# Complex numbers

Futhark does not have complex numbers, but they are fairly easy to implement as a module. First we define a module type describing the interface of complex numbers:

```
module type complex = {
-- | The type of the components of the complex number.
type real
-- | The type of complex numbers.
type complex
-- | Construct a complex number from real and imaginary components.
val mk: real -> real -> complex
-- | Construct a complex number from just the real component. The
-- imaginary part will be zero.
val mk_re: real -> complex
-- | Construct a complex number from just the imaginary component. The
-- real part will be zero.
val mk_im: real -> complex
-- | Conjugate a complex number.
val conj: complex -> complex
-- | The real part of a complex number.
val re: complex -> real
-- | The imaginary part of a complex number.
val im: complex -> real
-- | The magnitude (or modulus, or absolute value) of a complex number.
val mag: complex -> real
-- | The argument (or phase) of a complex number.
val arg: complex -> real
val +: complex -> complex -> complex
val -: complex -> complex -> complex
val *: complex -> complex -> complex
val /: complex -> complex -> complex
val sqrt: complex -> complex
val exp: complex -> complex
val log: complex -> complex
}
```

Then we define a parametric module that given a `real`

module (in practice `f32`

or `f64`

) will give us back a complex number module. We use *local opens* to perform operations using the operators from the `T`

module. For examle, `T.(x + y)`

means the same as `x T.+ y`

, but is more readable. The definitions here are straight out of a textbook, so we wonâ€™t elaborate them.

```
module mk_complex (T: real): (complex with real = T.t
with complex = (T.t, T.t)) = {
type real = T.t
type complex = (T.t, T.t)
def mk (a: real) (b: real) = (a,b)
def mk_re (a: real) = (a, T.i32 0)
def mk_im (b: real) = (T.i32 0, b)
def conj ((a,b): complex) = T.((a, i32 0 - b))
def re ((a,_b): complex) = a
def im ((_a,b): complex) = b
def ((a,b): complex) + ((c,d): complex) = T.((a + c, b + d))
def ((a,b): complex) - ((c,d): complex) = T.((a - c, b - d))
def ((a,b): complex) * ((c,d): complex) = T.((a * c - b * d,
b * c + a * d))def ((a,b): complex) / ((c,d): complex) =
T.(((a * c + b * d) / (c * c + d * d),
(b * c - a * d) / (c * c + d * d)))
def mag ((a,b): complex) =
T.(sqrt (a * a + b * b))def arg ((a,b): complex) =
T.atan2 b a
def sqrt ((a,b): complex) =
let gamma = T.(sqrt ((a + sqrt (a * a + b * b)) / i32 2))
let delta = T.(sgn b *
0 - a) + sqrt (a * a + b * b)) / i32 2))
sqrt (((i32 in (gamma, delta)
def exp ((a,b): complex) =
let expx = T.exp a
in mk T.(expx * cos b) T.(expx * sin b)
def log (z: complex) =
mk (T.log (mag z)) (arg z) }
```

We instantiate the complex number module using double-precision floats for the components:

`module c64 = mk_complex f64`

And try it out in the REPL:

```
> c64.exp (c64.mk 1 2)
(-1.1312043837568135f64, 2.4717266720048188f64)
```

The github.com/diku-dk/complex package (which you should use in real programs rather than rolling your own) is implemented in exactly this way.