Function Aliases

Mathematica has a series of useful function aliases that make coding more concise and readable.

Technically, these are called infix operators , a term that also encapsulates + , - , and * , but that’s unimportant for their usage here.

@ (Prefix)

The @ alias is like a pair of square brackets. The following illustrates its usage:

N@π

(*Out:*)

3.141592653589793`

It only applies to a single argument however. For example, Mathematica will not evaluate the following command

N@π,12

/@ (Map)

/@ takes a function and applies to every element in a list, returning the list of results. Example:

Sin/@(π*{0,1/4,1/2,3/4,1,1+1/4,1+1/2,1+3/4,2})

(*Out:*)

{0,,1,,0,-,-1,-,0}

When using this alias care should be taken that the right-hand side is wrapped in parentheses, as was done above, if it’s more than just a simple list.

@@ (Apply)

@@ applies a function to all the arguments in a list. That is, the elements of the list become the arguments to the function. Example:

N@@{π,12}

(*Out:*)

3.14159265358979323846264338327950419984`12.

// (Postfix)

// is more or less a reversed form of @ . Example:

π//N

(*Out:*)

3.141592653589793`

But there is a small difference in terms of how the two work with other operators.

Consider you’re trying to apply the function f to the list of results generated by using g/@{1,2,3,4} One might first try the following:

f@g/@{1,2,3,4}

(*Out:*)

{f[g][1],f[g][2],f[g][3],f[g][4]}

This clearly doesn’t work. f is applied to g before being mapped across {1,2,3,4} .

In this case there are two options:

f@(g/@{1,2,3,4})

(*Out:*)

f[{g[1],g[2],g[3],g[4]}]

g/@{1,2,3,4}//f

(*Out:*)

f[{g[1],g[2],g[3],g[4]}]

because /@ evaluates before // , this works. What is said is that // has a low precedence .

Precedences for any operator can be checked using the function Precedence , if the function name is known. Function names for a given operator can be found by selecting the operator and using ––F on Mac or F1 on Windows.

For example, knowing that // is an alias for the function Postfix

Precedence@Postfix

(*Out:*)

70.`

And we’ll compare that to the function Prefix for which @ is an alias

Prefix//Precedence

(*Out:*)

640.`

~ (Infix)

~ takes a function and applies it to the preceding and following arguments as follows:

π~N~12

(*Out:*)

3.14159265358979323846264338327950419984`12.

This is useful and worth knowing, but it can often be cleaner simply to use @@ , potentially in combination with //

`@*` (Composition)

@* takes a two functions and composes them. For example:

(f@*g)[1]

(*Out:*)

f[g[1]]

This is particularly useful when negating a function, that is, applying the Not function to it. We can make a simple NotTrueQ function by doing this:

NotTrueQ = (Not@*TrueQ);
NotTrueQ@1
NotTrueQ@False
NotTrueQ@True

(*Out:*)

True

(*Out:*)

True

(*Out:*)

False

Function Aliases