In the factorial example, the program returns 1 if the parameter is 0，and returns n * fact (n - 1) otherwise. In this way, the value returned by a function is represented by an expression. 1 in the first case is an expression representing the natural number $1$. n * fact (n - 1) is an expression involving variable n and function application. The basic principle of ML programming is

programming is done by defining an expression that represents the desired value.

An expression consist of the following components:

1. 1.

   constants such as 1,

2. 2.

   variables representing function parameters and defined values.

3. 3.

   function applications (function calls), and

4. 4.

   functions and other data structure constructions

The items 1 through 3 are the same as in mathematical expressions, we have leaned in school. For example, the sum of the arithmetic sequence $S{}_n=1{}^2+2{}^2+\mathrm{\cdots }+n{}^2$ is given by the following equation.

This is directly programmed as follows.

val Sn = (n * (n + 1) * (2 * n + 1)) div 6

* and div are integer multiplication and division, respectively. When $n$ is defined, it correctly computes $S{}_n$ as seen below.

# val n = 10;
val n = 10 : int
# val Sn = (n * (n + 1) * (2 * n + 1)) div 6;
val Sn = 385 : int