Timeline for Represent a real number without loss of precision
Current License: CC BY-SA 3.0
8 events
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| Jul 12, 2014 at 11:31 | comment | added | Andrej Bauer | Perhaps it does technically, but it misses the point, which is that there are perfectly reasonable infinite representations of real numbers. And that's nothing strange: a TCP/IP connection, or a Skype call, or a video feed from a camera are all examples of (potentially) infinite amount of data. There is no a priori limitation on how much information they can provide. There is only a limitation on how much information you can get out of it in a finite amount of time. | |
| Jul 12, 2014 at 9:13 | comment | added | David Richerby | @Andrej Does the edit remove the falsity to your satisfaction? | |
| Jul 12, 2014 at 9:12 | history | edited | David Richerby | CC BY-SA 3.0 |
Restricted to finite representations.
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| Jul 12, 2014 at 9:03 | comment | added | Andrej Bauer | This answer is false. Alan Turing in his first paper on machines, the ones in which he invents Turing machines, talks about representing reals as infinite strings of data. This leads to the idea of so-called "Type II Turing machine", and there is a very succesful theory of real-number computation based on the idea. It is also implemented in practice, see my answer. | |
| Jul 11, 2014 at 23:34 | comment | added | kasperd | It is indeed quite clear that computing equality on such numbers is equivalent to solving the halting problem. Given a TM one can define a real number, which starts with a lot of decimals that are zero, exactly as many as the running time of the TM, and then followed by a one. Comparing that number to zero is equivalent to solving the halting problem for the original TM. | |
| Jul 11, 2014 at 22:41 | comment | added | David Richerby | @kasperd They're called computable reals. Unfortunately, things like equality aren't computable over the computable reals. | |
| Jul 11, 2014 at 22:30 | comment | added | kasperd | One could restrict the requirement from representing every real number to only restricting those real numbers, which could be the output of a turing machine. That would only be a countable number of real numbers, but would still cover every number you would ever want to represent. But I don't think you could do efficient computations with such numbers. | |
| Jul 11, 2014 at 17:26 | history | answered | David Richerby | CC BY-SA 3.0 |