Question

What are the sign, mantissa, and exponent bit strings for double 11? Assemble these bit strings into an IEEE 754 format and then show how the value stored in memory is a match for this bit string.

Answer 1

Converting 11.0 to binary
   Convert decimal part first, then the fractional part
   > First convert 11 to binary
   Divide 11 successively by 2 until the quotient 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 1011
   So, 11 of decimal is 1011 in binary
   > Now, Convert 0.0 to binary
      > Multiply 0.0 with 2.     Since 0.0 is < 1. then add 0 to result
      > This is equal to 1, so, stop calculating
   0.0 of decimal is .0 in binary
   so, 11.0 in binary is 1011.0
11.0 in simple binary => 1011.0
so, 11.0 in normal binary is 1011.0 => 1.011 * 2^3

single precision:
--------------------
sign bit is 0(+ve)
exp bits are (127+3=130) => 10000010
   Divide 130 successively by 2 until the quotient is 0
      > 130/2 = 65, remainder is 0
      > 65/2 = 32, remainder is 1
      > 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 10000010
   So, 130 of decimal is 10000010 in binary
frac bits are 01100000000000000000000

so, 11.0 in single-precision format is 0 10000010 01100000000000000000000
in hexadecimal it is 0x41300000

sign bit = 0
mantissa bits = 10000010
exponent bits = 01100000000000000000000

11 is stored in memory as 0 10000010 01100000000000000000000 in hexadecimal it is 0x41300000