-1

Are there any transcendentals whose real or imaginary components have not been found in exact form?

Nonsematter
  • 167
  • 2
    You should look up "non-computable numbers". Indeed most number are of this type (and forever will be) – Pax Jul 21 '16 at 13:51
  • True, like $\pi$ or the Feigenbaum constant. Are there any particularly interesting ones for which no name has (yet) been assigned? – Nonsematter Jul 21 '16 at 13:53
  • 2
    $\pi$ is computable, as is the Feigenbaum constant. Computable numbers are numbers where it is impossible (in a certain techincal sense) to compute the number. An example, I believe, is the probablility a ramdomly selected Turing Machine will hault. We don't know this number, and it is impossible to ever know this number. There are a variety of other examples, and infact most numbers have this property. – Pax Jul 21 '16 at 14:00
  • True, I've looked up the number - it's called Chaitin's number/the halting probability - I like it! – Nonsematter Jul 21 '16 at 14:21
  • Note that if $\alpha$ is a complex number expressed in some suitable fashion, then $\mathrm{Re}(\alpha)$ and $\mathrm{Im}(\alpha)$ are explicit formulas for its real and imaginary parts. –  Jul 21 '16 at 14:35
  • 1
    Maybe the question should be interpreted as follows. Suppose there exists a complex number $c$ defined by $f(c)=0$, where $f$ is known. Suppose $c$ is equivalently defined by functions $g,h$: $g(\Re(c))=0,h(\Im(c))=0$. Could it be that $f$ is known, but $g$ and $h$ are not? – Wouter Jul 21 '16 at 15:02

1 Answers1

3

In any reasonable interpretation, "exact form" means "expressible in a finite number of symbols from a fixed finite alphabet".

The number of finite words on a fixed finite alphabet is countable.

The set of real numbers is not countable.

The set of real algebraic numbers is countable.

The set of transcendental numbers is not countable.

Therefore, there are transcendental numbers that are not expressible in exact form. (Most of them!)

lhf
  • 216,483