Question

1. Using the simple model for representing binary floating point numbers
   • A floating-point number is 14 bits in length.
   • The exponent field is 5 bits.
   • The significand field is 8 bits.
   • The bias is 15

Represent -32.50₁₀ in the simple model.

Answer 1

1 10101 10000010

Explanation:
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-32.5
Converting 32.5 to binary
   Convert decimal part first, then the fractional part
   > First convert 32 to binary
   Divide 32 successively by 2 until the quotient 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 100000
   So, 32 of decimal is 100000 in binary
   > Now, Convert 0.50000000 to binary
      > Multiply 0.50000000 with 2.  Since 1.00000000 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.5 of decimal is .1 in binary
   so, 32.5 in binary is 00100000.1
-32.5 in simple binary => 100000.1
so, -32.5 in normal binary is 100000.1 => 0.1000001 * 2^6

14-bit format:
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sign bit is 1(-ve)
exponent bits are (15+6=21) => 10101
   Divide 21 successively by 2 until the quotient is 0
      > 21/2 = 10, remainder is 1
      > 10/2 = 5, remainder is 0
      > 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 10101
   So, 21 of decimal is 10101 in binary
frac/significant bits are 10000010

so, -32.5 in 14-bit format is 1 10101 10000010