fixednumbers

Fixed numbers, often called fixed-point numbers, are a way of representing real numbers using a fixed scaling factor. They store numbers as integers that implicitly represent fractions.

In fixed-point form, a chosen scale factor (for example 2^n or 10^d) determines where the decimal or binary point lies. Common formats include Qm.n notation, where m is the number of integer bits and n is the number of fractional bits.

Example: with 8 fractional bits (a Qm.n format with n = 8) the value 1.5 is stored as 1.5 × 2^8 = 384. Arithmetic operates on the underlying integers with scaling preserved, so addition and subtraction are straightforward, while multiplication and division require re-scaling to maintain the fixed point.

Advantages include deterministic results, exact representation of increments aligned with the scaling, and efficient use of hardware resources, especially in embedded systems and real-time environments. They avoid the rounding behavior associated with floating-point when the range and precision are well chosen.

Limitations include a fixed dynamic range, potential overflow, and the need for careful scaling during multiplication and division. Converting to and from real numbers can require rounding, and changing precision often requires scaling adjustments or data type changes.

Applications span digital signal processing, financial calculations in currencies with fixed decimals, and hardware design. Variants include binary fixed-point and decimal fixed-point, the latter avoiding binary rounding issues in some financial computations.

Fixed numbers contrast with floating-point representations, which offer a much larger dynamic range but introduce rounding variability. They are also contrasted with arbitrary-precision arithmetic.

In computing literature, fixed-point approaches are documented under fixed-point arithmetic and Q-format schemes.