# Connected on Benford's law

The following clip is from the excellent Netflix documentary Connected: The Hidden Science of Everything Season 1, 4th episode (2020). That episode covered Benford's law. I had included Benford's law in my course Math E320 since 2014, triggered by This article. (Some Slides from 2017 show the Benford part. I included it this year also in the data lecture of Math 1a. I had been intrigued by Benford's law since 1984, when reading Arnold's book "Mathematical methods of classical mechanics" which had been suggested by Juerg Froehlich, who taught us Theoretical mechanics. Fröhlich had been moving back in 1982 from IHES to Zuerich and it was great to hear his lectures in Hoenggerberg. I must have lost those lecture notes but I have some exam preparation notes (there is something on page 336-337 while reading an appendix of Arnold. Arnold has given the exercise to prove that the first significant digit of 2n is given by the Benford distribution P[X=k] = log10(k+1)-log10(k). He gave this 2 line exercise however without hint about Benford but as the section was about averaging, one could see the reason: the doubling map in logarithmic coordinates is an irrational rotation which has the unique invariant measure dx. The Benford law is just the pull back to the original coordinates.