Normalize the number .431. using a IEEE style float format.
    1 bit sign, 7 bit exponent, 16 bit significand
    (real 1,8,23 format makes math difficult) 

 
First convert to binary.  We need to go at least 16 places past the
most significant 1

  . .431
  0 .862
  1 .724  # start counting here.
  1 .448
  0 .896

  1 .793
  1 .586
  1 .172
  0 .344

  0 .688
  1 .376
  0 .752
  1 .504

  1 .008
  0 .016
  0 .032
  0 .064

  0 .128
  0 .256 # we have 16 digits past the most signficant 1.
  ...

  .0110 1110 0101 1000 00 

Represent in the equivalent of scientific notation,.

  1.10 1110 0101 1000 00 * 2^-2

 * We left the exponent in decimal because we haven't added the bias yet.

In standard scientific notation:
  Integer has range from 1 - 9 and and one digit must be preserved in 
  the integer portion of the scientific representation.

However in binary form, only possible value is 1 and can be implied.
  So only decimal fraction portion needs to be stored.

  .1011100101100000

  This is called the significand and allows for 1 extra digit of precission.

With 16 bit precision, a very close approximation can be found.


Generating a IEEE 754 bias.

  Because values can be greater or less than 1
    Exponent must represent both positive and negative values.

  Problem: an exponent of zero used to flag special conditions.

  Solution: shift (bias) representation of zero exponent up the number line.

Using single precision float with 7 bit exponent.
  Giving a range of 0 to 127 that should represent a signed exponent.

  Need to reserve 0 and 127 (1111111b) as flag values.

  Giving a number range between 1 and 126 to represent an exponent.
    That represents an exponent in the range of -63 to 62.


Shift representation of exponent 0 to middle of range.
  Use formula 2^(n-1) - 1 where n is the size of the exponent.

  For our example using 7 bit exponent, calculate bias

    2^(7-1)-1 = 63 bias

Recall that we converted the original value to a binary form of scientific
notation.

  1.10 1110 0101 1000 00 * 2^-2
    So, our exponent is -2.

Biasing this exponent give :

  63 - 2 = 61 = 011 1101


Finally, we need to record the sign of the whole value  .431 is positive.

Sign bit 0 (positive)
Biased Exponent 011 1101
Signficand 1011 1001 0110 0000

Floating point 0 0111101 1011100101100000