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The hexadecimal notation requires sixteen symbols; the symbols 0 through 9 are therefore augmented by the letters ABCDEF. The codes owe their popularity to the special relationship which they have with three- and four-bit binary numbers. 2a. 2b. 2: Write down the octal and hex equivalents of 179 10 . 3 that the eight possible combinations of three binary digits can be represented by the eight symbols of the octal system. Similarly, the sixteen combinations of four bits can be represented by the sixteen symbols 46 FUNDAMENTALS OF MODERN DIGITAL SYSTEMS sz a' 80 Decimal 0 0 0 0 0 1 0 1 0 0 0 0 1 7 1 7 0 1 8 0 1 7 0 7 0 63 64 ..

Normally it is arranged that numbers are interpreted throughout as either integers or fractions and any scaling necessary is left to the user. However, we must be able to represent positive or negative and the most significant digit is reserved for this purpose. Consider a 4-bit register. 5 shows the various ways in which the bit pattern can be interpreted. It is interesting to note that the allocation of a bit to represent the sign has not resulted in the loss of one-half of the number capacity, but has merely shifted the axis on the number scale.

Such combinations give rise to don't care conditions, in which we can assume the output value to be either zero or one, whichever is the more convenient in our design. Because they introduce the element of choice, don't care conditions are often helpful in providing simpler expressions. Bearing these points in mind, a careful inspection of the truth table can lead to an optimal circuit. 17: In a certain system with four input signals A, B, C, D, only ten combinations of the signals ever occur. 29.

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