Question

Assume we are using the simple model for floating-point representation as given in the book (the representation uses a 14-bit format, 5 bits for the exponent with a bias of 16, a normalized mantissa of 8 bits, and a single sign bit for the number). Show how the computer would represent the numbers 96.5 and -0.75 using this floating-point format. [10]

Answer 1

a)
Converting 96.5 to binary
   Convert decimal part first, then the fractional part
   > First convert 96 to binary
   Divide 96 successively by 2 until the quotient is 0
      > 96/2 = 48, remainder is 0
      > 48/2 = 24, remainder is 0
      > 24/2 = 12, remainder is 0
      > 12/2 = 6, remainder is 0
      > 6/2 = 3, remainder is 0
      > 3/2 = 1, remainder is 1
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 1100000
   So, 96 of decimal is 1100000 in binary
   > Now, Convert 0.50000000 to binary
      > Multiply 0.50000000 with 2.  Since 1.00000000 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.5 of decimal is .1 in binary
   so, 96.5 in binary is 1100000.1
96.5 in simple binary => 1100000.1
so, 96.5 in normal binary is 1100000.1 => 1.1000001 * 2^6

14-bit format:
--------------------
sign bit is 0(+ve)
exp bits are (16+6=22) => 10110
   Divide 22 successively by 2 until the quotient is 0
      > 22/2 = 11, remainder is 0
      > 11/2 = 5, remainder is 1
      > 5/2 = 2, remainder is 1
      > 2/2 = 1, remainder is 0
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10110
   So, 22 of decimal is 10110 in binary
frac bits are 10000010

so, 96.5 in 14-bit format is 0 10110 10000010

b)
Converting 0.75 to binary
   Convert decimal part first, then the fractional part
   > First convert 0 to binary
   Divide 0 successively by 2 until the quotient is 0
   Read remainders from the bottom to top as 
   So, 0 of decimal is  in binary
   > Now, Convert 0.75000000 to binary
      > Multiply 0.75000000 with 2.  Since 1.50000000 is >= 1. then add 1 to result
      > Multiply 0.50000000 with 2.  Since 1.00000000 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.75 of decimal is .11 in binary
   so, 0.75 in binary is .11
-0.75 in simple binary => .11
so, -0.75 in normal binary is .11 => 1.1 * 2^-1

14-bit format:
----------------
sign bit is 1(-ve)
exp bits are (16-1=15) => 01111
   Divide 15 successively by 2 until the quotient is 0
      > 15/2 = 7, remainder is 1
      > 7/2 = 3, remainder is 1
      > 3/2 = 1, remainder is 1
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 1111
   So, 15 of decimal is 1111 in binary
frac bits are 10000000

so, -0.75 in 14-bit format is 1 01111 10000000