This repository provides a library to do basic arithmetic with highly precise rational numbers and to convert between rational numbers and the usual number types supported by Forth.

Rational numbers are expressed as fractions with a single-precision signed integer numerator and single-precision signed integer denominator.

Rational numbers are able to exactly express many “floating point” decimal numbers that cannot be expressed exactly as fixed or floating point numbers. For example, 0.1 can only be approximately expressed as a floating or fixed point number but can be exactly expressed as a rational number: 1/10.

This library supports 2-cell rational numbers consisting of a single-precision signed numerator and a single-precision unsigned denominator.

The largest number that can be expressed is the same as a single-precision signed integer (2,147,483,647 with a 32-bit cell), with 1 in the denominator. The smallest fraction that can be expressed is the inverse of the largest single-precision signed integer (1/2147483647), with 1 (or -1) in the numerator and the maximum integer in the denominator.

There is an inherent trade-off between precision and range. Precision and range vary inversely to each other depending on the size of the denominator. If the denominator is at the minimum, 1, the range is the entire single-precision integer range (-2,147,483,648 to 2,147,483,647) but there is only integer precision. If the denominator is 1000, there is 0.001 precision, but 1/1000 of the single-precision integer range (-2,147,483 to 2.147,483). When the denominator is at its maximum, precision is very high and the range is reduced to -1 to 1.

I realize that the limited range may be a concern for some applications. I intend to address the range issue by creating a 3-word “mixed rational” type that includes a single-precision integer and a single-precision rational. This will expand the range to the entire single-precision integer range while retaining full rational precision.