Transcendental Numbers

Jean-François COLONNA
[Contact me]

www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

[Site Map and Help [Plan du Site et Aide]]
[The Y2K Bug [Le bug de l'an 2000]]
[Are we ready for the Year 2038 [Notre informatique est-elle prête pour l'An 2038]?]
[Real Numbers don't exist in Computers and Floating Point Computations aren't safe. [Les Nombres Réels n'existent pas dans les Ordinateurs et les Calculs Flottants ne sont pas sûrs.]]
[Please, visit A Virtual Machine for Exploring Space-Time and Beyond, the place where you can find more than 10.000 pictures and animations between Art and Science]
(CMAP28 WWW site: this page was created on 12/12/2024 and last updated on 05/08/2026 20:16:25 -CEST-)

[en français/in french]

Keywords: A2M, An 2000, Bogue, Bug, Bugix, Millenium Bug, Year 2000 Problem, Y2K, Y2K Bug, Anaglyphs, Art and Science, Artistic Creation, Autostereograms, Celestial Mechanics, Computer Graphics, Deterministic Chaos, Fractal Geometry, Intertwinings, Mathematics, Natural Phenomenon Synthesis, Numerical Simulation, Physics, Quantum Mechanics, Rounding-off Errors, Scientific Visualization, Sensitivity to Rounding-Off Errors, Software Engineering, Stereograms, Texture Synthesis, Virtual Experimentation, Virtual Space-Time Travel.

The dictionary Le Petit Robert explains that something is transcendental when it rises above a given level, but also when it is sublime or superior. This implies, in everyday life, that something transcendental is rare, exceptional...

However, in Mathematics, it is quite different. Indeed:

The set R of real numbers can be divided into two subsets:
- The subset A of algebraic numbers, which are the real solutions of polynomial equations with integer coefficients. For example, the first-degree equation:
```
7.X - 22 = 0
```
  defines the rational number 22/7. As for the second-degree equation:
```
X²- X - 1 = 0
```
  it defines the numbers (1-sqrt(5))/2 and (1+sqrt(5))/2 (the golden ratio -phi-).
- The subset T of transcendental numbers, which are numbers that are not algebraic. The most famous among them is certainly the number π...
and thus:
```
R = A ∪ T
A ∩ T = 0
```
Regarding the respective cardinalities of N (the integers), Q (the rational numbers), A and R, it is easily demonstrated that:
- N is described as countable.
- Q is countable due to the existence of a bijection with N
- A is countable due to the existence of a bijection with Q
- R is uncountable and is described as continuous.

In summary:
```
R(Uncountable) = A(Countable) ∪ T
```
implies that T , like R, is uncountable, and thus T is "much larger" than A. Therefore, it is the algebraic numbers (in A) that are rare, while the transcendental numbers (in T) are the most "abundant" in R.

Transcendental Numbers

Jean-François COLONNA [Contact me]

www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

Copyright © Jean-François COLONNA, 2024-2026. Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2024-2026.

Jean-François COLONNA
[Contact me]

Copyright © Jean-François COLONNA, 2024-2026.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2024-2026.