Support "UnifiedReal" class designed for the Android calculator

[gwhitney]

There is a JavaScript library that implements the numeric system discussed in Towards an API for the real numbers. The virtues of this system include:

• exact representation for all rational numbers as well as rational multiples of many commonly-used irrational constants.
• essentially exact representation for all square roots of rational numbers
• representation as "computable reals" (i.e., function generating approximations as closely as desired) for essentially anything you could try to calculate.

Note that the equality problem on computable reals is not decidable in general. However, the UnifiedReals do support nearlyEqual at any desired level of precision. They also allow you to ask whether it happens to be capable of exactly comparing two real numbers, and recognizes many many cases in which it can.

This numeric system was developed for the sake of the Android calculator, so that (for example) it could decide whether the result of a calculation was exactly an integer (in which case it would be displayed without a decimal point) or merely very very close to an integer (in which case it would be displayed with trailing zeros).

Thus, the design of the UnifiedReal class seems extremely consonant with the design goals of mathjs itself, in that mathjs seeks in particular to support its use for practical calculation in a web context.

Hence, I recommend that mathjs seek to support UnifiedReal as one of the types it can use in computation. If it goes well, it should perhaps be the primary/default non-ordinary-JavaScript-number type employed by mathjs. Many of the points we struggle with in mathjs (nearlyEqual, controlling precision for BigNumber, and so on) are rather thoroughly worked out in UnifiedReal.

[josdejong]

It would be awesome to play around with these Unified Real numbers!

Here is another article explaining the story behind it: https://chadnauseam.com/coding/random/calculator-app

A big reason for using BigNumbers, bigint, and Fractions is to work around the rounding-off issues and limited digits that we have with floating point numbers. So indeed UnifiedReal may make all of them redundant in many real-world cases.