6 ms·

My favorite thing about this constant is the counter-intuitive fact that if you uniformly randomly choose a real number, there's a 100% chance it will have the

My favorite thing about this constant is the counter-intuitive fact that if you uniformly randomly choose a real number, there's a 100% chance it will have the Khinchin behavior, and yet there are almost no known numbers which actually do. Or as Wikipedia puts it: > Although almost all numbers satisfy this property, it has not been proven for any real number not specifically constructed for the purpose.

benkuykendall 10y ago
>> if you uniformly randomly choose a real number There is no uniform probability distribution over all real numbers.
- tgb 10y ago
  Since this value is independent of the floor a_0, you can easily restrict to the real numbers 0 to 1.
- goldenkey 10y ago
  Isnt the problem moreover that the reals have infinite digits?
  - Retra 10y ago
    How many digits a number has is entirely a matter of what representation you use for it.
    - goldenkey 10y ago
      Right but isnt the Komogrolov complexity of the reals on average, incompressable?
      - Retra 10y ago
        The vast majority of real numbers are never used for anything, so their aggregate properties are only important insofar as you don't have a 'smaller' system with those same properties.
- kilovoltaire 10y ago
  Thanks good point, I should've just said the unit interval.
pash 10y ago
This reflects basic properties of the real numbers, e.g., that almost all of them are irrational, almost all of them are transcendental, almost all of them are uncomputable, etc. The familiar real numbers, those that people (including mathematicians) actually use, and that our machines use, are atypical of the set of real numbers as a whole in many important respects. The real numbers are a formal abstraction to an extent that few people realize.
- api 10y ago
  IMHO this is a special case of something much more profound. I studied biologically inspired and evolutionary computing very intensely in college, and I reached the conclusion fairly early that the way we do compute is very unlike the natural world. Our computers are designed to compute finite values using instruction sets that are explicitly designed to be precise, which has the side effect of these instruction sets and encodings being highly brittle and un-evolvable. This is what makes things like neural nets and genetic programming hard-- making these systems work involves creating virtual anti-computers within our anti-biological existing ones that are fuzzy, mutable, evolvable, etc. I speculate that this is because we've built computers explicitly to do things we are not good at: crunching massive amounts of data very precisely. We are pretty good at more "organic" forms of cognition so there was little evolutionary "market need" for us to augment ourselves in that way.
  - j1vms 10y ago
    Though for this specific discussion (which illustrates the uncountably infinite nature of the set of real numbers), it is useful to note that the class of realizable computers includes both analog and digital variants, though usually we are more familiar with the digital in the form of our desktop/laptop computers or servers. Analog computers can be indeed be used for these types of problems, and their usage in the field predates their digital siblings. Sampling of the result of an analog calculation is of course limited by analog noise (e.g. thermal noise), which we must contend with in this universe. On the other hand, processing in the analog domain gets us further than quantization noise and heat dissipation (due to clock signal & switching) would in the digital domain. Edit: If you are interested further, here is a discussion of the application of analog computing in solving differential equations: http://chalkdustmagazine.com/features/analogue-computing-fun-differential-equations/ http://chalkdustmagazine.com/features/analogue-computing-fun... Also, someone on here once posted a link to a recent dissertation comparing the efficiency of solution finding for a problem tackled analog vs. digital, wherein given the problem's nature, the analog computer outperformed its digital counterpart both in terms of performance and energy consumption. Edit2: Here is the dissertation http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf And the comment it was from: https://news.ycombinator.com/item?id=11728186 https://news.ycombinator.com/item?id=11728186

benkuykendall 10y ago

>> if you uniformly randomly choose a real number There is no uniform probability distribution over all real numbers.

tgb 10y ago
Since this value is independent of the floor a_0, you can easily restrict to the real numbers 0 to 1.
goldenkey 10y ago
Isnt the problem moreover that the reals have infinite digits?
- Retra 10y ago
  How many digits a number has is entirely a matter of what representation you use for it.
  - goldenkey 10y ago
    Right but isnt the Komogrolov complexity of the reals on average, incompressable?
    - Retra 10y ago
      The vast majority of real numbers are never used for anything, so their aggregate properties are only important insofar as you don't have a 'smaller' system with those same properties.
kilovoltaire 10y ago
Thanks good point, I should've just said the unit interval.

pash 10y ago

This reflects basic properties of the real numbers, e.g., that almost all of them are irrational, almost all of them are transcendental, almost all of them are uncomputable, etc. The familiar real numbers, those that people (including mathematicians) actually use, and that our machines use, are atypical of the set of real numbers as a whole in many important respects. The real numbers are a formal abstraction to an extent that few people realize.

api 10y ago
IMHO this is a special case of something much more profound. I studied biologically inspired and evolutionary computing very intensely in college, and I reached the conclusion fairly early that the way we do compute is very unlike the natural world. Our computers are designed to compute finite values using instruction sets that are explicitly designed to be precise, which has the side effect of these instruction sets and encodings being highly brittle and un-evolvable. This is what makes things like neural nets and genetic programming hard-- making these systems work involves creating virtual anti-computers within our anti-biological existing ones that are fuzzy, mutable, evolvable, etc. I speculate that this is because we've built computers explicitly to do things we are not good at: crunching massive amounts of data very precisely. We are pretty good at more "organic" forms of cognition so there was little evolutionary "market need" for us to augment ourselves in that way.
- j1vms 10y ago
  Though for this specific discussion (which illustrates the uncountably infinite nature of the set of real numbers), it is useful to note that the class of realizable computers includes both analog and digital variants, though usually we are more familiar with the digital in the form of our desktop/laptop computers or servers. Analog computers can be indeed be used for these types of problems, and their usage in the field predates their digital siblings. Sampling of the result of an analog calculation is of course limited by analog noise (e.g. thermal noise), which we must contend with in this universe. On the other hand, processing in the analog domain gets us further than quantization noise and heat dissipation (due to clock signal & switching) would in the digital domain. Edit: If you are interested further, here is a discussion of the application of analog computing in solving differential equations: http://chalkdustmagazine.com/features/analogue-computing-fun-differential-equations/ http://chalkdustmagazine.com/features/analogue-computing-fun... Also, someone on here once posted a link to a recent dissertation comparing the efficiency of solution finding for a problem tackled analog vs. digital, wherein given the problem's nature, the analog computer outperformed its digital counterpart both in terms of performance and energy consumption. Edit2: Here is the dissertation http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf And the comment it was from: https://news.ycombinator.com/item?id=11728186 https://news.ycombinator.com/item?id=11728186