Randomness of digits of irrational numbers.

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SUMMARY

The digits of irrational numbers, such as π (pi = 3.14159...) and e, are not definitively proven to be random or normal. A normal number is defined as one where every possible sequence of digits appears with a frequency of 1/10^n. While it is established that "almost all" numbers are normal, specific examples like π and e remain unproven in this regard. Further exploration into the concept of normal numbers is essential for understanding this topic.

PREREQUISITES
  • Understanding of irrational numbers and their properties
  • Familiarity with the concept of normal numbers
  • Basic knowledge of probability theory
  • Awareness of mathematical constants such as π and e
NEXT STEPS
  • Research the definition and properties of normal numbers
  • Explore the mathematical significance of π and e in number theory
  • Investigate the implications of randomness in digit sequences
  • Study the concept of measure theory in relation to "almost all" numbers
USEFUL FOR

Mathematicians, number theorists, and anyone interested in the properties of irrational numbers and randomness in mathematics.

Fallen Seraph
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How random are the digits of irrational numbers? Can it be said of them (i.e. pi=3.14159...) that given any arbitrarily long string of digits it must occur at some point in any irrational number? And would anyone know of anywhere I could find out more on this topic?
 
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Fallen Seraph said:
How random are the digits of irrational numbers? Can it be said of them (i.e. pi=3.14159...) that given any arbitrarily long string of digits it must occur at some point in any irrational number? And would anyone know of anywhere I could find out more on this topic?

Except for a few "made up" examples that are defined by using random numbers, no one really knows. In particular it is not known whether \pi or e or other familiar irrational numbers are "normal numbers": numbers such that every possible list of n numbers occurs, on average, 1/10n of the time: exactly what you would expect of a set of random numbers.

It can be shown, however, that, in a very specific sense, "almost all" numbers are normal! For more information, look up "normal numbers".
 
Thanks a lot.
 

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