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Yeah, this was disappointed. Information about e is widely available, but I don't recall ever really reading much about transcendental numbers in general, othe

Yeah, this was disappointed. Information about e is widely available, but I don't recall ever really reading much about transcendental numbers in general, other than a brief introduction to e. Definitely a missed opportunity.

lisper 5y ago
OK, here's a fun fact: most numbers are transcendental, which is to say, if you choose a real number uniformly at random in any finite interval the probability of choosing a transcendental number is 1. (The proof is easy and left as an exercise.) But actually describing a transcendental number is very hard. The first transcendental number other than pi and e was not described until the mid 19th century, and even today only a few dozen classes of transcendental numbers are known.
- JackFr 5y ago
  Possible spoiler (I don't know if my proof is correct)-- Algebraic numbers are countable, since their description is a countable combination of rationals which are also countable. The measure of a countable set is zero, so the measure of its complement is 1, and thus the probability of choosing a transcendental is 1.
  - lisper 5y ago
    Exactly right!
- 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 ago
    But 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 ago
      You 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 ago
      So, 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 ago
    If 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 ago
      Sorry, 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?
- AceyMan 5y ago
  On this line of thought, the Numberphile YT video titled "All the Numbers" hits on this and other interesting truths. (I'd never learned the term 'normal number' until watching this.) Runtime 14m27s. Highly recommended.
- colanderman 5y ago
  Not just very hard, but actually impossible for almost all transcendentals. Reason being, any description must be finite in length; descriptions therefore form a countable set; but transcendentals are uncountable. The subset of reals which are describable are known as "definable real numbers" [1]. (Notably, all algebraic numbers -- the reals of which form the complement of transcendentals -- are definable by definition.) [1] https://en.wikipedia.org/wiki/Definable_real_number https://en.wikipedia.org/wiki/Definable_real_number
  - lisper 5y ago
    Yes, that's true. But what I intended to convey was that it's very hard in the sense that we have only described a few dozen classes of transcendentals. So the actual limiting factor has to be something other than this counting argument. The hard part is finding descriptions of transcendentals within the space of describable numbers.

lisper 5y ago

OK, here's a fun fact: most numbers are transcendental, which is to say, if you choose a real number uniformly at random in any finite interval the probability of choosing a transcendental number is 1. (The proof is easy and left as an exercise.) But actually describing a transcendental number is very hard. The first transcendental number other than pi and e was not described until the mid 19th century, and even today only a few dozen classes of transcendental numbers are known.

JackFr 5y ago
Possible spoiler (I don't know if my proof is correct)-- Algebraic numbers are countable, since their description is a countable combination of rationals which are also countable. The measure of a countable set is zero, so the measure of its complement is 1, and thus the probability of choosing a transcendental is 1.
- lisper 5y ago
  Exactly right!
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 ago
  But 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 ago
    You 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 ago
    So, 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 ago
  If 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 ago
    Sorry, 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?
AceyMan 5y ago
On this line of thought, the Numberphile YT video titled "All the Numbers" hits on this and other interesting truths. (I'd never learned the term 'normal number' until watching this.) Runtime 14m27s. Highly recommended.
colanderman 5y ago
Not just very hard, but actually impossible for almost all transcendentals. Reason being, any description must be finite in length; descriptions therefore form a countable set; but transcendentals are uncountable. The subset of reals which are describable are known as "definable real numbers" [1]. (Notably, all algebraic numbers -- the reals of which form the complement of transcendentals -- are definable by definition.) [1] https://en.wikipedia.org/wiki/Definable_real_number https://en.wikipedia.org/wiki/Definable_real_number
- lisper 5y ago
  Yes, that's true. But what I intended to convey was that it's very hard in the sense that we have only described a few dozen classes of transcendentals. So the actual limiting factor has to be something other than this counting argument. The hard part is finding descriptions of transcendentals within the space of describable numbers.