Benford law

Again a topic not so related to patents, though still of a puzzling beauty:

I 've just stumbled upon the mystery of Benford law in one of the video's of the Khan Academy: http://www.khanacademy.org/video/vi-and-sal-talk-about-the-mysteries-of-benford-s-law?playlist=New+and+Noteworthy

This law dictates that the first digit distribution of naturally occurring random numbers is not at all randomly distributed, instead occurs in a certain distribution. For curiosity's sake, i have made an excel matrix with in each cell only the first digit of the product of any number of 1-99 (lined up in rows) and any number of 1-9 (lined up in columns)..

much to my surprise, these random numbers (with random first digits) multiplied with random numbers (also with random first digits) results in a mysterious mathematical distribution of the first digit....

Below I 've showed the matrix with the results. If you want to expand the matrix, for each cell I 've used the function VALUE (LEFT($A2*B$1;1)) returning the first digit of the product of A2 and B1. After adding some color, the following art piece is the result:

When plotting the frequency of each of the first digits of the occurring results, it becomes clear that the randomness of this distribution is gone. In this graph, each line represents the frequency distribution of first digits in a column of the above depicted matrix.

From this graph, it is shown that in order to destroy randomness of the first digit, multiplication with 2 will do best, more than 50% of the results obtains 1 as first digit. On the other hand, multiplication with 9 the least disturbs the randomness of the first digit.

The total frequency of first digits of the entire matrix is:

which closely resembles the Benford Law chart shown by Sal....

Fascinating.....