What does # mean in Mathematica?

Question

Does anyone know What # in for example Root[-1 - 2 #1 - #1^2 + 2 #1^3 + #1^4 &, 1] means in Mathematica?

Then what does Root[-1 - 2 #1 - #1^2 + 2 #1^3 + #1^4 &, 1] exactly mean?

Thanks.

Answer 1

#1 represents the first argument in a pure function.

If you have multiple arguments #1, #2, #3... refer to the first, second, third argument and so on.

Answer 2

Notation # is (as stated above) used to mean "a variable goes here" in a pure function ("closure" for you traditional developers). It must always be followed at the end by &.

Best example is this: f[x_]:=x+5. This creates a delayed set, that any time a value is passed into a symbol reference f as a functional parameter, that value will be given a local context function-specific name of x (not affecting the global definition of x, if there is one). Then the expression x+5 will be evaluated using this new variable/value. The above process requires that symbol f be initialized, local variable x created, and expression x+5 is permanently held in memory, unless you clear it.

Side note: f=5 and f[x_]:=5 both work with a "Symbol" f. f can be referred to as a function, when square brackets are used to extract its value, and f[x_] can peacefully co-exist with f[x_,y_] without overriding each other. One will be used when one parameter is sent, and another when 2 parameters are sent.

Some times you just need a quick function and do not need to define it and leave it hanging. So, (someValue + 5) becomes (#+5)&, where & says "I'm a pure function, and will work with whatever you send me", and # says "I'm the parameter (or a parameter list) that was sent to the pure function". You can also use #1, #2, #3, etc, if you're sending it more than 1 parameter.

Example of multi-parameter pure function in common use:

Let's say mydata is a list of lists, which you need to sort by median of the lists (e.g. housing price data from various US cities):

Sort[ myData , Median[#1] > Median[#2]& ]

Quick tip, if you're applying a function to a single value, it may look neater and cleaner, and uses less typing to use @ instead of [], which essentially means Prefix. Do not confuse with Map (/@) or Apply(@@). The above command then becomes:

Sort[ myData , Median@#1 > Median@#2 & ] 

You can chain @ as such: Reverse@Sort@DeleteDuplicates[...]

Answer 3

It's a placeholder for a variable.

If you want to define a y(x)=x^2 function, you just could do:

  f = #^2 & 

The & "pumps in" the variable into the # sign. That is important for pairing & and # when you have nested functions.

  In: f[2]  
  Out: 4   

If you have a function operating on two vars, you could do:

 f = #1 + #2 &

So

  In: f[3,4]  
  Out: 7  

Or you may have a function operating in a list, so:

 f = #[[1]] + #[[2]] &

So:

  In: f[{3,4}]
  Out: 7

About Root[]

According to Mathematica help:

Root[f,k] represents the exact kth root of the polynomial equation f[x]==0  .

So, if your poly is x^2 - 1, using what we saw above:

        f = #^2 - 1 &

In[4]:= Root[f, 1]  

Out[4]= -1  (* as we expected ! *)

And

In[5]:= Root[f, 2]  

Out[5]= 1  (* Thanks God ! *)

But if we try with a higher order polynomial:

         f = -1 - 2 #1 - #1^2 + 2 #1^3 + #1^4 &  

In[6]:= Root[f, 1]

Out[6]= Root[-1 - 2 #1 - #1^2 + 2 #1^3 + #1^4 &, 1]

That means Mathematica doesn't know how to caculate a symbolic result. It's just the first root of the polynomial. But it does know what is its numerical value:

In[7]:= N@Root[-1 - 2 #1 - #1^2 + 2 #1^3 + #1^4 &, 1]

Out[7]= -2.13224

So, Root[f,k] is a kind of stenographic writing for roots of polynomials with order > 3. I save you from an explanation about radicals and finding polynomial roots ... for the better, I think

Answer 4

How to find out what any built-in syntax means in Mathematica:

1. Copy expression
2. Do TreeForm[Hold[paste the expression here]].
3. Mouse-over parts of the tree to identify the syntax in question, in this case Slot
4. Enter "?Slot"