Chapter 3

  1. Four bits are required to store a single decimal digit. Many codes could be used. This one uses the binary number system.

    digit code digit code
    0 0000 5 0101
    1 0001 6 0110
    2 0010 7 0111
    3 0011 8 1000
    4 0100 9 1001

  2. Binary addition

     Let carry = 0
 Repeat for each i = 0,...,(n - 1)  // starting in ones place
     sum<sub>i</sub> = (x<sub>i</sub> + y<sub>i</sub>) % 2           // remainder
     carry = (x<sub>i</sub> + y<sub>i</sub>) / 2         // integer division

  3. Hexadecimal addition

     Let carry = 0
 Repeat for each i = 0,...,(n - 1)  // starting in ones place
     sum<sub>i</sub> = (x<sub>i</sub>) + y<sub>i</sub>) % 16           // remainder
     carry = (x<sub>i</sub> + y<sub>i</sub>) / 16         // integer division

  4. Binary subtraction

     Let borrow = 0
 Repeat for i = 0,··· ,(N − 1)
 If y<sub>i</sub> ≤ x<sub>i</sub> 
     Let difference<sub>i</sub> = x<sub>i</sub> − y<sub>i</sub>
 Else
     Let j = i + 1
     While (x<sub>i</sub> = 0) and (j < N)
         Add 1 to j
     If j = N
         Let borrow = 1
         Subtract 1 from j
         Add 2 to x<sub>i</sub>
     While j > i
         Subtract 1 from x<sub>i</sub>
         Subtract 1 from j
         Add 2 to x<sub>i</sub>
     Let difference<sub>i</sub> = x<sub>i</sub> − y<sub>i</sub>

  5. Hexadecimal subtraction

     Let borrow = 0
 Repeat for i = 0,··· ,(N − 1)
 If y<sub>i</sub> ≤ x<sub>i</sub> 
     Let difference<sub>i</sub> = x<sub>i</sub> − y<sub>i</sub>
 Else
     Let j = i + 1
     While (x<sub>i</sub> = 0) and (j < N)
         Add 1 to j
     If j = N
         Let borrow = 1
         Subtract 1 from j
         Add 16 to x<sub>i</sub>
     While j > i
         Subtract 1 from x<sub>i</sub>
         Subtract 1 from j
         Add 16 to x<sub>i</sub>
     Let difference<sub>i</sub> = x<sub>i</sub> − y<sub>i</sub>

  6. Signed decimal to two’s complement binary

     If x >= 0
     Convert x to binary
 Else
     Negate x
     Convert the result to binary
     Compute the 2s complement of the result in the binary domain

  7. Two’s complement in binary to signed decimal

     If high-order bit of x is 0
     Convert x to decimal
 Else
     Compute the 2s complement of x
     Compute the decimal equivalent of the result
     Place a minus sign in front of the decimal equivalent
  8. Two’s complement binary to signed decimal
    1. 0x1234 = +4660
    2. 0xffff = -1
    3. 0x8000 = -32768
    4. 0x7fff = +32767
  9. Signed decimal to 2s complement binary
    1. +1024 = 0x0400
    2. -1024 = 0xfc00
    3. -256 = 0xff00
    4. -32767 0x8001
  10. Three-bit arithmetic using Decoder Ring
    1. Start at the tic mark for 1, move 3 tic marks CW, giving 100 = 4. We did not pass the tic mark at the top, so C = 0, and the result is right.
    2. Start at the tic mark for 3, move 4 tic marks CW, giving 111 = 7. We did not pass the tic mark at the top, so C = 0, and the result is right.
    3. Start at the tic mark for 5, move 6 tic marks CW, giving 011 = 3. We did pass the tic mark at the top, so C = 1, and the result is wrong.
    4. Start at the tic mark for +1, move 3 tic marks CW, giving 101 = -3. We did pass the tic mark at the bottom, so V = 1, and the result is wrong.
    5. Start at the tic mark for -3, move 3 tic marks CCW, giving 010 = +2. We did pass the tic mark at the bottom, so V = 1, and the result is wrong.
    6. Start at the tic mark for +3, move 4 tic marks CCW, giving 111 = -1. We did not pass the tic mark at the bottom, so V = 0, and the result is right.
  11. Eight-bit addition, unsigned and signed
    1. 0x55 + 0xaa = 0xff, unsigned right, signed right
    2. 0x55 + 0xf0 = 0x45, unsigned wrong (C), signed right
    3. 0x80 + 0x7b = 0xfb, unsigned right, signed right
    4. 0x63 + 0x7b = 0xde, unsigned right, signed wrong (V)
    5. 0x0f + 0xff = 0x0e, unsigned wrong (C), signed right
    6. 0x80 + 0x80 = 0x00, unsigned wrong (C), signed wrong (V)
  12. Sixteen-bit addition, unsigned and signed
    1. 0x1234 + 0xedcc = 0x0000, unsigned wrong (C), signed right
    2. 0x1234 + 0xfedc = 0x1110, unsigned wrong (C), signed right
    3. 0x8000 + 0x8000 = 0x0000, unsigned wrong (C), signed wrong (V)
    4. 0x0400 + 0xffff = 0x03ff, unsigned wrong (C), signed right
    5. 0x07d0 + 0x782f = 0x7fff, unsigned right, signed right
    6. 0x8000 + 0xffff = 0x7fff, unsigned wrong (C), signed wrong (V)