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Find the arrival times of the first records for w n , focusing on minima rather than maxima this time. Let t n denotes the arrival time of the n -th record. Again, we have a similar connection with music theory and the convergents of q.

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See here for more. Even though no one has ever fully proved that SQRT 2 has its binary digits uniformly distributed, a much deeper result is "known" but of course, also unproved and published here for the first time:. The proportion q n of binary digits of SQRT 2 or any other normal number for that matter, say Pi or log 2 , among the first n digits, that are equal to one, satisfies.

In short, this is a consequence of a refined version of the Berry-Esseen theorem , itself a second-order refinement of the central limit theorem.

The time series in the above chart looks like a Brownian motion. However, it is not, because it is bounded. In a true Brownian motion, the variance grows infinitely over time. It would be very interesting to study the distribution of the records as well as their values, in that time series. Currently, other than conjectures, very little is known for sure about the distribution of the binary digits of SQRT 2. Some weak results have been proved though, see for instance here. To not miss our future articles more on this topic coming soon , subscribe to our newsletter, here. Views: Share Tweet Facebook.

Join Data Science Central. Interesting comment by Jean Fivaz :. I have encountered this in my efforts to automate 3 phase transformer design. The spec often requires a voltage ratio which is a multiple of sqrt 3 , but must be accomplished using whole number winding ratios. The optimum core steel and copper mass will determine the starting point, but then the algorithm has to find these sweet spots where the turns ratio is near the ideal voltage ratio, yet still close to the optimum starting point.

## Applications of continued fractions of irrational numbers

So I end up with these odd number adjustments which I couldn't really see a pattern to. Unexpectedly insightful. It is amazing to see the connection with "harmonious music", and other applications such as this one. I have been working for many years, spending a good part of my life working on this. Millions of great mathematicians tried over hundreds of years. I just have to finish it, I can't afford not to do it.

Also, the most fascinating part here, in my opinion and from a theoretical perspective, is about the last part of the exercise mentioned in the article.

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