210 Notes on Fixed Point Numbers

The electronic circuitry used for adding unsigned integers can also be used to add fractional numbers. As with signed integers, we need to impose a new interpretation on the bit patterns to achieve this.

One part of the computer word is chosen as the fractional part, and the other part is the integer part.

Example: 8-bit integer. Decide to treat last 4 bits as fraction part:

1011.0110 = 1*2^3 + 0*2^2 + 1*2^1 + 1*2^0
          + 0*2^-1 + 1*2^-2 + 1*2^-3 + 0*2^-4

I.e. the number is 11.375

Powers of 2:
3	8
2	4
1	2
0	1
-1	0.5
-2	0.25
-3	0.125
-4	0.0625

Adding fixed-point numbers

  1011.0110	  11.375	[  182
+ 0011.1010	+  3.625        +  58
= 1111.0000	= 15.000	 = 240 ]

Great! Ordinary unsigned arithmetic works.

[ Can you make signed fixed-point numbers? ]

Multiplication

Need to right-shift answer 4 places (or shift numbers before multiply)

  0011.1010	   3.625
x 0000.0100	 x 0.250
  1110.1000 (requires right shift 4 places)
= 0000.1110	 = 0.875

Actual answer is 0.90625; we lost a low-order bit in the right shift.

Advantage

fixed point arithmetic is as fast as integer operations
no special hardware required (although a multiply-and-shift instruction would be useful)

Disadvantages

limited range of values (e.g. 32-bit fixed point with 8-bit fraction gives MAXFIXED of about 8 million)
massive loss of precision if intermediate result exceeds MAXINT
most languages do not support fixed point (programmer must normalise results manually)