## Types

You have a `Type`

, `Integer`

. `Integer`

is an integer, representing intergers, just like real life.

Other types are `Char`

, which are characters such as `'a'`

, `'b'`

etc. The one we will look at in detail is `Maybe`

.

## Type variable, constructors

A data of type `Maybe`

can only be made if a *type variable* is specified. This means that you can make a `Maybe Integer`

, `Maybe Char`

, but a `Maybe`

isn’t a complete type.

The way to make a data of type `Maybe`

is to use a *type constructor*. `Maybe`

has two type constructors, `Just`

and `Nothing`

. `Just`

is like `Maybe`

, it can only be made if a type variable is specified. `Nothing`

can be made simply on its own, it is a complete type.

Let’s try making `Maybe`

s in GHCI:

```
> let best_friends = Just 1
> best_friends
Just 1
```

You have just used the type constructor `Just`

, passed in the `Int`

1, to make a type of `Maybe Int`

. To verify the type of `best_friends`

:

```
> :t best_friends
best_friends :: Maybe Integer
```

`::`

can be read as **is of type**. How about make a `Maybe`

using `Nothing`

?

```
> let girlfriend = Nothing
> girlfriend
Nothing
```

## Functor

Suppose you have twice as many best friends as I do, how can I find out how many best friends you have?

```
> let your_best_friends = 1 * 2 --hard coded value for the number best friends I have
> your_best_friends
2
> let your_best_friends = best_friends * 2 --fails terribly
```

You get a crazy error message. It means that you cannot multiply 2 with best_friends, because `best_friends`

is not a type that can be multiplied with 2. Clearly there is something going on here, because `best_friends`

can contain an `Integer`

, and we can definitely multiply `Integer`

s, what we can do is to define a function that helps us do so.

```
> :{
| let calc_friends :: Maybe Integer -> Maybe Integer
| calc_friends Nothing = Nothing
| calc_friends (Just x) = Just (x * 2)
> :}
> let your_best_friends = calc_friends best_friends
> your_best_friends
Just 2
```

What we did was to define a function that can calculate the number of best friends you have, given the number of best friends I have. By applying this function to `best_friends`

, you get the number of best friends you have, wrapped in a `Just`

. What if I told you that you had twice as many girlfriends as I do?

```
> let your_girlfriends = calc_friends girlfriend
> your_girlfriends
Nothing
```

Hahaha.

Perhaps you grow more popular, and you actually had 3 times more friends than I do. Since we hard coded the value `2`

, we have to make another function with the value `3`

, or we can actually generalize the function as such.

```
> :{
| let calc_friends :: (Integer -> Integer) -> Maybe Integer -> Maybe Integer
| calc_friends _ Nothing = Nothing
| calc_friends f (Just x) = Just (f x)
> :}
> let your_best_friends = calc_friends (* 3) best_friends
> your_best_friends
Just 3
> let your_girlfriends = calc_friends (* 2) girlfriend
> your_girlfriends
Nothing
```

We define `calc_friends`

as a function that takes in another function, `f`

, and this function `f`

takes in an `Integer`

and returns another `Integer`

. An example of this function would be `(* 3)`

. `calc_friends`

also takes in a `Maybe Integer`

, which it will apply the function `f`

to, and returning a resut of type `Maybe Integer`

What if instead of number of best friends, an `Integer`

, we wanted to compare height, a `Double`

, and you are 1 metre taller than me, this looks pretty similar to calc_friends, so let’s reuse that:

```
> let height = Just 1.7
> let calc_height = calc_friends (+ 1)
> calc_height height
```

Throws you an error about incompatible types. This is becase we defined `calc_height`

in terms of `calc_friends`

, which only worked with `Maybe Integer`

, but our height was a `Maybe Double`

. So let’s generalize our function using *type variables*, just like how `Maybe`

is defined.

```
> :{
| let calc_stuff :: (a -> a) -> Maybe a -> Maybe a
| calc_stuff _ Nothing = Nothing
| calc_stuff f (Just x) = Just (f x)
> :}
> calc_stuff (* 3) best_friends
Just 9
> calc_stuff (+ 1) height
Just 2.7
> calc_stuff (* 2) girlfriend
Nothing
```

The last bit to understanding `Functor`

is a method called `fmap`

, all types that are instances of the *typeclass* `Functor`

have to implement a method called `fmap`

, which has the signature:

```
fmap :: (a -> b) -> f a -> f b
calc_stuff :: (a -> a) -> Maybe a -> Maybe a
```

I’m putting them one above the other to let you see how similar they are. `fmap`

is just `calc_stuff`

generalized

– generalizing the function that `calc_stuff`

takes in and

– generaizing the type `Maybe`

to other types that take in a single type variable

And the `f`

we see in the `fmap`

is the Functor, a typeclass of which `Maybe`

is an instance of.