Question

convert -297.875 to single precision and double precision.
Please show all the steps(atleast 500 words)

Answer 1

1)
-297.875
Converting 297.875 to binary
   Convert decimal part first, then the fractional part
   > First convert 297 to binary
   Divide 297 successively by 2 until the quotient is 0
       > 297/2 = 148, remainder is 1
       > 148/2 = 74, remainder is 0
       > 74/2 = 37, remainder is 0
       > 37/2 = 18, remainder is 1
       > 18/2 = 9, remainder is 0
       > 9/2 = 4, remainder is 1
       > 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 100101001
   So, 297 of decimal is 100101001 in binary
   > Now, Convert 0.87500000 to binary
       > Multiply 0.87500000 with 2.    Since 1.75000000 is >= 1. then add 1 to result
       > 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
   0.875 of decimal is .111 in binary
   so, 297.875 in binary is 100101001.111
-297.875 in simple binary => 100101001.111
so, -297.875 in normal binary is 100101001.111 => 1.00101001111 * 2^8

single precision:
--------------------
sign bit is 1(-ve)
exponent bits are (127+8=135) => 10000111
   Divide 135 successively by 2 until the quotient is 0
       > 135/2 = 67, remainder is 1
       > 67/2 = 33, remainder is 1
       > 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 10000111
   So, 135 of decimal is 10000111 in binary
frac/significant bits are 00101001111000000000000

so, -297.875 in single-precision format is 1 10000111 00101001111000000000000
in hexadecimal it is 0xC394F000

2)
-297.875
Converting 297.875 to binary
   Convert decimal part first, then the fractional part
   > First convert 297 to binary
   Divide 297 successively by 2 until the quotient is 0
       > 297/2 = 148, remainder is 1
       > 148/2 = 74, remainder is 0
       > 74/2 = 37, remainder is 0
       > 37/2 = 18, remainder is 1
       > 18/2 = 9, remainder is 0
       > 9/2 = 4, remainder is 1
       > 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 100101001
   So, 297 of decimal is 100101001 in binary
   > Now, Convert 0.87500000 to binary
       > Multiply 0.87500000 with 2.    Since 1.75000000 is >= 1. then add 1 to result
       > 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
   0.875 of decimal is .111 in binary
   so, 297.875 in binary is 100101001.111
-297.875 in simple binary => 100101001.111
so, -297.875 in normal binary is 100101001.111 => 1.00101001111 * 2^8

64-bit format:
--------------------
sign bit is 1(-ve)
exponent bits are (1023+8=1031) => 10000000111
   Divide 1031 successively by 2 until the quotient is 0
       > 1031/2 = 515, remainder is 1
       > 515/2 = 257, remainder is 1
       > 257/2 = 128, remainder is 1
       > 128/2 = 64, remainder is 0
       > 64/2 = 32, remainder is 0
       > 32/2 = 16, remainder is 0
       > 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 10000000111
   So, 1031 of decimal is 10000000111 in binary
frac/significant bits are 0010100111100000000000000000000000000000000000000000

so, -297.875 in 64-bit format is 1 10000000111 0010100111100000000000000000000000000000000000000000
in hexadecimal it is 0xC0729E0000000000