Infix operators

Prelude

Logical

&& is logical AND, || is logical OR.

== is equality, /= non-equality, < / <= lesser and > / >= greater operators.

Arithmetic operators

The numerical operators +, - and / behave largely as you’d expect. (Division works only on fractional numbers to avoid rounding issues – integer division must be done with quot or div). More unusual are Haskell’s three exponentiation operators:

[`^`](http://hackage.haskell.org/packages/archive/base/latest/doc/html/Prelude.html#v:-94-) takes a base of any number type to a non-negative, integral power. This works simply by ([fast](https://en.wikipedia.org/wiki/Exponentiation_by_squaring)) iterated multiplication. E.g.

4^5  ≡  (4*4)*(4*4)*4

[`^^`](http://hackage.haskell.org/packages/archive/base/latest/doc/html/Prelude.html#v:-94-) does the same in the positive case, but also works for negative exponents. E.g.

3^^(-2)  ≡  1 / (2*2)

Unlike ^, this requires a fractional base type (i.e. 4^^5 :: Int will not work, only 4^5 :: Int or 4^^5 :: Rational).

[`**`](http://hackage.haskell.org/package/base/docs/Prelude.html#v:-42--42-) implements real-number exponentiation. This works for very general arguments, but is more computionally expensive than `^` or `^^`, and generally incurs small floating-point errors.

2**pi  ≡  exp (pi * log 2)

Lists

There are two concatenation operators:

`:` (pronounced [cons](https://en.wikipedia.org/wiki/Cons)) prepends a single argument before a list. This operator is actually a constructor and can thus also be used to **pattern match** (“inverse construct”) a list.

[`++`](http://hackage.haskell.org/packages/archive/base/latest/doc/html/Prelude.html#v:-43--43-) concatenates entire lists.

[1,2] ++ [3,4]  ≡  1 : 2 : [3,4]  ≡  1 : [2,3,4]  ≡  [1,2,3,4]

!! is an indexing operator.

[0, 10, 20, 30, 40] !! 3  ≡  30

Note that indexing lists is inefficient (complexity O(n) instead of O(1) for arrays or O(log n) for maps); it’s generally preferred in Haskell to deconstruct lists by folding ot pattern matching instead of indexing.

Control flow

[`$`](http://hackage.haskell.org/package/base/docs/Prelude.html#v:-36-) is a function application operator.

f $ x  ≡  f x
       ≡  f(x)  -- disapproved style

This operator is mostly used to avoid parentheses. It also has a strict version $!, which forces the argument to be evaluated before applying the function.

[`.`](http://hackage.haskell.org/package/base/docs/Prelude.html#v:-36-) composes functions.

(f . g) x  ≡  f (g x)  ≡  f $ g x

[`>>`](http://hackage.haskell.org/packages/archive/base/latest/doc/html/Prelude.html#v:-62--62-) sequences monadic actions. E.g. `writeFile "foo.txt" "bla" >> putStrLn "Done."` will first write to a file, then print a message to the screen.

[`>>=`](http://hackage.haskell.org/packages/archive/base/latest/doc/html/Prelude.html#v:-62--62--61-) does the same, while also accepting an argument to be passed from the first action to the following. `readLn >>= \x -> print (x^2)` will wait for the user to input a number, then output the square of that number to the screen.

Finding information about infix operators

Because infixes are so common in Haskell, you will regularly need to look up their signature etc.. Fortunately, this is just as easy as for any other function:

The Haskell search engines [Hayoo](https://hayoo.fh-wedel.de/) and [Hoogle](https://www.haskell.org/hoogle/) can be used for infix operators, like for anything else that's defined in some library.

In GHCi or IHaskell, you can use the `:i` and `:t` (**i**nfo and **t**ype) directives to learn the basic properties of an operator. For example,

Prelude> :i +
class Num a where
  (+) :: a -> a -> a
  ...
      -- Defined in ‘GHC.Num’
infixl 6 +
Prelude> :i ^^
(^^) :: (Fractional a, Integral b) => a -> b -> a
      -- Defined in ‘GHC.Real’
infixr 8 ^^

This tells me that ^^ binds more tightly than +, both take numerical types as their elements, but ^^ requires the exponent to be integral and the base to be fractional.
The less verbose :t requires the operator in parentheses, like

Prelude> :t (==)
(==) :: Eq a => a -> a -> Bool

Custom operators

In Haskell, you can define any infix operator you like. For example, I could define the list-enveloping operator as

(>+<) :: [a] -> [a] -> [a]
env >+< l = env ++ l ++ env

GHCi> "**">+<"emphasis"
"**emphasis**"

You should always give such operators a fixity declaration, like

infixr 5 >+<

(which would mean >+< binds as tightly as ++ and : do).

Remarks

Most Haskell functions are called with the function name followed by arguments (prefix notation). For functions that accept two arguments like (+), it sometimes makes sense to provide an argument before and after the function (infix).