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I think i am close to this answer but still to confirm can we create a turing machine(At least in Principle) which can work on real number computation and give exact results?**For example finding square root of an integer.(whose output would be a real number) My logic that we can't develop such a machine is that the real numbers are uncountably infinite and for uncountably infinite languages we can't create a turing machine.

bashrc
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  • Yet any given real number can be specified with countably many digits... – Kerrek SB Nov 28 '11 at 03:12
  • That's the point of confusion.The real numbers can be specified with countably many digits but the set of **all real numbers are uncountably infinite**.In language of automata,the language corresponding to real numbers is made up of uncountably infinite strings... – bashrc Nov 28 '11 at 03:19
  • But you're never working on *all* real numbers at once. If you allow a countably long tape (maybe two-ended, trailing off the left for the input and off the right for the output), you could compute the square root of one number I reckon. You wouldn't stop in finitely many steps, but in countably many. – Kerrek SB Nov 28 '11 at 03:21
  • and when would the machine halt (if there is any) for example if the input to the machine is 5(11111 in unary notation)? – bashrc Nov 28 '11 at 03:33
  • At time omega-naught of course :-) – Kerrek SB Nov 28 '11 at 03:36
  • I doubt your logic...That way we can say that the http://en.wikipedia.org/wiki/Halting_problem is solvable too by the turing machine.... – bashrc Nov 28 '11 at 03:42
  • No no, it's not an actual Turing machine, because it doesn't finish in *finite* time. You'd have to extend the notion to a "transfinite Turing machine". You can certainly *never* stop in finite time. I was just saying that you can probably do the next best thing though and stop in one-more-than-finite time. – Kerrek SB Nov 28 '11 at 12:10

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I think the Turing machine can be made if you put some restriction on Precision (i.e. answer up to 4 or 5 decimal place). Then it is possible. Otherwise I feel it can't be made.

T.Rob
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Amol Sharma
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