This is a hard question to ask. But I've been contemplating transcendental numbers. I know that there are infinitely many; simply multiply a known transcendental (like pi) by every rational number. So are there any transcendental numbers that we cannot conceptually denote? That is, they don't represent constants of nature or mathematics like pi an e. This question seems obviously true to me, but a follow up question is, is there a classification for these numbers?

The idea of such numbers fascinates me. They are like points on number line that cannot be reached even conceptually. Even pi and e can be represented geometrically. I want to know what these numbers are called so that I can research them.


This is number as I understand it:

Rational -> Algebraic Irrational -> Transcendental Irrational

I am wanting to divide the transcendental irrationals into two groups: the ones that can be represented conceptually, and the ones that cannot. It is this second group that I am interested in. They are transcendentals, but I'm wondering if they have some other classification indicating their inability to be conceptually reached.

Further Clarification:

I'm looking simply for a name for the classification of transcendentals that are beyond any ability to represent conceptually. For instance, you can represent pi by relating a circumference of a circle to its diameter. You can conceive of e as the base of the natural logarithm. These are transcendentals, but we have access to them conceptually. Any multiple of these by a rational or algebraic irrational can also be accessed conceptually. But there have to be numbers that are not tethered to some conceptual anchor. Not only are they transcendental they are inaccessible. We now they are there, but they represent holes in our numberline points that we can never get to. Has no one been as fascinated by this to give this classification a name?

Non-computable is the best answer I've gotten. That's what I'll go with I suppose.


closed as unclear what you're asking by José Carlos Santos, Xander Henderson, Crostul, Will Jagy, ahulpke May 17 '18 at 19:43

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    $\begingroup$ You might be interested in computable numbers (this is a related idea). Essentially, the collection of numbers that can be described in a finite number of symbols from a finite alphabet is at most countable. This implies that an uncountable number of real numbers cannot even be described in a finite way. $\endgroup$ – Xander Henderson May 17 '18 at 18:29
  • $\begingroup$ How is this different from algebraic irrationals like phi and 2^-1? $\endgroup$ – Jordan May 17 '18 at 18:36
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    $\begingroup$ The set of computable numbers is a strict superset of the set of algebraic numbers. $\endgroup$ – Xander Henderson May 17 '18 at 18:37
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    $\begingroup$ math.stackexchange.com/questions/462790/… $\endgroup$ – Xander Henderson May 17 '18 at 18:44
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    $\begingroup$ Related: en.wikipedia.org/wiki/Ring_of_periods $\endgroup$ – Chappers May 17 '18 at 18:49

Whatever method you use to denote numbers with a finite number of symbols drawn from a finite list, only countably infinitely many representations will be possible, so almost all transcendental numbers cannot be.

  • $\begingroup$ You can't represent them in decimal notation, but you can represent them geometrically. I'm looking for a classification of numbers that cannot be represented this way. Numbers that do not have a connection to our platonic universe even conceptually. $\endgroup$ – Jordan May 17 '18 at 18:32
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    $\begingroup$ @Jordan Any constructible number is computable, i.e. it can be represented by a finite number of symbols drawn from a finite list. $\endgroup$ – Xander Henderson May 17 '18 at 18:36
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    $\begingroup$ @Jordan My argument isn't limited to decimal form; you define $\pi$ a different way, but still with a finite string of symbols. $\endgroup$ – J.G. May 17 '18 at 18:39
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    $\begingroup$ @Jordan The point of the answer is that once you fix a method of representation, there will be numbers that "escape the net." $\endgroup$ – Noah Schweber May 17 '18 at 18:54

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