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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

Chapter 7

Arithmetic Operations and Circuits

1

7 1 binary arithmetic
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
Binary Arithmetic
  • Addition
    • General form A0 + B0 = 0 + Cout
      • Summation symbol ()
      • Carry-out (Cout)

6

binary arithmetic1
Binary Arithmetic
  • Carry-out is added to the next-more-significant column as a carry-in.

7

binary arithmetic2
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

binary arithmetic3
Binary Arithmetic
  • Subtraction
    • When A0 borrows from its left, A0 increases by 210.

10

binary arithmetic4
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

binary arithmetic5
Binary Arithmetic
  • Multiplication
    • Very similar to multiplying decimal numbers.

12

binary arithmetic6
Binary Arithmetic
  • Division
    • The same as decimal division.
    • This process is illustrated in Example 7-4.

13

7 2 two s complement representation
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

two s complement representation
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

decimal to two s complement conversion
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

two s complement to decimal conversion
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

discussion point
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

7 3 two s complement arithmetic
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

discussion point1
Discussion Point
  • Add the following numbers using two’s complement arithmetic:

19 + 27

18 – 7

21 – 13

59 – 96

22

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