Question

Show your work.
For 8-bit numbers, use 1 sign bit, 3 exponent bits, 4 mantissa bits:
1.)Convert 11.0 to 8-bit floating point format
2.)Convert -12.40625 to 8-bit floating point format

For 32-bit numbers, use 1 sign bit, 8 exponent bits, 23 mantissa bits:
3.)Convert 119.59375 to 32-bit floating point format
4.)Convert -67.1015625 to 32-bit floating point format

Answer 1

1)
11.0
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 0.00000000 to binary
      > Multiply 0.00000000 with 2.  Since 0.00000000 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

8-bit format:
--------------------
sign bit is 0(+ve)
exponent bits are (3+3=6) => 110
   Divide 6 successively by 2 until the quotient is 0
      > 6/2 = 3, remainder is 0
      > 3/2 = 1, remainder is 1
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 110
   So, 6 of decimal is 110 in binary
frac/significant bits are 0110

so, 11.0 in 8-bit format is 0 110 0110

2)
-12.40625
Converting 12.40625 to binary
   Convert decimal part first, then the fractional part
   > First convert 12 to binary
   Divide 12 successively by 2 until the quotient is 0
      > 12/2 = 6, remainder is 0
      > 6/2 = 3, remainder is 0
      > 3/2 = 1, remainder is 1
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 1100
   So, 12 of decimal is 1100 in binary
   > Now, Convert 0.40625000 to binary
      > Multiply 0.40625000 with 2.  Since 0.81250000 is < 1. then add 0 to result
      > 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
   0.40625 of decimal is .01101 in binary
   so, 12.40625 in binary is 1100.01101
-12.40625 in simple binary => 1100.01101
so, -12.40625 in normal binary is 1100.01101 => 1.1 * 2^3

8-bit format:
--------------------
sign bit is 1(-ve)
exponent bits are (3+3=6) => 110
   Divide 6 successively by 2 until the quotient is 0
      > 6/2 = 3, remainder is 0
      > 3/2 = 1, remainder is 1
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 110
   So, 6 of decimal is 110 in binary
frac/significant bits are 1000

so, -12.40625 in 8-bit format is 1 110 1000

3)
119.59375
Converting 119.59375 to binary
   Convert decimal part first, then the fractional part
   > First convert 119 to binary
   Divide 119 successively by 2 until the quotient is 0
      > 119/2 = 59, remainder is 1
      > 59/2 = 29, remainder is 1
      > 29/2 = 14, remainder is 1
      > 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 1110111
   So, 119 of decimal is 1110111 in binary
   > Now, Convert 0.59375000 to binary
      > Multiply 0.59375000 with 2.  Since 1.18750000 is >= 1. then add 1 to result
      > Multiply 0.18750000 with 2.  Since 0.37500000 is < 1. then add 0 to result
      > Multiply 0.37500000 with 2.  Since 0.75000000 is < 1. then add 0 to result
      > 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
   0.59375 of decimal is .10011 in binary
   so, 119.59375 in binary is 1110111.10011
119.59375 in simple binary => 1110111.10011
so, 119.59375 in normal binary is 1110111.10011 => 1.11011110011 * 2^6

single precision:
--------------------
sign bit is 0(+ve)
exponent bits are (127+6=133) => 10000101
   Divide 133 successively by 2 until the quotient is 0
      > 133/2 = 66, remainder is 1
      > 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 10000101
   So, 133 of decimal is 10000101 in binary
frac/significant bits are 11011110011000000000000

so, 119.59375 in single-precision format is 0 10000101 11011110011000000000000

4)
-67.1015625
Converting 67.1015625 to binary
   Convert decimal part first, then the fractional part
   > First convert 67 to binary
   Divide 67 successively by 2 until the quotient is 0
      > 67/2 = 33, remainder is 1
      > 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 1000011
   So, 67 of decimal is 1000011 in binary
   > Now, Convert 0.10156250 to binary
      > Multiply 0.10156250 with 2.  Since 0.20312500 is < 1. then add 0 to result
      > Multiply 0.20312500 with 2.  Since 0.40625000 is < 1. then add 0 to result
      > Multiply 0.40625000 with 2.  Since 0.81250000 is < 1. then add 0 to result
      > 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
   0.1015625 of decimal is .0001101 in binary
   so, 67.1015625 in binary is 1000011.0001101
-67.1015625 in simple binary => 1000011.0001101
so, -67.1015625 in normal binary is 1000011.0001101 => 1.0000110001101 * 2^6

single precision:
--------------------
sign bit is 1(-ve)
exponent bits are (127+6=133) => 10000101
   Divide 133 successively by 2 until the quotient is 0
      > 133/2 = 66, remainder is 1
      > 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 10000101
   So, 133 of decimal is 10000101 in binary
frac/significant bits are 00001100011010000000000

so, -67.1015625 in single-precision format is 1 10000101 00001100011010000000000