Product Operator

#beginner #intermediate

The product operator denoted with the mathematical symbol $Π$ is similar to the summation operator, but instead of summing elements, it multiplies them. The general form of the product operator is:

\begin{aligned} \prod_{i = 0}^{n} x_{i} = x_{1} \times x_{2} \times x_{3} \times \dots \times x_{n} \\ i = index of product operator \\ n = upper limit of product operator \end{aligned}

Translating to Code

In code, the product operator can be implemented using a for loop, where you multiply the elements in each iteration.

float product = 1;
float[] x;
int n = x.Length;

for (int i = 0; i <= n; i++)
{
    product *= x[i];
}

Let's take a simple example:

\begin{aligned} P = \prod_{i = 1}^{5} i = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 & = 120 \\ = 5! \end{aligned}

float product = 1;
int n = 5;

for (int i = 1; i <= n; i++)
{
    product *= i;
}

Often, we need to compute the product of a function applied to a sequence of elements. For example:

P = \prod_{i = 0}^{n} f (x_{i}) = f (x_{0}) \cdot f (x_{1}) \cdot \dots \cdot f (x_{n})

float product = 1;
float[] x;
int n = x.Length;

for (int i = 0; i < n; i++)
{
    float hitDamage = x[i];
    
    product *= CalculateComboScore(hitDamage);
}

Console.WriteLine("Total Combo Score: " + product);

Additional Resources

For more learning on the topic: