Question

convert 1827.75 into IEEE-754 single precision, double precision and 16-bits.

Show all the calculations (Need atleast 500 words on the explanation)

Answer 1

a)
1827.75
Converting 1827.75 to binary
   Convert decimal part first, then the fractional part
   > First convert 1827 to binary
   Divide 1827 successively by 2 until the quotient is 0
      > 1827/2 = 913, remainder is 1
      > 913/2 = 456, remainder is 1
      > 456/2 = 228, remainder is 0
      > 228/2 = 114, remainder is 0
      > 114/2 = 57, remainder is 0
      > 57/2 = 28, remainder is 1
      > 28/2 = 14, remainder is 0
      > 14/2 = 7, remainder is 0
      > 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 11100100011
   So, 1827 of decimal is 11100100011 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, 1827.75 in binary is 11100100011.11
1827.75 in simple binary => 11100100011.11
so, 1827.75 in normal binary is 11100100011.11 => 1.110010001111 * 2^10

single precision:
--------------------
sign bit is 0(+ve)
exponent bits are (127+10=137) => 10001001
   Divide 137 successively by 2 until the quotient is 0
      > 137/2 = 68, remainder is 1
      > 68/2 = 34, remainder is 0
      > 34/2 = 17, remainder is 0
      > 17/2 = 8, remainder is 1
      > 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 10001001
   So, 137 of decimal is 10001001 in binary
frac/significant bits are 11001000111100000000000

so, 1827.75 in single-precision format is 0 10001001 11001000111100000000000
in hexadecimal it is 0x44E47800

b)
1827.75
Converting 1827.75 to binary
   Convert decimal part first, then the fractional part
   > First convert 1827 to binary
   Divide 1827 successively by 2 until the quotient is 0
      > 1827/2 = 913, remainder is 1
      > 913/2 = 456, remainder is 1
      > 456/2 = 228, remainder is 0
      > 228/2 = 114, remainder is 0
      > 114/2 = 57, remainder is 0
      > 57/2 = 28, remainder is 1
      > 28/2 = 14, remainder is 0
      > 14/2 = 7, remainder is 0
      > 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 11100100011
   So, 1827 of decimal is 11100100011 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, 1827.75 in binary is 11100100011.11
1827.75 in simple binary => 11100100011.11
so, 1827.75 in normal binary is 11100100011.11 => 1.110010001111 * 2^10

64-bit format:
--------------------
sign bit is 0(+ve)
exponent bits are (1023+10=1033) => 10000001001
   Divide 1033 successively by 2 until the quotient is 0
      > 1033/2 = 516, remainder is 1
      > 516/2 = 258, remainder is 0
      > 258/2 = 129, remainder is 0
      > 129/2 = 64, remainder is 1
      > 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 10000001001
   So, 1033 of decimal is 10000001001 in binary
frac/significant bits are 1100100011110000000000000000000000000000000000000000

so, 1827.75 in 64-bit format is 0 10000001001 1100100011110000000000000000000000000000000000000000
in hexadecimal it is 0x409C8F0000000000