Question

One day Tom found a weird-looking computer in his basement. After studying it, Tom figured out that this computer stores floating point numbers in a way similar to the IEEE 754 standard. However, it uses 30 bits. Among these bits, the most significant (i.e. the leftmost) bit is for the sign (same as IEEE 754), the following 7 bits are for the biased exponent with the bias as 63, and the rest are for the significand. If the real number 8.78125 is stored on the computer, what are the 30 bits stored? Show the result in the order of sign, biased exponent and significand. Separate them by comma.

Answer 1

Answer

Answer:
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0,1000010,0001100100000000000000

Explanation:
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Converting 8.78125 to binary
   Convert decimal part first, then the fractional part
   > First convert 8 to binary
   Divide 8 successively by 2 until the quotient 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 1000
   So, 8 of decimal is 1000 in binary
   > Now, Convert 0.78125000 to binary
      > Multiply 0.78125000 with 2.  Since 1.56250000 is >= 1. then add 1 to result
      > Multiply 0.56250000 with 2.  Since 1.12500000 is >= 1. then add 1 to result
      > Multiply 0.12500000 with 2.  Since 0.25000000 is < 1. then add 0 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
   0.78125 of decimal is .11001 in binary
   so, 8.78125 in binary is 1000.11001
8.78125 in simple binary => 1000.11001
so, 8.78125 in normal binary is 1000.11001 => 1.00011001 * 2^3

single precision:
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sign bit is 0(+ve)
exp bits are (63+3=66) => 1000010
   Divide 66 successively by 2 until the quotient is 0
      > 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 1000010
   So, 66 of decimal is 1000010 in binary
frac bits are 0001100100000000000000

so, 8.78125 in single-precision format is 0 1000010 0001100100000000000000

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