Example created by our ta, Logan Manning, and modified by me. 

55.575  -  significand is 8 bits,  exponent is 7 bits.

1st convert integer portion. Using divide by 2 and collecting the remainders.

 55
 27  1
 13  1
  6  1
  3  0
  1  1
  0  1

  110111 

Remember, the most significant bit is at the bottom of the column.

We don't store the most significant bit, but we do need to store other 5. Our
significand is 8 bits wide, so we have room for 3 more bits.

To convert the decimal, we multiply by two and collect the integer portion
of the product. Remember, the integer portion gets set aside after each 
multiplication.

  .575 * 2 =
 1.150 * 2 =
 0.300 * 2 =
 0.600 * 2 =
 1.200 * 2 = etc.

We only needed 3 bits, so we can stop now.

In a real system, it would have rounded the 4th bit into the answer, but for
our assignment, we will truncate.

Recombine the integer and fractional portions of the number.

110111.1001

State in scientific style notation. 

1.101111001 * 2^5


Calculate the bias and add the exponent to it.

Given a 7 bit exponent :
 2^(7-1) -1 = 2^6 -1 = 63

 63 + 5 = 68


 68
 34  0
 17  0
  8  1
  4  0
  2  0
  1  0
  0  1
 
   1000100  - biased exponent in binary format.



Remember our conversion so far in scientific notation :   1.101111001 * 2^5

Number is positive so sign flag is 0.  Putting it together :
  sign  biased     significand
        exponent
  0     100 0100   1011 1100 


When you assemble it back you convert 100 0100 from binary to 68 ... you see 
there are 7 spots so you do the

 2^(7-1) -1 = 2^6 -1 = 63, then 68-63 = your exponent (5)
 
Reassemble the scientific form from the significand. Remember to add the
1. at the beginning.

 1.10111100  x 2 ^5 = 110111.100  

Convert back integer portion.

  1  1  0  1  1  1
  1
     1+2
        0+6
           1+12
              1+26
                 1+54
                   55

Convert fractional portion.

  . 
  1  * 2^-1 (.5)
  0  * 2^-2 (.25)
  0  * 2^-3 (.125)

  .5
 
  55.5