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Number Representation Systems Explained in One Picture

Back to the basics. Here we are dealing with the oldest data set, created billions of years ago -- the set of integers -- and mostly the set consisting of two numbers: 0 and 1.  All of us have learned how to write numbers even before attending primary school. Yet, it is attached to the most challenging unsolved mathematical problems of all times, such as the distribution of the digits of Pi in the decimal system. The table below reflects this contrast, being a blend of rudimentary and deep results. It is a reference for statisticians, number theorists, data scientists, and computer scientists, with a focus on probabilistic results. You will not find it in any textbook. Some of the systems described here (logistic map of exponent p = 1/2, nested square roots, auto-correlations in continued fractions) are research results published here for the first time.  

This material is described in this article, including how to derive all the results, and the equivalence between the base-2 system and the standard  logistic map (with p = 1) mentioned in exercise 7. It includes applications to cryptography, random number generation, high performance computing (HPC), population growth modeling, financial markets, BlockChain, and more.

A higher resolution of the table below is available here.

Many other systems are not described in this table, including:

  • The iterated exponential map, possibly one of the most mysterious chaotic systems. See exercise 4 here for details. 
  • The nested cubic root, a generalization of the nested square root.
  • The generalized continued fraction of power p, especially p = 2, defined by

Other interesting facts:

For the standard logistic map (p = 1) a closed-form formula is available for x(n):

Also, if x(n) is the base-2 sequence for x, then sin^2(Pi x(n)) is the logistic map sequence for sin^2(Pi x).    

Not all number representation systems are as well-behaved as those describe here. For examples of more complex systems, see here.

For related articles from the same author, click here or visit www.VincentGranville.com. Follow me on Twitter at @GranvilleDSC or on LinkedIn.

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