Chapter 7 Arithmetic Operations and Circuits 1
Feb 23, 2016
Chapter 7
Arithmetic Operations and Circuits
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7-1 Binary Arithmetic
• Addition– When the sum exceeds 1, carry a 1 over to the
next-more-significant column.– 0 + 0 = 0 carry 0– 0 + 1 = 1 carry 0– 1 + 0 = 1 carry 0– 1 + 1 = 0 carry 1
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Binary Arithmetic• Addition
– General form A0 + B0 = 0 + Cout
• Summation symbol ()• Carry-out (Cout)
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Binary Arithmetic– Carry-out is added to the next-more-significant
column as a carry-in.
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Binary Arithmetic
• Subtraction– 0 0 = 0 borrow 0– 0 1 = 1 borrow 1– 1 0 = 1 borrow 0– 1 1 = 0 borrow 0
• General form A0 B0 = R0 + Bout
– Remainder is R0
– Borrow is Bout
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Binary Arithmetic
• Subtraction– When A0 borrows from its left, A0 increases by
210.
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Binary Arithmetic
• Multiplication– Multiply the 20 bit of the multiplier times the
multiplicand.– Multiply the 21 bit of the multiplier times the
multiplicand. Shift the result one position to the left.
– Repeat step 2 for the 22 bit of the multiplier, and for all remaining bits.
– Take the sum of the partial products to get the final product.
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Binary Arithmetic• Multiplication
– Very similar to multiplying decimal numbers.
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Binary Arithmetic• Division
–The same as decimal division. –This process is illustrated in
Example 7-4.
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Example 7-4
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Example 7-4 (Continued)
7-2 Two’s-Complement Representation• Both positive and negative numbers can be
represented• Binary subtraction is simplified• Groups of eight• Most significant bit (MSB) signifies positive or
negative
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Two’s-Complement Representation
• Sign bit– 0 for positive– 1 for negative
• Range of positive numbers (8-bit)– 0000 0000 to 0111 1111 (0 to 127)– Maximum positive number: 2N-1-1
• Range of negative numbers (8-bit)– 1111 1111 to 1000 0000 (-1 to -128)– Minimum negative number: -2N-1
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Decimal-to-Two’s-Complement Conversion
• If a number is positive, – the two’s complement number is the true binary
equivalent of the decimal number.• If a number is negative:
– Complement each bit (one’s complement)– Add 1 to the one’s complement
• The sign bit will always end up a 1.
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Two’s-Complement Representation
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Two’s-Complement-to-Decimal Conversion
• If the number is positive (sign bit = 0), convert directly
• If the number is negative:– Complement the entire two’s-complement
number– Add 1– Do the regular b-to-d conversion to get the
decimal numeric value– Result will be a negative number
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Discussion Point
• Convert the following numbers to two’s-complement form:3510
-3510
• Convert the following two’s-complement number to decimal:1101 1101
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7-3 Two’s-Complement Arithmetic
• Addition– Regular binary addition
• Subtraction– Convert number to be subtracted to a negative
two’s-complement number– Regular binary addition– Carry out of the MSB is ignored
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Discussion Point
• Add the following numbers using two’s complement arithmetic:
19 + 2718 – 721 – 1359 – 96
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