# What Is Sigma Notation?

The concept of sigma notation means to sum up all terms and uses three parts to form math statements, like *∑ _{i}* a

_{i}. The Greek letter

*∑*is the summation operator and means the sum of all,

*i*is called the index number, and a

_{i}refers to a series of terms to be added together. This mathematical notation is used to compactly write down the equations in which summing all terms is required. It can be used, for example, to show the addition of all employees’ hours at a company. If a

_{i}is the hours worked by a certain employee and there are

*n*employees, then

*∑*a

_{i}_{i}means to add

*a*.

_{1}+a_{2}+a_{3}+a_{4}…a_{n}Understanding the associative, distribution and commutative properties allows for more uses of these mathematics. The associative and commutative properties will allow any number to be multiplied by all terms of the summation. Instead of performing the multiplication for each term, it can be done once at the end with the sum of all terms. If every employee earned *k* per hour, the notation is written compactly as k *∑ _{i}* a

_{i}. The distribution property changes the sum of two series of numbers into two sigma notation formulas.

Sigma notation, often referred to as summation notation, can be used in many common situations. For example, it can be used to calculate the sum of deposits for a bank account. Banks add together all deposits and withdrawals to determine the current balance. A grocery receipt shows all the items to be added and subtracted to calculate a checkout total. All of these examples can be written in a short formula.

There are many complex examples of the use of the notation as well. Many college students need sigma notation to make equations to solve difficult problems. Computer programmers use sigma notation for finance, business and gaming software. Scientists use it often in statistical analysis of their experiments.

The history of sigma notation was changed by Carl Friedrich Gauss in the late 18th century. He was asked to calculate the sum of the first 100 integers. He returned moments later with the correct answer, 5050. He realized a new theorem, that *∑ _{i}* a

_{i}is the same as adding the first and last numbers, such as 100+1 then 99+2, which always gives the same answer, 50 times over. He was a young child when he discovered this theorem and went on to become a renowned mathematician.

## Discussion Comments

I had to do a paper on Carl Friedrich Gauss some time ago. Despite his mind for math, he was an interesting character. You want to imagine a Sheldon Cooper type of guy when thinking of a mathematician and he was, at least to some degree. He was a young genius, a perfectionist and a school kid who could misbehave and produce correct arithmetic answers within seconds. I’m sure his instructors loved him!One famous Gauss story is the one where he was interrupted in midsts of a problem to be told his wife was dying. Supposedly he replied,"Tell her to wait a moment till I'm done.” No doubt the guy loved his math!

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