Transcendental Numbers

Jean-François COLONNA
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www.lactamme.polytechnique.fr

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

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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.