The Pi Product Symbol For Multiplying Numbers

In an earlier post, I talked about adding numbers/quantities together with the sigma sum notation. We had <img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/83/ql_b1f9df64a5419438a588889300ac9183_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/83/ql_b1f9df64a5419438a588889300ac9183_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/83/ql_b1f9df64a5419438a588889300ac9183_l3.png 2x" /> , be replaced by <center><img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/a5/ql_829e98b96fc00fbef25b0a43c68b12a5_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/a5/ql_829e98b96fc00fbef25b0a43c68b12a5_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/a5/ql_829e98b96fc00fbef25b0a43c68b12a5_l3.png 2x" /> <center><a href="http://quicklatex.com/cache3/a5/ql_829e98b96fc00fbef25b0a43c68b12a5_l3.png" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"> In this post, the focus is on the multiplication case. Instead of having <img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/4a/ql_8b5334427aed0239c3a09584ce7d284a_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/4a/ql_8b5334427aed0239c3a09584ce7d284a_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/4a/ql_8b5334427aed0239c3a09584ce7d284a_l3.png 2x" /> , you would use <center><img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/58/ql_298c4221c24e34b053c829d593221558_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/58/ql_298c4221c24e34b053c829d593221558_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/58/ql_298c4221c24e34b053c829d593221558_l3.png 2x" /> <center><a href="http://quicklatex.com/cache3/58/ql_298c4221c24e34b053c829d593221558_l3.png" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"> The big Pi symbol above is used to represent a product of numbers. <h3>The Pi Product Notation <hr /> Suppose you have the expression <img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/9b/ql_c7ad7ced10bfdf7bfa07d22c76c5819b_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/9b/ql_c7ad7ced10bfdf7bfa07d22c76c5819b_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/9b/ql_c7ad7ced10bfdf7bfa07d22c76c5819b_l3.png 2x" />. In terms of product notation, this can be represented as: <center><img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/fb/ql_358c7ecf915ebc2de97c5b4edbc5dcfb_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/fb/ql_358c7ecf915ebc2de97c5b4edbc5dcfb_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/fb/ql_358c7ecf915ebc2de97c5b4edbc5dcfb_l3.png 2x" /> <center><a href="http://quicklatex.com/cache3/fb/ql_358c7ecf915ebc2de97c5b4edbc5dcfb_l3.png" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"> The starting number is when <code>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 <code>k = 1 instead of <code>k = 2.) Variables With Subscripts Case Consider the case where you multiply the following: <center><img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/f6/ql_6754e81adc1b2118a904e817f94b24f6_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/f6/ql_6754e81adc1b2118a904e817f94b24f6_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/f6/ql_6754e81adc1b2118a904e817f94b24f6_l3.png 2x" /> <center><a href="http://quicklatex.com/cache3/f6/ql_6754e81adc1b2118a904e817f94b24f6_l3.png" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"> The subscripts keep increasing by 1. The above can be represented in product notation as: <center><img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/5e/ql_4f433273eb03dcd28217bfe02ceb925e_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/5e/ql_4f433273eb03dcd28217bfe02ceb925e_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/5e/ql_4f433273eb03dcd28217bfe02ceb925e_l3.png 2x" /> <center><a href="http://quicklatex.com/cache3/5e/ql_4f433273eb03dcd28217bfe02ceb925e_l3.png" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"> I have used a different index variable which is j. (You could use other common letters like <code>i, or <code>k.) <h3>A Few Algebra Applications <hr /> Exponent Laws The following expression <center><img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/e1/ql_0df2694d0c983792fbd7fecc67d772e1_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/e1/ql_0df2694d0c983792fbd7fecc67d772e1_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/e1/ql_0df2694d0c983792fbd7fecc67d772e1_l3.png 2x" /> <center><a href="http://quicklatex.com/cache3/e1/ql_0df2694d0c983792fbd7fecc67d772e1_l3.png" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"> <code>ncan be expressed as <img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/c8/ql_187c678d2514172545e710a0ef9d4cc8_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/c8/ql_187c678d2514172545e710a0ef9d4cc8_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/c8/ql_187c678d2514172545e710a0ef9d4cc8_l3.png 2x" />. The represents the number of twos in the product. In Pi product notation, the above can be represented as:<img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/54/ql_ee5831fe803c86c1c153abe56539a154_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/54/ql_ee5831fe803c86c1c153abe56539a154_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/54/ql_ee5831fe803c86c1c153abe56539a154_l3.png 2x" /> If you have something like <img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/c0/ql_c02e5fa5e4c5440ca5dff6d772f328c0_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/c0/ql_c02e5fa5e4c5440ca5dff6d772f328c0_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/c0/ql_c02e5fa5e4c5440ca5dff6d772f328c0_l3.png 2x" />, it can be expressed as <center><img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/19/ql_2d4193d3cec408b2d2a8b524c2c80519_l3.png " srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/19/ql_2d4193d3cec408b2d2a8b524c2c80519_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/19/ql_2d4193d3cec408b2d2a8b524c2c80519_l3.png 2x" /> <center><a href="http://quicklatex.com/cache3/19/ql_2d4193d3cec408b2d2a8b524c2c80519_l3.png" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"> Logarithms In this example, I use the natural logarithm where <img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/1d/ql_568c0f4f8f14614671a312970c8d911d_l3.png" srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/1d/ql_568c0f4f8f14614671a312970c8d911d_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/1d/ql_568c0f4f8f14614671a312970c8d911d_l3.png 2x" /> . One property of logarithms is where the logarithm of a product is the sum of the logarithms with separate components. <center><img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/a9/ql_a43cf4ac44700d61f813dc9477e4eba9_l3.png " srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/a9/ql_a43cf4ac44700d61f813dc9477e4eba9_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/a9/ql_a43cf4ac44700d61f813dc9477e4eba9_l3.png 2x" /> <center><a href="http://quicklatex.com/cache3/a9/ql_a43cf4ac44700d61f813dc9477e4eba9_l3.png" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"> The general case for logartihms would be as follows: <center><img src="https://images.hive.blog/768x0/http://quicklatex.com/cache3/45/ql_b2febff985dce6d405310526a041fd45_l3.png " srcset="https://images.hive.blog/768x0/http://quicklatex.com/cache3/45/ql_b2febff985dce6d405310526a041fd45_l3.png 1x, https://images.hive.blog/1536x0/http://quicklatex.com/cache3/45/ql_b2febff985dce6d405310526a041fd45_l3.png 2x" /> <center><a href="https://hhttp://quicklatex.com/cache3/45/ql_b2febff985dce6d405310526a041fd45_l3.png" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link"> <ul> <li>A more complicated application that uses the Pi Product notation is the <a href="http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">Lagrange Interpolating Polynomial for fitting a polynomial to points. <hr /> This post was created in RMarkdown with the math text/images created with QuickLaTeX.com. (Markdown does not support LaTeX here.)