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I'm wodering about the relationship between the theory that studies irrational/transcendental numbers and computer science. For example, I found this paper (but was unable to get the full text) Pseudo-random number generator based on irrational numbers

I found the paper because I started looking for a RNG that uses the property that irrational/transc. numbers don't have repeating decimals. I wanted to know in what other ways we can use or have a motivation to study irrational/transcendental numbers, because of their use in CS.

I'm ignorant on the matter, so I appreciate if someone more knowledgeable can provide examples when the two subjects connect themselves. Like the paper above.

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On the surface that does not strike me as a high-quality paper. I wouldn't recommend it.

I don't think irrational/transcendental numbers are a good approach to building PRNGs. Building PRNGs is tricky business. There are standard PRNGs; I suggest using one of them, rather than trying to invent your own. They all ensure that the output is extremely unlikely to repeat within any reasonable time period.

There is no one theory of irrational/transcendental numbers, and not just one connection to CS; irrational numbers show up in a variety of areas of CS. It's not feasible to give you a complete list; such a list would be far too long. If you just want one example, you might read about the Fibonacci numbers and their connection to the golden ratio (which is an irrational number).

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Sometime we can just identify real numbers with their binary expansions. This more or less works well as long as we can exclude the dyadic rationals. As binary expansions of dyadic rationals are clearly not pseudorandom, this means that talking about "(pseudo)random real numbers" makes sense.

Some topics that come up in this context are normal numbers and diophantine approximations. Ted Slaman is a renowned researcher working in this area, and he recently gave a talk on the topic which is available on Youtube here:

https://www.youtube.com/watch?v=V1i6Mn4C5I4&t=6s

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