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On peut également utiliser une bombe atomique pour tuer une mouche : "si x_0 est une solution de l'équation x^n + a_1.x^(n-1) + ... + a_(n-1).x + a_n = 0, alors x_0 est soit entier, soit irrationnel.
Posons x_0 = rac(5), alors x_0 est solution x² - 5 = 0 et c'est gagné.
Ce théorème est plus marrant pour x = rac(p) + rac(q) qui n'est pas entier :
x² = p + q + 2.rac(pq)
x^4 = (p + q)² + 4pq + 4(p+q).rac(pq)
d'où x^4 - 2(p+q).x² = -(p-q)².
Donc x est solution de x^4 - 2(p+q).x² + (p-q)² = 0.
Et rac(2)^rac(2), c'est rationnel ou irrationnel ? -
${\sqrt2}^{\sqrt2}$ est transcendant (théorème de Gel'fond-Schneider, 1934). C'est la solution au 7ème problème d'Hilbert. Pour une preuve: Lecture notes in mathematics 402, Waldschmidt
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Plus généralement, $\alpha^\beta$ est transcendant si $\alpha$ est algébrique $\ne0,1$ et $\beta$ est algébrique non-rationnel.
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Salut,
je profite de l'occasion pour poser la question suivante:
c'est classique la démonstration que pour n qui tend vers l'infini:
$$\sum_{k=1}^{n}\frac{1}{k}-ln(n)=\gamma$$
où $\gamma\sim0,57...$ appelée constante d'Euler.
ma question est:
$\gamma$ est-elle rationnelle ou irrationnelle?
Rmq: dans un livre de prépa imprimé en 2000, l'auteur affirme que ce nombre demeurait de nature inconnue.
je sais pas donc si ce nombre est toujours inconnue aujourd'hui?
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En effet, on ne sait toujours pas si $\gamma$ est rationnel ou non.
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Il y a toutefois un résultat récent qui affirme que s'il n'est pas rationnel alors sa mesure d'irrationalité n'est pas n'importe laquelle.....Il me semble qu'on en a déjà parlé ici il y a quelques mois ?
lolo -
Bonjour
Je prends l'intervalle d extrémité 0 et 1, ouvert en 0 et fermé en 1.
La longueur de cet intervalle n’est pas 1, c’est quoi alors ?
Merci pour votre aide. -
je reviens à la "descente infinie"
si on applique le raisonnement de oumpapah a $\sqrt{4}$=$\frac{p}{q}$
alors on déduit que $\sqrt{4}$ est irrationnel, il faut donc préciser quelque chose. -
Bonsoir
à vue de nez, 4 n'est pas un nombre premier et cela ne remet pas en cause l'argumentation pour sqrt(5)
les fractions continues donnent l'irrationnalité également, je lis actuellement : le fabuleux destin de "sqrt"(2)), j'espère y lire des démonstrations exotiques (j'ai vu g-adique... dans le sommaire)
Les maths sont si belles
aimablement
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