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 + 27

18 – 7

21 – 13

59 – 96

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