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Stephen Wolfram
An Elementary Introduction to the Wolfram Language
Second Edition (2017)
25Ways to Apply Functions
When you write f[x] it means “apply the function f to x”. An alternative way to write the same thing in the Wolfram Language is f@x.
f@x is the same as f[x]:
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It’s often convenient to write out chains of functions using @:
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Avoiding the brackets can make code easier to type, and read:
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There’s a third way to write f[x] in the Wolfram Language: as an “afterthought”, in the form x//f.
Apply f “as an afterthought” to x:
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You can have a sequence of “afterthoughts”:
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The functions here read in the order they are applied:
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A particularly common use of // is to apply N (for numerical evaluation) “as an afterthought”.
Apply numerical evaluation “as an afterthought”:
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In working with the Wolfram Language, a powerful notation that one ends up using all the time is /@, which means “apply to each element”.
Apply f to each element in a list:
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f usually would just get applied to the whole list:
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Framed is a function that displays a frame around something.
Display x framed:
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Applying Framed to a list just puts a frame around the whole list.
Apply Framed to a whole list:
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@ does exactly the same thing:
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Now use /@ to apply Framed to each element in the list:
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The same thing works with any other function. For example, apply the function Hue separately to each number in a list.
/@ applies Hue separately to each number in the list:
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Here’s what the /@ is doing:
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It’s the same story with Range, though now the output is a list of lists.
/@ applies Range separately to each number, producing a list of lists:
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Here’s the equivalent, all written out:
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Given a list of lists, /@ is what one needs to do an operation separately to each sublist.
Apply PieChart separately to each list in a list of lists:
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You can use exactly the same idea with lots of other functions.
Apply Length to each element, getting the length of each sublist:
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Applying Length to the whole list just gives the total number of sublists:
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Apply Reverse to each element, getting three different reversed lists:
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Apply Reverse to the whole list, reversing its elements:
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As always, the form with brackets is exactly equivalent:
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Some calculational functions are listable, which means they automatically apply themselves to elements in a list.
N is listable, so you don’t have to use /@ to get it applied to each element in a list:
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The same is true with Prime:
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A function like Graphics definitely isn’t listable.
This makes a single graphic with three objects in it:
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This gives three separate graphics, with Graphics applied to each object:
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When you enter f/@{1,2,3}, the Wolfram Language interprets it as Map[f,{1,2,3}]. f/@x is usually read as “map f over x”.
The internal interpretation of f/@{1, 2, 3}:
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Vocabulary
f@x equivalent to f[x]
x//f equivalent to f[x]
f/@{a,b,c} apply f separately to each element of the list
Map[f,{a,b,c}] alternative form of /@
Framed[expr] put a frame around something
25.1Use /@ and Range to reproduce the result of Table[f[n], {n, 5}]. »
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25.2Use /@ twice to generate Table[f[g[n]], {n, 10}]. »
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25.3Use // to create a[b[c[d[x]]]]. »
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25.6Use /@ to draw separate maps of each country in the G5. »
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25.7Binarize each flag in Europe, and make an image collage of the result. »
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25.8Find a list of the dominant colors in images of the planets, putting the results for each planet in a column. »
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25.9Find the total of the letter numbers given by LetterNumber for the letters in the word “wolfram”. »
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Q&A
Why not always use f@x instead of f[x]?
f@x is a fine equivalent to f[x], but the equivalent of f[1+1] is f@(1+1), and in that case, f[1+1] is shorter and easier to understand.
Why is /@ called Map?
It comes from math. Given a set {1, 2, 3}, f/@{1, 2, 3} can be thought of as mapping of this set to another one.
How does one say "//" and "/@"?
Typically “slash slash” and “slash at”.
When do I need to use parentheses with @, // and /@?
It’s determined by the precedence or binding of different operators. @ binds tighter than +, so f@1+1 means f[1]+1 not f@(1+1) or f[1+1]. // binds looser than +, so 1/2+1/3//N means (1/2+1/3)//N. In a notebook you can find how things are grouped by repeatedly clicking on your input, and seeing how the selection expands.
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