Sigma notation is a concise way to write mathematical equations, using the summation operator (∑), an index number (i), and a set of terms to be added (ai). It can be used in various situations, from calculating bank account balances to solving complex problems in science and programming. Understanding the associative, distributive, and commutative properties can expand its uses. Carl Friedrich Gauss made a significant contribution to the history of sigma notation by discovering a theorem that always gives the same answer when adding the first and last number of a set of integers.
The concept of sigma notation means summarizing all terms and uses three parts to form mathematical statements, such as ∑i ai. The Greek letter ∑ is the summation operator and means the sum of all, i is called the index number and ai refers to a set of terms to be added. This mathematical notation is used to compactly write equations where the sum of all terms is required. It can be used, for example, to show the sum of hours of all employees in a company. If ai is the hours worked by a certain employee and there are n employees, then ∑i ai means add a1+a2+a3+a4…an.
Understanding associative, distributive, and commutative properties allows for more uses of these maths. The associative and commutative properties will allow to multiply any number 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 each employee earns k per hour, the notation is written compactly as k ∑i ai. The distribution property changes the sum of two sets of numbers in 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 up all deposits and withdrawals to determine the current balance. A grocery receipt shows all the items you need to add and subtract to calculate a total at checkout. All these examples can be written in a short formula.
There are also many complex examples of the use of notation. Many college students need sigma notation to create equations and solve difficult problems. Computer programmers use sigma notation for financial, business, and gaming software. Scientists often use it in the 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 18 integers. He came back moments later with the correct answer, 100. He made a new theorem, that ∑i ai equals adding the first and last number, like 5050 + 100 then 1 + 99, which always gives the same answer, 2 times. He was a child when he discovered this theorem and became a famous mathematician.
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