Fixed point arithmetic

In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after (and sometimes also before) the radix point (after the decimal point '.' in English decimal notation). Fixed-point number representation can be compared to the more complicated (and more computationally demanding) floating-point number representation.

Fixed-point numbers are useful for representing fractional values, usually in base 2 or base 10, when the executing processor has no floating point unit (FPU) or if fixed-point provides improved performance or accuracy for the application at hand. Older low-cost embedded microprocessors and microcontrollers do not have an FPU.

A value of a fixed-point data type is essentially an integer that is scaled by an implicit 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 1,230,000 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.

To convert a number from a fixed point type with scaling factor R to another type with scaling factor S, the underlying integer must be multiplied by R and divided by S; that is, multiplied by the ratio R/S. Thus, for example, to convert the value 1.23 = 123/100 from a type with scaling factor R=1/100 to one with scaling factor S=1/1000, the underlying integer 123 must be multiplied by (1/100)/(1/1000) = 10, yielding the representation 1230/1000. If S does not divide R (in particular, if the new scaling factor S is greater than the original R), the new integer will have to be rounded. The rounding rules and methods are usually part of the language's specification.

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