4 ms·

> It gets rid of many machine-particularities, like floats not being real numbers (due to limited precision). I will have to doubt that for the simple fact tha

> It gets rid of many machine-particularities, like floats not being real numbers (due to limited precision). I will have to doubt that for the simple fact that it is impossible to have a real number type on a computer.

gnramires 6y ago
Real numbers are represented as abstract data structures, not as infinite series of digits (although that indeed fits in a Turing machine? :p) So for example, '2' is recognized as an integer and is represented using arbitrary size integers. You can also however write 'x = sqrt(2)' (or just 'sqrt(2)'), which has no finite digital (irrational), 'x' is a real number. You can then ask for finitely many digits of x, with 'x.n(5)' (gives 5 digits), or write something like 'y=x^3+3x-4', which gives another real number, represented as this polynomial data structure itself. The only problem with this is irreducibility. You can compose arbitrarily many operations on floats and still get a float of the same size. With this approach, it may not be possible to simplify a series of operations so the representation can grow unbounded. edit: Fun fact, there are (real) numbers that indeed cannot be represented in a finite computer no matter what -- but they cannot be represented in paper or uniquely represented in any finite abstract form either! This follows from the pigeonhole principle: finite expressions may represent numbers, but there are uncountably infinite (2^(N0)) real numbers, and only countably infinitely many expressions. So indeed almost every real cannot be represented. You can think of those as not being identifiable with any property, so there's no finite expression to describe them. They're more or less random.
- dependenttypes 6y ago
  There are real numbers that can't be represented in an "infinite" computer either.
  - gnramires 6y ago
    Every real number can be associated with an infinite series of digits, so a theoretical infinite tape could hold any one (or countably infitely many).
    - dependenttypes 6y ago
      > so a theoretical infinite tape could hold any one (or countably infitely many). I am not sure what you mean with that. You could have an infinite tape that holds the number 1 and an "infinitely sized" irrational number at the same time.

gnramires 6y ago

Real numbers are represented as abstract data structures, not as infinite series of digits (although that indeed fits in a Turing machine? :p) So for example, '2' is recognized as an integer and is represented using arbitrary size integers. You can also however write 'x = sqrt(2)' (or just 'sqrt(2)'), which has no finite digital (irrational), 'x' is a real number. You can then ask for finitely many digits of x, with 'x.n(5)' (gives 5 digits), or write something like 'y=x^3+3x-4', which gives another real number, represented as this polynomial data structure itself. The only problem with this is irreducibility. You can compose arbitrarily many operations on floats and still get a float of the same size. With this approach, it may not be possible to simplify a series of operations so the representation can grow unbounded. edit: Fun fact, there are (real) numbers that indeed cannot be represented in a finite computer no matter what -- but they cannot be represented in paper or uniquely represented in any finite abstract form either! This follows from the pigeonhole principle: finite expressions may represent numbers, but there are uncountably infinite (2^(N0)) real numbers, and only countably infinitely many expressions. So indeed almost every real cannot be represented. You can think of those as not being identifiable with any property, so there's no finite expression to describe them. They're more or less random.

dependenttypes 6y ago
There are real numbers that can't be represented in an "infinite" computer either.
- gnramires 6y ago
  Every real number can be associated with an infinite series of digits, so a theoretical infinite tape could hold any one (or countably infitely many).
  - dependenttypes 6y ago
    > so a theoretical infinite tape could hold any one (or countably infitely many). I am not sure what you mean with that. You could have an infinite tape that holds the number 1 and an "infinitely sized" irrational number at the same time.