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Another thing that tingles my mind: it's easy to describe numbers that are likely to be transendential (just define a turing machine that spews out digits by so
by GolDDranks 5y ago
Another thing that tingles my mind: it's easy to describe numbers that are likely to be transendential (just define a turing machine that spews out digits by some more or less chaotic algorithm – most would agree that it seems extremely unlikely, that the output would correspond to some algebraic number by chance!), it's just hard to PROVE that they are.
- ladams 5y agoBut you have to let the Turing machine run forever for this to work! Every number with a terminating decimal representation in rational. Maybe it's not so easy to describe a transcendental number, at least in our lifetimes...
- Kranar 5y agoYou don't have to let the Turing machine run forever anymore than you have to actually fully write out every digit of PI. It's possible to determine strictly from the description of the Turing Machine that the value it outputs will converge to a specific number equal to the output of some other process, such the process of writing out 4 - 4/3 + 4/5 - 4/7 + 4/9 + ... The equivalence class of all such representations is the number we call PI.
- GolDDranks 5y agoSo, I was thinking that the Turing machine itself is a description of the number, because a description of the machine uniquely determines the output. If you require the digits to be spelled out, then yeah, that's impossible.
- 613style 5y agoIf such a program ever terminates, then the result is rational. If it doesn't terminate, then pausing at any point will always describe an infinite number of irrationals.
- GolDDranks 5y agoSorry, I caught your comment only a few days later. What do you mean by "infinite number of irrationals"? We are talking about a deterministic machine here, so surely it describes only one number?