General Computation Methods as Higher-Order Functions

So far, we’ve mainly defined functions whose arguments or return values are numbers, boolean values, or strings. But recall that functions are first-class citizens in functional programming languages. Thanks to this, we can easily define a function that accepts other functions as arguments or returns another function as a result. Such a function is called a higher-order function. High-order functions are powerful because they enable us to formulate computation patterns that work with different functions.

Summation as a higher-order function

A good way to appreciate the power of functions operating on other functions is to look at summation in mathematics. For instance, mathematicians often study summations, sums of a sequence of numbers, like the sum of all natural numbers from 1 to $n$:

$1 + 2 + 3 + \ldots + n$

Or the sum of squares of natural numbers from 1 to $n$:

$1^2 + 2^2 + 3^2 + \ldots + n^2$ ...