Question

1. Convert the decimal real number -100.756 to IEEE 754 single precision (32 bit) floating point in hexadecimal representation of the binary value  (Hint: consider the field or bit-by-bit structure of an IEEE 754 value to decide which of the choices is correct, without necessarily constructing the full encoding of the value.)

   		32C98312
   		42C98312
   		C2C98312
   		52C98313

Answer 1

c) C2C98312

-100.756
Converting 100.756 to binary
   Convert decimal part first, then the fractional part
   > First convert 100 to binary
   Divide 100 successively by 2 until the quotient is 0
       > 100/2 = 50, remainder is 0
       > 50/2 = 25, remainder is 0
       > 25/2 = 12, remainder is 1
       > 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 1100100
   So, 100 of decimal is 1100100 in binary
   > Now, Convert 0.75600000 to binary
       > Multiply 0.75600000 with 2.    Since 1.51200000 is >= 1. then add 1 to result
       > Multiply 0.51200000 with 2.    Since 1.02400000 is >= 1. then add 1 to result
       > Multiply 0.02400000 with 2.    Since 0.04800000 is < 1. then add 0 to result
       > Multiply 0.04800000 with 2.    Since 0.09600000 is < 1. then add 0 to result
       > Multiply 0.09600000 with 2.    Since 0.19200000 is < 1. then add 0 to result
       > Multiply 0.19200000 with 2.    Since 0.38400000 is < 1. then add 0 to result
       > Multiply 0.38400000 with 2.    Since 0.76800000 is < 1. then add 0 to result
       > Multiply 0.76800000 with 2.    Since 1.53600000 is >= 1. then add 1 to result
       > Multiply 0.53600000 with 2.    Since 1.07200000 is >= 1. then add 1 to result
       > Multiply 0.07200000 with 2.    Since 0.14400000 is < 1. then add 0 to result
       > Multiply 0.14400000 with 2.    Since 0.28800000 is < 1. then add 0 to result
       > Multiply 0.28800000 with 2.    Since 0.57600000 is < 1. then add 0 to result
       > Multiply 0.57600000 with 2.    Since 1.15200000 is >= 1. then add 1 to result
       > Multiply 0.15200000 with 2.    Since 0.30400000 is < 1. then add 0 to result
       > Multiply 0.30400000 with 2.    Since 0.60800000 is < 1. then add 0 to result
       > Multiply 0.60800000 with 2.    Since 1.21600000 is >= 1. then add 1 to result
       > Multiply 0.21600000 with 2.    Since 0.43200000 is < 1. then add 0 to result
       > Multiply 0.43200000 with 2.    Since 0.86400000 is < 1. then add 0 to result
       > Multiply 0.86400000 with 2.    Since 1.72800000 is >= 1. then add 1 to result
       > Multiply 0.72800000 with 2.    Since 1.45600000 is >= 1. then add 1 to result
       > Multiply 0.45600000 with 2.    Since 0.91200000 is < 1. then add 0 to result
       > Multiply 0.91200000 with 2.    Since 1.82400000 is >= 1. then add 1 to result
       > Multiply 0.82400000 with 2.    Since 1.64800000 is >= 1. then add 1 to result
       > Multiply 0.64800000 with 2.    Since 1.29600000 is >= 1. then add 1 to result
       > Multiply 0.29600000 with 2.    Since 0.59200001 is < 1. then add 0 to result
       > Multiply 0.59200001 with 2.    Since 1.18400002 is >= 1. then add 1 to result
       > Multiply 0.18400002 with 2.    Since 0.36800003 is < 1. then add 0 to result
       > Multiply 0.36800003 with 2.    Since 0.73600006 is < 1. then add 0 to result
       > Multiply 0.73600006 with 2.    Since 1.47200012 is >= 1. then add 1 to result
       > Multiply 0.47200012 with 2.    Since 0.94400024 is < 1. then add 0 to result
       > Multiply 0.94400024 with 2.    Since 1.88800049 is >= 1. then add 1 to result
       > Multiply 0.88800049 with 2.    Since 1.77600098 is >= 1. then add 1 to result
       > Multiply 0.77600098 with 2.    Since 1.55200195 is >= 1. then add 1 to result
       > Multiply 0.55200195 with 2.    Since 1.10400391 is >= 1. then add 1 to result
       > Multiply 0.10400391 with 2.    Since 0.20800781 is < 1. then add 0 to result
       > Multiply 0.20800781 with 2.    Since 0.41601562 is < 1. then add 0 to result
       > Multiply 0.41601562 with 2.    Since 0.83203125 is < 1. then add 0 to result
       > Multiply 0.83203125 with 2.    Since 1.66406250 is >= 1. then add 1 to result
       > Multiply 0.66406250 with 2.    Since 1.32812500 is >= 1. then add 1 to result
       > Multiply 0.32812500 with 2.    Since 0.65625000 is < 1. then add 0 to result
       > Multiply 0.65625000 with 2.    Since 1.31250000 is >= 1. then add 1 to result
       > Multiply 0.31250000 with 2.    Since 0.62500000 is < 1. then add 0 to result
       > Multiply 0.62500000 with 2.    Since 1.25000000 is >= 1. then add 1 to result
       > Multiply 0.25000000 with 2.    Since 0.50000000 is < 1. then add 0 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
   0.7560000000000002 of decimal is .110000011000100100110111010010111100011010101 in binary
   so, 100.756 in binary is 1100100.110000011000100100110111010010111100011010101
-100.756 in simple binary => 1100100.110000011000100100110111010010111100011010101
so, -100.756 in normal binary is 1100100.110000011000100100110111010010111100011010101 => 1.1001001100000110001001 * 2^6

single precision:
--------------------
sign bit is 1(-ve)
exponent bits are (127+6=133) => 10000101
   Divide 133 successively by 2 until the quotient is 0
       > 133/2 = 66, remainder is 1
       > 66/2 = 33, remainder is 0
       > 33/2 = 16, remainder is 1
       > 16/2 = 8, remainder is 0
       > 8/2 = 4, remainder is 0
       > 4/2 = 2, remainder is 0
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10000101
   So, 133 of decimal is 10000101 in binary
frac/significant bits are 10010011000001100010010

so, -100.756 in single-precision format is 1 10000101 10010011000001100010010
in hexadecimal it is 0xC2C98312