Binary Number System: Representations

IEEE 32-Bit Floating Point Format - EXAMPLE 2

The previous example was a little simplistic because the fractional portion of the number was a sum of powers of 1/2. This means that the binary representation will be an exact representation of the decimal value. Often, though, this is not possible. Consider, for example, the decimal value 42.42 expressed in IEEE Float.

1. Convert to binary:

First convert the integer portion to an unsigned binary integer:

4210 = 1010102
Next convert the fractional portion. This involves a technique which is the inverse of that for the integer portion. Begin with the value to be represented:
0.42
then successively multiply by 2. The integer portion of each result is used to create the binary fraction, the fractional portion of each result is used in the next step. Repeat until the result is 0 or the number of bits allowed is exceeded. Example: 
	0.42 x 2 = 0.84
	0.84 x 2 = 1.68
	0.68 x 2 = 1.36
	0.36 x 2 = 0.72
	0.72 x 2 = 1.44
	0.44 x 2 = 0.88
	0.88 x 2 = 1.76
	0.76 x 2 = 1.52
	0.52 x 2 = 1.04
	0.04 x 2 = 0.08
	0.08 x 2 = 0.16
	0.16 x 2 = 0.32
	0.32 x 2 = 0.64
	0.64 x 2 = 1.28
	0.28 x 2 = 0.56
	0.56 x 2 = 1.12
	0.12 x 2 = 0.24
	0.24 x 2 = 0.48
	0.48 x 2 = 0.96
	0.96 x 2 = 1.92
	0.92 x 2 = 1.84 <--- repeats here
	0.84 x 2 = 1.68  
	0.68 x 2 = 1.36

Assembling the bits we have:
0.4210 = 0.01101011100001010001111
Note that the first few 1's are most significant:
0.25 + 0.125 + 0.03125 + 0.0078125 = 0.414065
Now we have:
42.4210 = 101010.011010111000010100011112
2. Convert to normalized binary scientific notation:
1.01010011010111000010100011112 x 25
3. Determine s, e and f: 
	s = 0
	e = 5
	e+127  = 5 + 127 = 132 = 100001002
	f = 010100110101110000101002  (note that some bits are dropped off
the right)
4. Assemble the 32 bits: 
	s     e+127       1.fffffffffffffffffffffff 
	0     10000100      01010011010111000010100

	01000010001010011010111000010100
5. Convert to hex: 
	0100 0010 0010 1001 1010 1110 0001 0100

	4229AE14

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