Ivan Shishkin, Rye (1878)

Problems/OtherUnreviewed

0.999999…=10.999999 \ldots=1

by Uettechat·translated by visitor·
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.
·
English
EnglishFrançais
Unreviewed. This problem has not been reviewed by trusted users yet.

Show that the number 0.99999999...0.99999999... (where the nines continue forever) is in fact equal to 1.

I solved itMark it doneAdd to my listKeep it in your list
I like this problem2 likes

Liked by

Show related problems0

No related problems yet.

Solutions

2
Reveal solutionsAre you sure? Give it a try first.

Solution by UettechatFR

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 budleyaFR

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.

Report

For an unclear, ambiguous, or possibly incorrect statement, please use the Discussion tab on the right. Report content that needs moderator intervention, such as dangerous, clearly non-mathematical, or plagiarized content.