John L's Formulas – Chapter One
Page 2, continued from the first page.
Page 2 was updated January 15, 2013. The content of the first page will always remain as originally posted.
To summarize the development of the equation introduced on the first page along with some examples, we have this whiteboard:
This equation appears to perform its initially-intended function in finding special numbers (represented by X) that each fulfill the following requirement: When that number (X) is set forth a certain number of times (n) and then multiplied by itself, the product must equal the sum when the multiplication operation is substituted by addition. And vice versa!
The classic example is 2 + 2 = 2 × 2. The equation works when n is an integer which is equal to or greater than 2. Note the examples in the whiteboard image above when n = 2, 3 and 4.
As X can actually be any number (not just an integer) – as can n – this algebraic equation will generate a value for X for various values of n, and a graph can be made accordingly. Graphing calculators on the web (cited on Page 4 where these thoughts continue) have no difficulty with this equation which has to be made workable by being expressed as the equivalent, "X=n^(1/(n-1))" but put into standard x,y form as "y=x^(1/(x-1))." For purposes of consistency on these web pages, the original n,X form remains (although I do tend to use a small x at times).
|This page was last modified on 1/15/13 at 12:30 PM, CST.
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University of Wisconsin-Madison