# The Universe of Discourse

Fri, 04 Nov 2022

I keep getting lost in the maze of Haskell's numeric types. Here's the map I drew to help myself out. (I think there might have been something like this in the original Haskell 1998 report.)

(PNG version) (Original DOT file (The SVG above is hand-edited graphviz output))

Ovals are typeclasses. Rectangles are types. Black mostly-straight arrows show instance relationships. Most of the defined functions have straightforward types like !!\alpha\to\alpha!! or !!\alpha\to\alpha\to\alpha!! or !!\alpha\to\alpha\to\text{Bool}!!. The few exceptions are shown by wiggly colored arrows.

## Basic plan

After I had meditated for a while on this picture I began to understand the underlying organization. All numbers support !!=!! and !!\neq!!. And there are three important properties numbers might additionally have:

• Ord : ordered; supports !!\lt\leqslant\geqslant\gt!! etc.
• Fractional : supports division
• Enum: supports ‘pred’ and ‘succ’

Integral types are both Ord and Enum, but they are not Fractional because integers aren't closed under division.

Floating-point and rational types are Ord and Fractional but not Enum because there's no notion of the ‘next’ or ‘previous’ rational number.

Complex numbers are numbers but not Ord because they don't admit a total ordering. That's why Num plus Ord is called Real: it's ‘real’ as constrasted with ‘complex’.

## More stuff

That's the basic scheme. There are some less-important elaborations:

Real plus Fractional is called RealFrac.

Fractional numbers can be represented as exact rationals or as floating point. In the latter case they are instances of Floating. The Floating types are required to support a large family of functions like !!\log, \sin,!! and π.

You can construct a Ratio a type for any a; that's a fraction whose numerators and denominators are values of type a. If you do this, the Ratio a that you get is a Fractional, even if a wasn't one. In particular, Ratio Integer is called Rational and is (of course) Fractional.

## Shuff that don't work so good

Complex Int and Complex Rational look like they should exist, but they don't really. Complex a is only an instance of Num when a is floating-point. This means you can't even do 3 :: Complex Int — there's no definition of fromInteger. You can construct values of type Complex Int, but you can't do anything with them, not even addition and subtraction. I think the root of the problem is that Num requires an abs function, and for complex numbers you need the sqrt function to be able to compute abs.

Complex Int could in principle support most of the functions required by Integral (such as div and mod) but Haskell forecloses this too because its definition of Integral requires Real as a prerequisite.

You are only allowed to construct Ratio a if a is integral. Mathematically this is a bit odd. There is a generic construction, called the field of quotients, which takes a ring and turns it into a field, essentially by considering all the formal fractions !!\frac ab!! (where !!b\ne 0!!), and with !!\frac ab!! considered equivalent to !!\frac{a'}{b'}!! exactly when !!ab' = a'b!!. If you do this with the integers, you get the rational numbers; if you do it with a ring of polynomials, you get a field of rational functions, and so on. If you do it to a ring that's already a field, it still works, and the field you get is trivially isomorphic to the original one. But Haskell doesn't allow it.

I had another couple of pages written about yet more ways in which the numeric class hierarchy is a mess (the draft title of this article was "Haskell's numbers are a hot mess") but I'm going to cut the scroll here and leave the hot mess for another time.

[ Addendum: Updated SVG and PNG to version 1.1. ]