Understanding the fundamentals of summation symbol (sigma) in mathematic

Summation Symbol ($\sum$) §

In mathematic the summation symbol capital sigma is generally used to reqpresents summation of similar terms.

Simple Sum §

For example, adding a sequence of number up to the n number starting from 1 can be compactly describe in this manner:

$$\begin{array}{r}\sum _{i=1}^{n}i=1+2+3+...+n\\ \text{where,}\\ \text{i = lowerbound of the summation}\\ \text{n = upperbound of the summation}\end{array}$$

More generally, the expression ${\sum }_{i=1}^n{x}_i$ represents the sum of n terms

$$x_1+x_2+x_3+...+x_n$$

A few more examples:

$$\begin{array}{rl}\sum _{i=1}^{n}4x_i& =4x_1+4x_2+4x_3+...+4x_n\\ & =4(x_1+x_2+x_3+...+x_n)\end{array}$$

Thus,

$$\sum _{i=1}^{n}4x_i=4(\sum _{i=1}^n{x}_i)$$

Another example:

$$\begin{array}{rl}\sum _{i=1}^3(x_i+y_i)& =(x_1+y_1)+(x_2+y_2)+(x_3+y_3)\\ & =x_1+x_2+x_3+y_1+y_2+y_3\end{array}$$

Thus,

$$\sum _{i=1}^n(x_i+y_i)=\sum _{i=1}^n{x}_i+\sum _{i=1}^n{y}_i$$

Attention §

We must not confuse the following expressions pairs. Because the expression:

$$\sum _{i=1}^n{x}_i{y}_i$$

is not the same as:

$$(\sum _{i=1}^n{x}_i)(\sum _{i_1}^n{y}_i)$$

And the expression

$$\sum _{i=1}^n{x}_i^2$$

is not the same as:

$$(\sum _{i=1}^n{x}_i{)}^2$$

Let’s explorer further on why that is by subsituting some arbitrary numbers into the variables.

For example, given $i=1$, $n=3$, $x_1=4$, $x_2=2$, $x_3=7$ and $y_1=2$, $y_2=2$, $y_3=3$:

Conclusion 1,

$$\begin{array}{rl}\sum _{i=1}^n{x}_i{y}_i& \ne (\sum _{i=1}^n{x}_i)(\sum _{i=1}^n{y}_i)\\ x_1{y}_1+x_2{y}_2+x_3{y}_3& \ne (x_1+x_2+x_3)(y_1+y_2+y_3)\\ (4\ast 2)+(2\ast 2)+(7\ast 3)& \ne (4+2+7)\ast (2+2+3)\\ 31& \ne 91\end{array}$$

Conclusion 2,

$$\begin{array}{rl}\sum _{i=1}^n{x}_i^2& \ne (\sum _{i=1}^n{x}_i{)}^2\\ x_1^2+x_2^2+x_3^2& \ne (x_1+x_2+x_3{)}^2\\ 4^2+2^2+7^2& \ne (4+2+7{)}^2\\ 69& \ne 169\end{array}$$

Double Sums §

Summation doubles are just simply summation of a summation, the below expression are the ones you’ll likely encounter when you’re dealing with double sums.

The basic double sum expression:

$$\begin{array}{rl}\sum _{i=1}^m{x}_i\sum _{j=1}^n{y}_j& =\sum _{i=1}^m\sum _{j=1}^n{x}_i{y}_j\end{array}$$

Let’s replace some arbitrary numbers into upperbounds of the summations to explore them:

$$\begin{array}{rl}\sum _{i=1}^3\sum _{j=1}^2{x}_i{y}_j& =\sum _{i=1}^3(x_i{y}_1+x_i{y}_2)\\ & =x_1{y}_1+x_1{y}_2+x_2{y}_1+x_2{y}_2+x_3{y}_1+x_3{y}_2\end{array}$$

Less general case:

$$\begin{array}{rl}(\sum _{i=1}^n{x}_i{)}^2& =\sum _{i=1}^n{x}_i\sum _{j=1}^n{x}_j\\ & =\sum _{i=1}^n\sum _{j=1}^n{x}_i{x}_j\\ & =\sum _{i=1}^n{x}_i^2+\sum _{i=1}^n\sum _{j=1}^n{x}_i{x}_j(j\ne i)\end{array}$$

Explanations for above relations with an example of $n=3$:

Let’s sort out the $({\sum }_{i=1}^3{x}_i{)}^2$ first:

$$\begin{array}{rl}(\sum _{i=1}^3{x}_i{)}^2& =(x_1+x_2+x_3{)}^2\\ & =(x_1^2+x_2^2+x_3^2)+(2x_1{x}_2+2x_1{x}_3+2x_2{x}_3)\\ & =x_1^2+x_2^2+x_3^2+x_1{x}_2+x_1{x}_3+x_2{x}_3+x_1{x}_2+x_1{x}_3+x_2{x}_3\end{array}$$

Exploring relation 1:

$$\begin{array}{rl}\sum _{i=1}^3{x}_i\sum _{j=1}^3{x}_j& =(x_1+x_2+x_3)(x_1+x_2+x_3)\\ & =x_1^2+x_2^2+x_3^2+x_1{x}_2+x_1{x}_3+x_2{x}_3+x_1{x}_2+x_1{x}_3+x_2{x}_3\end{array}$$

Exploring relation 2:

$$\begin{array}{rl}\sum _{i=1}^3\sum _{j=1}^3{x}_i{x}_j& =\sum _{i=1}^3(x_i{x}_1+x_i{x}_2+x_i{x}_3)\\ & =(x_1{x}_1+x_1{x}_2+x_1{x}_3)+(x_2{x}_1+x_2{x}_2+x_2{x}_3)+(x_3{x}_1+x_3{x}_2+x_3{x}_3)\\ & =x_1^2+x_2^2+x_3^2+x_1{x}_2+x_1{x}_3+x_2{x}_3+x_1{x}_2+x_1{x}_3+x_2{x}_3\end{array}$$

Explanation for last relation:

$$\begin{array}{rl}\sum _{i=1}^3{x}_i^2+\sum _{i=1}^3\sum _{j=1}^n{x}_i{x}_j(i\ne j)& =(x_1^2+x_2^2+x_3^2)+x_1(x_2+x_3)+x_2(x_1+x_3)+x_3(x_1+x_2)\\ & =x_1^2+x_2^2+x_3^2+x_1{x}_2+x_1{x}_3+x_2{x}_3+x_1{x}_2+x_1{x}_3+x_2{x}_3\end{array}$$

There are few other complex double sums examples but we’ll not dive into them in this article.

Double Index §

Sometime in computer science and afew other areas where we need to represent the data of a table or a matrix. In the following example we’ll explorer double index notations where in $x_{i}j$ the first index (i) represent the number of the row where the data is located and the second (j) to the column.

$x_{49}$ represents the data at row 4 and column 9

Example:

Given

$$\begin{array}{r}{x}_{11}=2,x_{12}=4,x_{13}=5,x_{14}=1\\ x_{21}=1,x_{22}=8,x_{23}=3,x_{24}=2\\ x_{31}=11,x_{32}=9,x_{33}=2,x_{34}=3\end{array}$$

Let’s visualize the data with a table.

x	$x_{j=1}$	$x_{j=2}$	$x_{j=3}$	$x_{j=4}$
$x_{i=1}$	2	4	5	1
$x_{i=2}$	1	8	3	2
$x_{i=3}$	11	9	2	3

To figure out the sum of the terms of a row, we fix the index of the row and vary for all possible values, the index of the column.

For exmaple:

$$\sum _{j=1}^3{x}_{1j}=x_{11}+x_{12}+x_{13}+x_{14}=2+4+5+1=12$$

To carry out the summation of the whole of the table, we vary both indices and use double sum:

$$\begin{array}{rl}\sum _{i=1}^3\sum _{j=4}^4& =\sum _{i=3}^3{x}_{i1}+x_{i2}+x_{i3}+x_{i4}\\ & =x_{11}+x_{12}+x_{13}+x_{14}+x_{21}+x_{22}+x_{23}+x_{24}+x_{31}+x_{32}+x_{33}+x_{34}\\ & =2+4+5+1+1+8+3+2+11+9+2+3\\ & =51\end{array}$$