Pi Notation (Product Notation)

 

The Pi symbol, \prod, is a capital letter in the Greek alphabet call “Pi”, and corresponds to “P” in our alphabet. It is used in mathematics to represent the product (think of the starting sound of the word “product”: Pppi = Ppproduct) of a bunch of factors.

If you are not familiar or comfortable with Sigma Notation, I suggest you read my post on Sigma Notation first, then come back to this one – because Pi Notation is very similar.

Once you understand the role of the index variable in Sigma Notation, you will see it used exactly the same way with Pi Notation, except that it describes a factor number instead of a term number:

\displaystyle\prod_{k=3}^{7}k\\*~\\*=(3)(4)(5)(6)(7)

\displaystyle\prod_{n=0}^{3}(n+x)\\*~\\*~\\*=(0+x)(1+x)(2+x)(3+x)

\displaystyle\prod_{i=1}^{2}\displaystyle\prod_{j=4}^{6}(3ij)\\*~\\*~\\*=\displaystyle\prod_{i=1}^{2}((3i\cdot4)(3i\cdot5)(3i\cdot6))\\*~\\*~\\*=((3\cdot1\cdot4)(3\cdot1\cdot5)(3\cdot1\cdot6)) ((3\cdot2\cdot4)(3\cdot2\cdot5)(3\cdot2\cdot6))

Summary

Pi Notation, or Product Notation, is used in mathematics to indicate repeated multiplication. Pi notation provides a compact way to represent many products.

To make use of it you will need a “closed form” expression (one that allows you to describe each factor’s value using its factor number) that describes all factors in the product. Pi Notation saves much paper and ink, as do other math notations, and allows fairly complex ideas to be described in a relatively compact notation.

Whit Ford's avatar

By Whit Ford

Math tutor since 1992. Former math teacher, product manager, software developer, research analyst, etc.

10 comments

      1. M M Krady – I have never actually seen Pi Notation used. This is probably because exponentiation also summarizes repeated multiplication (by the same factor each time), and that takes care of the vast majority of situations in physics. Other situations (dot products, cross products) typically only involve two to four factors, which will be easier to read and understand when written out. Sigma Notation gets a lot of use in working with infinite Series, where each term can be described in a handy way using the index variable. I am not aware of situations in which infinite products arise, and I have not seen any situations where an index variable would be useful in describing each factor in a multi-factor product. Sorry to not have a more satisfactory answer. Perhaps you could ask a physicist?

    1. Have you read the post on Sigma notation (https://mathmaine.com/2018/03/04/sigma-notation/)? It provides additional examples of how the index variable (under the Pi or Sigma symbol) is has the factor/term number substituted in repeatedly, once for each iteration, resulting in a series of factors (Pi notation), or terms (Sigma notation). Anything to the right of the Pi symbol which is NOT the index variable will stay as it is in each factor.

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