question about flooring a certain kind of numbers

Asked Aug 17 '21 at 12:57

Active Aug 17 '21 at 14:01

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Our math teacher was teaching us about the floor function and after finishing all the important parts we started to do some exercises by ourselves and there was this problem which we all did wrong: $$\lfloor2.\overline{9}\rfloor=??$$ We all thought that since $2.\overline{9}< 3$ the answer would be $2$, but the teacher said the answer is $3.$ He tried to explain us that since we can't find any number between $2.\overline{9}$ and $3$ it can be mathematically proved that $2.\overline{9} = 3.$

He said the full proof of that is very hard and long so I am curious about this subject and I want to know the reason behind this.

thanks in advance

edited Aug 17 '21 at 14:01

Andria

asked Aug 17 '21 at 12:57

infinite

https://en.wikipedia.org/wiki/0.999... – Steven01123581321 Aug 17 '21 at 13:01
1

Duplicate of this question, which has 89 (!) answers. – TonyK Aug 17 '21 at 13:03
The proof is not very hard or long. It is immediate if you have an understanding of limits, and well-explainable if you accept this argument: $x=2.\overline{9},10x-x=27,9x=27\implies x=3$. This works because a decimal string such as $2.\overline{9}$ must converge to a real number – FShrike Aug 17 '21 at 14:40

question about flooring a certain kind of numbers

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