Question

Convert 1.8125 to IEEE-754 representation. Show all your work.

Answer 1

1.8125 in IEEE-754 single precision binary format is 00111111111010000000000000000000

sign bit:   0
exponent bits:  01111111
mantissa bits:  11010000000000000000000

Explanation:
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1.8125
Converting 1.8125 to binary
   Convert decimal part first, then the fractional part
   > 1 in binary is 1, simple!
   > Now, Convert 0.81250000 to binary
      > Multiply 0.81250000 with 2.  Since 1.62500000 is >= 1. then add 1 to result
      > Multiply 0.62500000 with 2.  Since 1.25000000 is >= 1. then add 1 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
   0.8125 of decimal is .1101 in binary
   so, 1.8125 in binary is 1.1101
1.8125 in simple binary => 1.1101
so, 1.8125 in normal binary is 1.1101 => 1.1101 * 2^0
=>  1(.1101 this is the fractional part) * 2^(127+0 this is the exponent bits)

single precision:
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sign bit is 0(+ve)
exponent bits are (127+0=127) => 01111111
   Divide 127 successively by 2 until the quotient is 0
      > 127/2 = 63, remainder is 1
      > 63/2 = 31, remainder is 1
      > 31/2 = 15, remainder is 1
      > 15/2 = 7, remainder is 1
      > 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 1111111
   So, 127 of decimal is 1111111 in binary
frac/significant bits are 11010000000000000000000

so, 1.8125 in single-precision format is 0 01111111 11010000000000000000000