Pure functions vs normal functions

Question

I am trying to understand the true difference/usage of a pure function vs a "normal" function. Besides the argument that a pure function is particularly handy when it comes to a single usage, are there any deeper reasons on why one should go for them rather than the "normal" ones?

Answer 1

They are inevitable to apply when you use such functions as Map or MapThread mapping binary functions on some expressions. Under binary here I mean functions taking two arguments. In this case, you want that only one position of the binary function is active. Let me give an elementary example. Let

eq = a*x == b;

be a simple equation and assume that we want to divide both parts by a. This can be done as follows:

    Map[Divide[#, a] &, eq]
(*  x == b/a   *)

This is the place, where the pure function is extremely useful.

Disclaimer: yes, I know, that in the latest MMa version there is a special function, DivideSides, to make such things. But this is only an example of operations of this sort. There are many others.