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The amount of information in a real number is not determined by the number of digits. For example, 1/3 = 0.333333… but we wouldn’t say 1/3 has “infinite informa
by chongli 3y ago
The amount of information in a real number is not determined by the number of digits. For example, 1/3 = 0.333333… but we wouldn’t say 1/3 has “infinite information.”
Perhaps a better way to characterize the amount of information in a number is to look at the length of the program necessary to compute it. Of course, this depends on the programming language, but I think it’s fair to say that any particular program to compute a number is in some loose sense an upper bound on the information in the number.
So, most of the irrational numbers we’re familiar with (pi, e, phi) are rather low on information since they may be computed by very short programs, despite having infinitely many non-repeating digits.
- rnhmjoj 3y agoYes, but the real numbers that can be computed to arbitrary precision with an algorithm are countable: this means almost all the reals contain an infinite amount of information. The real numbers are useful to reason and construct proofs, but as far as the physical world go, only very few of them are actually meaningful.
- chongli 3y agoThe non-computable reals are also not uniquely representable. It’s this property which I think is the most reasonable explanation for why these real numbers don’t cause an information black hole.
- Aardwolf 3y agoBut 1/3 counts as discrete in a sense, you can describe it with a finite amount of symbols. To move from one point to another, a particle has to move through all of them though, including the non computable ones
- chongli 3y agoIf you're talking about a particle in real life, then you're talking about quantum mechanics, which is discrete, so the real numbers don't even come into play in that realm. Then, the use of real numbers in physics is a conceptual and calculational convenience which is not quite correct.