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Chapter 7

Chapter 7. Arithmetic Operations and Circuits. 1. 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. 5. Binary Arithmetic. Addition

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Chapter 7

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  1. Chapter 7 Arithmetic Operations and Circuits 1

  2. 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 5

  3. Binary Arithmetic • Addition • General form A0 + B0 = 0 + Cout • Summation symbol () • Carry-out (Cout) 6

  4. Binary Arithmetic • Carry-out is added to the next-more-significant column as a carry-in. 7

  5. 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 8

  6. 9

  7. Binary Arithmetic • Subtraction • When A0 borrows from its left, A0 increases by 210. 10

  8. 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. 11

  9. Binary Arithmetic • Multiplication • Very similar to multiplying decimal numbers. 12

  10. Binary Arithmetic • Division • The same as decimal division. • This process is illustrated in Example 7-4. 13

  11. Example 7-4 14

  12. Example 7-4 (Continued) 14

  13. 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 15

  14. 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 16

  15. 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. 18

  16. Two’s-Complement Representation 18

  17. 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 19

  18. Discussion Point • Convert the following numbers to two’s-complement form: 3510 -3510 • Convert the following two’s-complement number to decimal: 1101 1101 20

  19. 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 21

  20. Discussion Point • Add the following numbers using two’s complement arithmetic: 19 + 27 18 – 7 21 – 13 59 – 96 22

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