Sigma and Pi Notation
is a capital letter from the Greek alphabet called “Sigma”… it corresponds to “S” in our alphabet (think of the starting sound of the word “sigma”). It is used in mathematics to represent the sum of a bunch of terms (think of the starting sound of the word “sum”: Sssigma = Sssum).
Sigma can be used all by itself to represent a generic sum… the general idea of a sum, of an unspecified number of unspecified terms:
But this is not something that can be evaluated to produce a specific answer, as we have not been told how many terms to include in the sum, nor have we been told how to determine the value of each term.
A more typical use of Sigma notation will include an integer below the Sigma (the “starting term number”), and an integer above the Sigma (the “ending term number”). In the example below, the exact starting and ending numbers don’t matter much since we are being asked to add the same value, two, repeatedly. All that matters in this case is the difference between the starting and ending term numbers… that will determine how many twos we are being asked to add, one two for each term number.
Sigma notation is not usually worth the extra ink to describe simple sums such as the one above… multiplication could do that more simply.
Sigma notation is most useful when the “term number” can be used in some way to calculate each term. To facilitate this, a variable is usually listed below the Sigma with an equal sign between it an the starting term number. If this variable appears in the expression being summed, then the current term number should be substituted in for the variable:
Note that it is possible to have a variable below the Sigma, but never use it. In such cases, just as in the example that resulted in a bunch of twos above, the term being added never changes:
The “starting term number” need not be 1. It can be any value, including 0. For example:
That covers what you need to know to begin working with Sigma notation. However, since Sigma notation will usually have more complex expressions after the Sigma symbol, here are some further examples to give you a sense of what is possible:
Note that the previous example illustrates that, using the commutative property of addition, a sum of multiple terms can be broken up into multiple sums:
And lastly, this notation can be nested:
is a capital letter from the Greek alphabet call “Pi”… it corresponds to “P” in our alphabet (think of the starting sound of the word “pi”). It is used in mathematics to represent the product of a bunch of terms (think of the starting sound of the word “product”: Pppi = Ppproduct). It is used in the same way as the Sigma notation described above, except that succeeding terms are multiplied instead of added:
Sigma and Pi notation are used in mathematics to indicate repeated operations. Sigma notation provides a compact way to represent many sums, while Pi notation provides a compact way to represent many products.
To make use of them you will need a “closed form” expression (one that allows you to describe each term’s value using the term number) that describes all terms in the sum or product (just as you often do when working with sequences and series). Sigma and Pi notation save much paper and ink, as do other math notations, and allow fairly complex ideas to be described in a relatively compact notation.