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Unit I From Fundamentals of Logic Design by Roth and Kinney

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1.1 Digital Systems and Switching Circuits Basic Switching Circuit—Figure 1.1, page 7

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1.2 Number Systems and Conversions Representations: Power series –Let R be the base –R n then n is a whole number (integer) –a n is the coefficient of the power series term. To convert numbers into decimal, use power series.

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Conversions Example, page 10, use the division method to convert a whole number decimal, into another base. Example, page 11, use the multiplication method to convert a fraction in decimal, into another base. Example, page 10 (second example) illustrates that the fraction process does not always terminate. Example, page 12 illustrates how to convert a whole number and a fraction.

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1.3 Binary Arithmetic Basic Operations –0+0 =0; 0+1 =1; 1+0=1; 1+1=0 and carry 1 to next column. Example : Binary Addition—page 13 Example: Binary Subtraction—page 13. Note that 1’s are used to indicate carries and borrows. (Note that borrows can propagate.)

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Binary Subtraction Can be confusing; when you borrow from a column n, you subtract 1 from that column and add 2 to the next column, n-1; this is because of the power series concepts.

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1.4 Representation of Negative Numbers (Table 1.1, page16) Sign and Magnitude 0=+, 1= - Two’s complement –Let N be a binary number. –Complement each bit and add 1 –Addition—add the numbers and ignore any carries from the sign position. One’s complement –Complement each bit; in addition add the last carry.

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Overflows Overflow—page 16-- “if an operation, such as addition and subtraction, is performed on two numbers and the result is outside the range of representation, then we say that an overflow has occurred.” Example: Addition of two positive numbers results in a negative value.

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1.5 Binary Codes Binary Coded Decimal (BCD) or 8-4-2-1 –Note that only 10 values are valid. Table 1-2, page 22 shows BCD and other codes. Table 1-3 is ASCII.

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Negative Number Sign-Magnitude: left-most bit as the sign bit –16 bits –Example: 4-bit numbers is given by is given by ’s complement:

Negative Number Sign-Magnitude: left-most bit as the sign bit –16 bits –Example: 4-bit numbers is given by is given by ’s complement:

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