# Natural Numbers

We can define a recursive variant that acts like numbers, demonstrating that we don't really have to have numbers built into OCaml! (For sake of efficiency, though, it's a good thing they are.)

A natural number is either zero or the successor of some other natural number. This is how you might define the natural numbers in a mathematical logic course, and it leads naturally to the following OCaml type nat:

type nat = Zero | Succ of nat


We have defined a new type nat, and Zero and Succ are constructors for values of this type. This allows us to build expressions that have an arbitrary number of nested Succ constructors. Such values act like natural numbers:

let zero  = Zero
let one   = Succ zero
let two   = Succ one
let three = Succ two
let four  = Succ three


When we ask the compiler what four is, we get

# four;;
- : nat = Succ (Succ (Succ (Succ Zero)))


Now we can write functions to manipulate values of this type. We'll write a lot of type annotations in the code below to help the reader keep track of which values are nat versus int; the compiler, of course, doesn't need our help.

let iszero (n : nat) : bool =
match n with
| Zero   -> true
| Succ m -> false

let pred (n : nat) : nat =
match n with
| Zero   -> failwith "pred Zero is undefined"
| Succ m -> m


Similarly we can define a function to add two numbers:

let rec add (n1:nat) (n2:nat) : nat =
match n1 with
| Zero -> n2
| Succ n_minus_1 -> add n_minus_1 (Succ n2)


We can convert nat values to type int and vice-versa:

let rec int_of_nat (n:nat) : int =
match n with
| Zero   -> 0
| Succ m -> 1 + int_of_nat m

let rec nat_of_int(i:int) : nat =
if i < 0 then failwith "nat_of_int is undefined on negative ints"
else if i = 0 then Zero
else Succ (nat_of_int (i-1))


To determine whether a natural number is even or odd, we can write a pair of mutually recursive functions:

let rec
even (n:nat) : bool =
match n with
| Zero   -> true
| Succ m -> odd m
and
odd (n:nat) : bool =
match n with
| Zero   -> false
| Succ m -> even m


You have to use the keyword and to combine mutually recursive functions like this. Otherwise the compiler would flag an error when you refer to odd before it has been defined.