Chapter 3

Page 42

  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
        sumi = (xi + yi) % 2           // remainder
        carry = (xi + yi) / 2         // integer division
  1. Hexadecimal addition
    Let carry = 0
    Repeat for each i = 0,...,(n - 1)  // starting in ones place
        sumi = (xi) + yi) % 16           // remainder
        carry = (xi + yi) / 16         // integer division
  1. Binary subtraction
    Let borrow = 0
    Repeat for i = 0,··· ,(N − 1)
    If yi ≤ xi 
        Let differencei = xi − yi
    Else
        Let j = i + 1
        While (xi = 0) and (j < N)
            Add 1 to j
        If j = N
            Let borrow = 1
            Subtract 1 from j
            Add 2 to xi
        While j > i
            Subtract 1 from xi
            Subtract 1 from j
            Add 2 to xi
        Let differencei = xi − yi
  1. Hexadecimal subtracton
    Let borrow = 0
    Repeat for i = 0,··· ,(N − 1)
    If yi ≤ xi 
        Let differencei = xi − yi
    Else
        Let j = i + 1
        While (xi = 0) and (j < N)
            Add 1 to j
        If j = N
            Let borrow = 1
            Subtract 1 from j
            Add 16 to xi
        While j > i
            Subtract 1 from xi
            Subtract 1 from j
            Add 16 to xi
        Let differencei = xi − yi

Page 49

  1. 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
  1. 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
  1. Two’s complement binary to signed decimal
    • 0x1234 = +4660
    • 0xffff = -1
    • 0x8000 = -32768
    • 0x7fff = +32767
  2. Signed decimal to 2s complement binary
    • +1024 = 0x0400
    • -1024 = 0xfc00
    • -256 = 0xff00
    • -32767 0x8001

Page 54

  1. Three-bit arithmetic using Decoder Ring
    • 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 CF = 0, and the result is right.
    • 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 CF = 0, and the result is right.
    • 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 CF = 1, and the result is wrong.
    • 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 OF = 1, and the result is wrong.
    • 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 OF = 1, and the result is wrong.
    • 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 OF = 0, and the result is right.
  2. Eight-bit addition, unsigned and signed
    • 0x55 + 0xaa = 0xff, unsigned right, signed right
    • 0x55 + 0xf0 = 0x45, unsigned wrong (CF), signed right
    • 0x80 + 0x7b = 0xfb, unsigned right, signed right
    • 0x63 + 0x7b = 0xde, unsigned right, signed wrong (OF)
    • 0x0f + 0xff = 0x0e, unsigned wrong (CF), signed right
    • 0x80 + 0x80 = 0x00, unsigned wrong (CF), signed wrong (OF)
  3. Sixteen-bit addition, unsigned and signed
    • 0x1234 + 0xedcc = 0x0000, unsigned wrong (CF), signed right
    • 0x1234 + 0xfedc = 0x1110, unsigned wrong (CF), signed right
    • 0x8000 + 0x8000 = 0x0000, unsigned wrong (CF), signed wrong (OF)
    • 0x0400 + 0xffff = 0x03ff, unsigned wrong (CF), signed right
    • 0x07d0 + 0x782f = 0x7fff, unsigned right, signed right
    • 0x8000 + 0xffff = 0x7fff, unsigned wrong (CF), signed wrong (OF)