers of 2, there is a simple method for mapping between these bases. All we need do is group the binary representation into adjacent groups of three bits for base 8 and four bits for base 16.
Finally, we looked at the mechanics of binary addition and subtraction, using our intuition about how these operations work for base 10. For the addition operation, when the column sum exceeds the quantity that can be represented by a single digit, the "overflow" amount is carried over to the next higher-order column. Similarly for subtraction, when the column difference is negative, we must borrow an amount equal to the underlying base from the adjacent higher-order column. This guarantees that the column difference yields a nonnegative result. Both carries and borrows can cascade from the rightmost columns toward the left columns.
In this appendix, we concentrated on the representation of nonnegative numbers in a variety of alternative number bases. We will study number representations in more detail in Chapter 5. In particular, we will examine alternative methods for representing negative numbers in binary, as well as the detailed circuitry for implementing binary addition and subtraction.
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