Ivan Shishkin, Rye (1878)

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0.999999...=10.999999...=1

by Uettechat·
26
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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Montrer que le nombre 0.99999999...0.99999999... (où les 99 continuent jusqu’à l’infini) vaut en fait 11.

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Solution by Uettechat

Discussions1 useful vote

0.9999...=0.9+0.09+0.009+...=9(0.1+0.01+0.001+...)=9(10−1+10−2+10−3+...)=9∑n≥110−k=91011−1/10=10.9999...=0.9+0.09+0.009+...=9(0.1+0.01+0.001+...)=9(10^{-1}+10^{-2}+10^{-3}+...)=9\sum_{n\geq1}10^{-k}=\frac{9}{10}\frac{1}{1-1/10}=1

Solution by budleya

Discussions1 useful vote

Critiquer celle-ci : avec x=0.999…x=0.999\dots,
10x=9.999⋯=9+x10x=9.999\dots=9+xdonc 9x=99x=9 et x=1x=1.

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