<center><img src="http://www.clker.com/cliparts/B/f/F/I/j/O/lime-green-pi-md.png" /></center>
<center><a href="http://www.clker.com/cliparts/B/f/F/I/j/O/lime-green-pi-md.png">Featured Image</a></center>


In an earlier post, I talked about adding numbers/quantities together with the sigma sum notation. 

We had http://quicklatex.com/cache3/83/ql_b1f9df64a5419438a588889300ac9183_l3.png , be replaced by

<center><img src="http://quicklatex.com/cache3/a5/ql_829e98b96fc00fbef25b0a43c68b12a5_l3.png" /></center>
<center><a href="http://quicklatex.com/cache3/a5/ql_829e98b96fc00fbef25b0a43c68b12a5_l3.png"></a></center>


In this post, the focus is on the multiplication case. Instead of having http://quicklatex.com/cache3/4a/ql_8b5334427aed0239c3a09584ce7d284a_l3.png , you would use

<center><img src="http://quicklatex.com/cache3/58/ql_298c4221c24e34b053c829d593221558_l3.png" /></center>
<center><a href="http://quicklatex.com/cache3/58/ql_298c4221c24e34b053c829d593221558_l3.png"></a></center>

The big Pi symbol above is used to represent a product of numbers.


### The Pi Product Notation
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Suppose you have the expression http://quicklatex.com/cache3/9b/ql_c7ad7ced10bfdf7bfa07d22c76c5819b_l3.png.

In terms of product notation, this can be represented as:

<center><img src="http://quicklatex.com/cache3/fb/ql_358c7ecf915ebc2de97c5b4edbc5dcfb_l3.png" /></center>
<center><a href="http://quicklatex.com/cache3/fb/ql_358c7ecf915ebc2de97c5b4edbc5dcfb_l3.png"></a></center>

The starting number is when `k = 2` which would be just 2. Then you increase the index variable k by one each time you get the next number. When k is 3 you have the next number as 3. Continue this process until you have the upper limit of k = 10.

(The above example is an example of an ascending factorial. You can start the index at `k = 1` instead of `k = 2`.)


**Variables With Subscripts Case**

Consider the case where you multiply the following:

<center><img src="http://quicklatex.com/cache3/f6/ql_6754e81adc1b2118a904e817f94b24f6_l3.png" /></center>
<center><a href="http://quicklatex.com/cache3/f6/ql_6754e81adc1b2118a904e817f94b24f6_l3.png"></a></center>

The subscripts keep increasing by 1. The above can be represented in product notation as:

<center><img src="http://quicklatex.com/cache3/5e/ql_4f433273eb03dcd28217bfe02ceb925e_l3.png" /></center>
<center><a href="http://quicklatex.com/cache3/5e/ql_4f433273eb03dcd28217bfe02ceb925e_l3.png"></a></center>


I have used a different index variable which is j. (You could use other common letters like `i`, or `k`.)


### A Few Algebra Applications
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**Exponent Laws**

The following expression

<center><img src="http://quicklatex.com/cache3/e1/ql_0df2694d0c983792fbd7fecc67d772e1_l3.png" /></center>
<center><a href="http://quicklatex.com/cache3/e1/ql_0df2694d0c983792fbd7fecc67d772e1_l3.png"></a></center>

can be expressed as http://quicklatex.com/cache3/c8/ql_187c678d2514172545e710a0ef9d4cc8_l3.png. The `n` represents the number of twos in the product. In Pi product notation, the above can be represented as:http://quicklatex.com/cache3/54/ql_ee5831fe803c86c1c153abe56539a154_l3.png



If you have something like http://quicklatex.com/cache3/c0/ql_c02e5fa5e4c5440ca5dff6d772f328c0_l3.png, it can be expressed as

<center><img src="http://quicklatex.com/cache3/19/ql_2d4193d3cec408b2d2a8b524c2c80519_l3.png
" /></center>
<center><a href="http://quicklatex.com/cache3/19/ql_2d4193d3cec408b2d2a8b524c2c80519_l3.png
"></a></center>


**Logarithms**

In this example, I use the natural logarithm where http://quicklatex.com/cache3/1d/ql_568c0f4f8f14614671a312970c8d911d_l3.png .

One property of logarithms is where the logarithm of a product is the sum of the logarithms with separate components.

<center><img src="http://quicklatex.com/cache3/a9/ql_a43cf4ac44700d61f813dc9477e4eba9_l3.png
" /></center>
<center><a href="http://quicklatex.com/cache3/a9/ql_a43cf4ac44700d61f813dc9477e4eba9_l3.png
"></a></center>

The general case for logartihms would be as follows:

<center><img src="http://quicklatex.com/cache3/45/ql_b2febff985dce6d405310526a041fd45_l3.png
" /></center>
<center><a href="hhttp://quicklatex.com/cache3/45/ql_b2febff985dce6d405310526a041fd45_l3.png
"></a></center>



* A more complicated application that uses the Pi Product notation is the [Lagrange Interpolating Polynomial](http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html) for fitting a polynomial to points.

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