Fixed-point Arithmetic - Representation

Representation

A value of a fixed-point data type is essentially an integer that is scaled by a specific factor determined by the type. For example, the value 1.23 can be represented as 1230 in a fixed-point data type with scaling factor of 1/1000, and the value 1230000 can be represented as 1230 with a scaling factor of 1000. Unlike floating-point data types, the scaling factor is the same for all values of the same type, and does not change during the entire computation.

The scaling factor is usually a power of 10 (for human convenience) or a power of 2 (for computational efficiency). However, other scaling factors may be used occasionally, e.g. a time value in hours may be represented as a fixed-point type with a scale factor of 1/3600 to obtain values with one-second accuracy.

The maximum value of a fixed-point type is simply the largest value that can be represented in the underlying integer type, multiplied by the scaling factor; and similarly for the minimum value. For example, consider a fixed-point type represented as a binary integer with b bits in two's complement format, with a scaling factor of 1/2f (that is, the last f bits are fraction bits): the minimum representable value is −2b-1/2f and the maximum value is (2b-1-1)/2f.