Numbers with no finite representation on paper

Question

It occurred to me that there must be a lot of numbers without any form of finite representation on paper. Is there a name for these numbers?

For example...

Integers and rationals have a very simple representation e.g. 3/4

Irrational numbers obviously can also have a finite representation: 1.41421356... can be written as "the solution to the equation x^2 = 2"

Transcendental numbers can also have a finite representation: e can be written as "the limit of (1 + 1/n)^n as n approaches infinity"

In other words, with a finite amount of effort one can give the reader enough information to calculate the value of the specified number exactly (to any degree of accuracy the reader chooses)

However, there must be a lot of numbers where this simply is not possible.

Consider the number 1.2736358762987349862379358... where this is just a string of (genuinely) random digits. There is no way to provide a finite definition that will specify this number to an arbitrary degree of accuracy. Similarly, there is no equation to which this number is a solution (I think, although I don't know how one would prove this).

Does this mean there are "gaps" in the real numbers. The number above is definitely somewhere between 1.2 and 1.3 but there is no way I can specify the value of this number (without writing an infinite number of digits). The number exists on the number line but I will never be able to do anything with it.

Is there a name for these numbers? Can anyone point me to some interesting resources on this topic?

I'm only asking as an interested hobbyist so apologies if this question isn't very scientific.

Answer 1

That depends on what you mean by "representation." One way to cash this out is to talk about the definable numbers. These are more general than the computable numbers, but they are still countable because there are still only countably many possible descriptions of a number in a language over a finite alphabet.

Answer 2

I believe what you are describing are numbers which are not computable. I think that http://en.wikipedia.org/wiki/Computable_number explains it well enough. Actually, almost all numbers are not computable.

Answer 3

The number of finite names/words over a finite alphabet is countable, so you cannot name every irrational. Certainly, if you would use infinite words or infinite alphabet...

Answer 4

In addition to Daniel R.'s link, you might find the notion of Kolmogorov complexity (a.k.a "descriptive complexity") useful. As a very short summary, descriptive complexity measures how easy it is to write an exact description of a number; the numbers you're talking about don't have a description with a finite length, or any description "denser" than a simple list of the number's digits. By contrast, the transcendental number $e$ has no algebraic description, but it can be described very compactly by any one of its several unique properties (e.g., "the base of the natural logarithm").

Answer 5

You pick up an interesting subject. In my opinion you should do research into Math Logic (definability, etc.).

Assume you formally proved that some number is not definable (by means of symbols on a piece paper as you said). Then you come up with a paradox because your formal proof (a lot of finite number of symbols from a formal language) is already a some kind of "representation" of the number on paper.

It implies that even if such kind of number exists,formal proof of its existence is impossible.

That reminds me Continuum Hypothesis.