Скачать или смотреть A formula for writing all numbers and how to find them all

In this video I go over a formula for writing all numbers, that is, the natural numbers with respect to a given base. In other contexts this formula writes all the words, but in the base 10 case, this formula computes a number known as the Champernowne constant which is a pretty accessible example of a disjunctive sequence that is also normal. It turns out that all Champernowne constants of base b are disjunctive and normal with respect to their generating base. Due to their highly constructive nature, Champernowne constants of base b give normal number examples and examples of disjunctive sequences that are fairly accessible. There is also an algorithm for finding all numbers that appear in a Champernowne constant of base b with respect to that base and I spend some time going through an example of how to find them all using this algorithm. At the end of the video I also introduce some other questions that are out there, like "is pi normal?" among some other ideas. The video can be broken into the following sections:

00:00 Intro
00:25 Base Expansions Refresher
01:15 The Base 10 Case
02:24 Champernowne Constant
04:30 Normal and Disjunctive Numbers/Sequences
08:18 Locating numbers in C_10 an example of the algorithm
10:45 Other questions and related topics.

Resources related to this video:

◻️Videos:
‪@numberphile‬'s Strings and Loops in Pi video:    • Strings and Loops within Pi - Numberphile  

◻️Web References:
OEIS (A036903) Finding all the strings of a given length in π: http://oeis.org/A036903
Fabrice Bellard's Pi Computation Record Homepage (Links out to more recent records as well) : https://bellard.org/pi/pi2700e9/index...

◻️Papers:
Normality of C_b with respect to base b:
http://matwbn.icm.edu.pl/ksiazki/aa/a...
Constructing Absolutely Disjunctive Sequences: https://www.cs.auckland.ac.nz/~cristi...


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