Summation and Product Notation

Introduction

In this arti­cle we are going to talk about the sum­ma­tion and prod­uct nota­tion, what they mean and how to per­form the cal­cu­la­tions. At the end we will see how to pro­gram sum­ma­tion and prod­uct nota­tion in Java or C# language.

What is summation and product notation?

Sum¬≠ma¬≠tion and prod¬≠uct are ways of defin¬≠ing math¬≠e¬≠mat¬≠i¬≠cal oper¬≠a¬≠tions that con¬≠sist of sequences of sums and prod¬≠ucts respectively. 

Imag¬≠ine that we have to add or mul¬≠ti¬≠ply all the num¬≠bers start¬≠ing at 1 and end¬≠ing at 1000, but also mul¬≠ti¬≠ply each of them by 3. 

Using sym­bol­o­gy we can write long oper­a­tions like that in short. In addi­tion the prop­er­ties can help us to sim­pli­fy the expres­sion and to cal­cu­late the results.

Sigma Notation

In the next video you can see the con¬≠cept of the sig¬≠ma nota¬≠tion and an excer¬≠cise solved using pro¬≠gram¬≠ming tech¬≠niques in Unity.

Implement sigma notation in programming
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The sum­ma­tion is sym­bol­ized by the Greek let­ter Sig­ma in cap­i­tal letters.

Below the sum¬≠ma¬≠tion sym¬≠bol the index vari¬≠able is indi¬≠cat¬≠ed and from which val¬≠ue it starts. At the top of the sym¬≠bol is the last val¬≠ue to be tak¬≠en by the index.

To the right of the sym¬≠bol is the expres¬≠sion that deter¬≠mines all the terms of the sum¬≠ma¬≠tion, each term aris¬≠es from replac¬≠ing the index in the expres¬≠sion from the first to the last value.

In equa­tion 1 we see on the left side the expres­sion of the sum­ma­tion and on the right side the devel­oped operation.

(1)   \begin{equation*}    \sum_{i=1}^{5}{i}=1+2+3+4+5=15\end{equation*}

How to program a Summation?

To imple­ment the sum­ma­tion in some pro­gram­ming lan­guage is rel­a­tive­ly sim­ple, we must define a vari­able to accu­mu­late the sum that must be ini­tial­ized in 0 (neu­tral ele­ment for the sum). We have to take into account which numer­i­cal set the sum­mands belong to, this will depend on the type of vari­able to choose.

We define an inte­ger vari­able for the end of the summation.

We make a for loop that goes from the begin¬≠ning of the sum¬≠ma¬≠tion to the end and inside the loop we define the oper¬≠a¬≠tion to make. 

In fig¬≠ure 1 we see the imple¬≠men¬≠ta¬≠tion of the sum of equa¬≠tion 1 togeth¬≠er with the print¬≠ed result in console. 

programming of a summation of the first 5 whole numbers in java language, netbeans
Fig. 1: Imple¬≠men¬≠ta¬≠tion of the sum¬≠ma¬≠tion of equa¬≠tion 1 in Java, Net¬≠beans IDE.

Symbol of Product notación

The prod¬≠uct nota¬≠tion is sym¬≠bol¬≠ised by the let¬≠ter Pi in cap¬≠i¬≠tal letters. 

Under the pro¬≠duc¬≠er sym¬≠bol is indi¬≠cat¬≠ed the index vari¬≠able and from which val¬≠ue it starts. In the upper part of the sym¬≠bol is indi¬≠cat¬≠ed the last val¬≠ue that the index will take. 

To the right of the sym¬≠bol is the expres¬≠sion that deter¬≠mines all the terms of the pro¬≠duc¬≠er, each term aris¬≠es from replac¬≠ing the index in the expres¬≠sion from the first to the last value. 

In equa¬≠tion 2 we see on the left side the expres¬≠sion of the pro¬≠duc¬≠er and on the right side the devel¬≠oped operation. 

(2)   \begin{equation*}    \prod_{i=1}^{5}{i}= 1.2.3.4.5 = 120\end{equation*}

Implement the product notation in programming

We must define a vari­able to accu­mu­late the prod­uct that must be ini­tial­ized in 1 (neu­tral ele­ment of the prod­uct). We also ana­lyze the type of num­bers we are mul­ti­ply­ing to choose the type of the accu­mu­la­tion variable.

We define a whole vari­able for the end of the producer.

We make a for loop that goes from the begin¬≠ning of the pro¬≠duc¬≠tion to the end and inside the loop we define the oper¬≠a¬≠tion to be performed. 

In fig¬≠ure 2 we see the imple¬≠men¬≠ta¬≠tion of the pro¬≠duc¬≠er of equa¬≠tion 2 togeth¬≠er with the print¬≠ed result in console. 

programming of a producer of the first 5 whole numbers in java language, netbeans
Fig. 2: Imple¬≠men¬≠ta¬≠tion of the equa¬≠tion 2 pro¬≠duc¬≠er in Java, Net¬≠beans IDE. 

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