What is sigma notation?

What is a summation, often called “sigma” ($\sum$)? In this article, we start with the basics: what the symbol means, how its parts fit together, and which formulas are useful to know. Let’s take a look!

In mathematics, the symbol $\sum$, called sigma, provides a shorter way to write the sum of a sequence of numbers.

For example, adding the numbers from 1 to 5 gives:

We can express this using the $\sum$ symbol together with 1 and 5:

The parts of sigma notation

Sigma notation has three main parts:

1. The lower limit ($\colorbox{#fee2e2}{$i=m$}$) tells us where to start. It is written as “variable = starting value.” The variable indicates where each successive value is substituted in the expression being summed. The letter $i$ is common, but any letter can be used.
2. The upper limit ($\colorbox{#fef9c3}{$n$}$) tells us where to stop.
3. The summand ($\colorbox{#dcfce7}{$x_i$}$) is the expression in that variable that tells us what to add each time. If the variable is $i$, we can write the expression as $f(i)$.

More examples and how to evaluate them

Properties of summation

Let $m$ and $n$ be positive integers with $m<n$, and let $c$ be any real number.

Useful formulas

1. Adding the integers from 1 to n

To add the positive integers from 1 to n, use:

For example:

2. Adding a constant

Adding the same number or constant n times is equivalent to multiplication.

For example:

3. The difference plus one

We can also use summation to work with a difference. Although the result is not exactly the difference between the bounds, this technique is another way to construct numbers in mental arithmetic.

For example:

Sigma notation may look difficult at first, but once you understand it, you can see how useful it is. We hope you now have a clearer understanding of the symbol and can use it in your studies, understand equations in textbooks, and build hundreds of different numbers in mental arithmetic games.

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