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  1. #1 Notations 
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    I realize that, to some people, this is going to be trivial, but I would really like to understand the various different notations used in the math world. And, right now, my question is on Sigma(sum) notation and Pi(product) notation. Most specifically, in the equations:
    and
    I am curious what the and the in both examples means, exactly. I understand the symbols, and I know how to evaluate, but I just want to know the more intricate things that the notation means and, more specifically, how that leads to me being able to manipulate the notation to be more useful to me. Any help would be much appreciated.

    Oh, and one more thing I'm curious about, if I wanted to us Pi notation where I have a sequence of numbers denoted as but I want to exclude one of those elements of like, say where . Would this be represented as
    or would it be


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  3. #2  
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    Hmm, not really sure what you're asking for but I'll give it a shot.

    is the index variable and it must take on integer values. designates that the index begins at which is called the lower bound. k is the final index, or upper bound, meaning that you carry out the summation or product until n=k. Both k and must also take on integer values or infinity (in which case you are actually taking the limit as k or go to infinity). You probably know that, but quite honestly that's all there is to it. I'll give a few examples of "tricks" that can be used. Maybe that will give you more insight.

    First, can have two meanings. It can refer to the element of set A, or it can be a function such that

    Also the symbol you use for the index variable is arbitrary:



    The index variable, however, must match the index subscript. If the subscript does not match the index variable then you can move the variable out of the summation sign:



    If you have double summation then you can move things around as much as you want as long as the variable subscripted with a certain index variable stays behind the summation sign that sums over that index variable:



    Notice that I did not specify lower or upper bounds. If they are not specified then that means that the summation is carried over all index values.

    Another way to show that the index variable can be arbitrary is to change it, as long as it is changed everywhere. Thus:



    I've focused mostly on summation, but I believe that all of this also applies to production (sic).

    As for your second question, the one on the left is incorrect, not only in what you are trying to achieve but also in notation. The upper bound must be an integer.

    The notation on the right is correct as far your example is concerned (provided that you meant and not }. However if you wanted to carry out the product from k<m to n but exclude then the proper notation would be:



    or you could just do this:



    Sometimes, especially when doing paper and pencil computations, people write something like this:



    I don't know if strictly speaking such notation is correct, but it is used nonetheless.

    Hope this helps!

    P.S. I sorry that I used the in-line format for summation and product with the symbols next to the sigma and the pi instead of above and below them. I tried to get TeX to write it like that but I gave up after half an hour


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  4. #3  
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    that does answer my question, thank you! I was just looking for the accepted notation, so that I could properly display the notations without confusing the crap out of everyone. so could I write something like
    the sum of or, could I more simply write

    I just want to be as proper as possible, I get a little anal on my own notation at times.
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  5. #4 Re: Notations 
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    Quote Originally Posted by Arcane_Mathamatition
    I realize that, to some people, this is going to be trivial, but I would really like to understand the various different notations used in the math world. And, right now, my question is on Sigma(sum) notation and Pi(product) notation. Most specifically, in the equations:
    and
    I am curious what the and the in both examples means, exactly. I understand the symbols, and I know how to evaluate, but I just want to know the more intricate things that the notation means and, more specifically, how that leads to me being able to manipulate the notation to be more useful to me. Any help would be much appreciated.

    Oh, and one more thing I'm curious about, if I wanted to us Pi notation where I have a sequence of numbers denoted as but I want to exclude one of those elements of like, say where . Would this be represented as
    or would it be

    n is "dummy variable" and k is simply a limit. You are really trying to describe a set of integers, in this case a rather simple one -- all of the integers between 1 and k inclusive. Thus if you want to exclude one integer, call it m as in your last question you would write

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    I figured this would be a good place to post this instead of a new topic, but what does the function 'factorial', as in , really do? is it just an easier way of writing:

    when is an integer?

    or is there some other fancy way of saying what the function does, or does that just about sum it up?
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  7. #6  
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    Quote Originally Posted by Arcane_Mathamatition
    I figured this would be a good place to post this instead of a new topic, but what does the function 'factorial', as in , really do? is it just an easier way of writing:

    when is an integer?

    or is there some other fancy way of saying what the function does, or does that just about sum it up?
    Yes,
    k! =

    (You don't need to state that n is an integer since that is understood from the notation).

    But isn't it easier to write k! ? The factorial function comes up often enough to deserve its own notation. Combinatoric mathematics, for instance, would be a mess without that notation.
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