Is it possible to create a recursive data type with rational numbers like -> data Nat = Zero | Succ Nat?

Question

Is it possible to create a recursive data for rational numbers? I saw this one for natural numbers

data Nat = Zero | Succ Nat

I probably need to use two numbers for the ratio? But something like Zero,Zero doesnt work.

Thanks!

Answer 1

Positive rational numbers

We can define it like:

data Nat0 = One | Succ0 Nat0
data Nat = Zero | Succ Nat
data PosRational = PosRational Nat Nat0

so here we use two data types Nat0 ( excluding zero), and Nat (including zero) that are defined recursively, and a data type PosRational that takes a Nat and Nat0 (so a numerator, and a denominator).

Including negative rational numbers

We can also include negative rational numbers, for instance by defining:

data Z = Pos Nat | Neg Nat0

and then define our Rational as:

data Rational = Rational Z Nat0

Countability of ℚ

Nevertheless we actually do not need that (given we do not need fast access to the numerator and denominator). Rational numbers are countable . We can enumerate all of them with the following scheme [source] :

So we can simply define it as:

data Rational = Zero | NextRational Rational

Basically, every countable set can use this definition, so ℕ, ℤ, ℙ, ℚ, etc.