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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 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.

Sigma Notation (Summation Notation)

The Sigma symbol, \sum, is a capital letter in the Greek alphabet. It corresponds to “S” in our alphabet, and is used in mathematics to describe “summation”, the addition or sum of a bunch of terms (think of the starting sound of the word “sum”: Sssigma = Sssum).

The Sigma symbol can be used all by itself to represent a generic sum… the general idea of a sum, of an unspecified number of unspecified terms:

\displaystyle\sum a_i~\\*\\*=~a_1+a_2+a_3+...

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.

\displaystyle\sum_{1}^{5}2~\\*\\*=~2+2+2+2+2

Sigma notation, or as it is also called, summation 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 Continue reading Sigma Notation (Summation Notation)

One Step Linear Equation Problems

The solution to each of the following twenty problems is 12. Focus on finding the most helpful algebraic step to take a reader from the problem as stated to the solution, and be sure you can explain why that step leads to a solution.

  1.   x+7~=~19
  2.   3+x~=~15
  3.   11~=~-1+x
  4. Continue reading One Step Linear Equation Problems

Math: Pen vs Pencil

Many math students are given strict instructions by their teachers to do all their work in pencil. I disagree.

The advantage of doing work in pencil is that:

  • it is easier to erase, so students are less likely to be paralyzed by “I am not sure this is correct, so I don’t dare write it down”

The disadvantages of working in pencil are that: Continue reading Math: Pen vs Pencil

11 Ways To Do Better In Math

1) Reflect and Summarize
at the end of each class, at the end of each week, at the end of each month. Review your notes and/or think back over the material that has been covered, then decide which skills or ideas you think are most important. Summarize the material you are learning as concisely as you can, because summarizing helps you learn. Identify any skills or ideas that you are not confident about. Write your reflections and summarizations as part of your notes, with reminders about what needs more work.

2) Use scrap paper
Using scrap paper removes a source of anxiety when Continue reading 11 Ways To Do Better In Math