I try to illustrate the formation and derivation of algebraic numbers and transcendental numbers. I found that both categories of numbers can be made by continuous summation or division/fraction method. Are there some distinguishable features between algebraic and non-algebraic numbers, if you make them via summation or continued fraction method?

I thought I found an easy way to explain the difference between rational and irrational numbers by a continued fraction, which is a sort of precision made with a limit of infinity. But now this method doesn't seem to explain how the next categories of numbers, namely the imaginary, the transcendental and the complex numbers differ from each other. At least transcendental numbers can be expressed by a summation too.

I guess the same question can be asked if summation / continued fraction method can reveal if the number is algebraic?

Can we pinpoint the transcendentality of the number from the continued fraction or the summation notation in mathematics?

  • $\begingroup$ In general, the continued fraction does not help to check whether a given number is algebraic irrational or transcendental. In practice, it is even worse. If the continued fraction does not terminate, in general there is no proof that the given number is actually irrational. This is probably the case with the Euler-Mascheroni-constant. $\endgroup$ – Peter Aug 16 '18 at 11:40
  • $\begingroup$ Moreover, not for every transcendental number we will find an infinite closed-form sum converging to the given number. Even worse, there are transcendental numbers that are not even computable. $\endgroup$ – Peter Aug 16 '18 at 11:44
  • $\begingroup$ That is somewhat sad. Is there any alternative way to illustrate difference than "integer coefficient polynomial root" thing? $\endgroup$ – MarkokraM Aug 16 '18 at 12:06
  • $\begingroup$ Usually , even irrationality proofs are extremely difficult. Of course, this becomes not better for transcendentality-proofs. But some numbers can be proven to be transcendental. Google "Liouville-numbers" , "Gelfond-Schneider-theorem" and "Baker's theorem" to get a feeling of the progress that was made. Despite this progress, we still do not much when it comes to irrationality-questions. This would change dramatically if "Schnauel's conjecture" would be proven. Currently, we do not even know, whether , for example , $e+\pi$ is irrational. $\endgroup$ – Peter Aug 16 '18 at 12:19
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    $\begingroup$ We also do not know whether $e\cdot \pi$ is irrational, but it might be of interest that it is easy to prove that at least one of the numbers $e+\pi$ and $e\cdot \pi$ is transcendental (probably both are). $\endgroup$ – Peter Aug 16 '18 at 12:25

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