Does $[0.9999....]=1$?

Question

We all know that $0.99999...=1$

So does that imply $[0.99999...]=1?$

Or do we consider it as $0?$

My doubt is: any gif of the form $[0.xyz...]=0$. If $[0.99999...]=1$ won't that be contradicting?

There isn't much clear explanation in the previous post.

Here $[.]$ is the greatest integer function. I couldn't find a post containing this query on MSE. Please help :)

Relevant Article

Answer 1

If $x = y$ then $[x] = [y]$. How could they not?

So as $0.9999.... = 1$ it has to be that $[0.9999...] = [1]$.

Equality means they are the same thing. If they are the same thing whatever you do to them will have the same result.[*]

[*](Provided what you do to them is based on their value. It's possible for things to not be "well-defined" which means they will not have consistent results based on circumstances that aren't about their value. Operations that are not "well-defined" are not considered to be valid. Usually. There are always exceptions.)

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Ah, I didn't see the $[0.xyz...] = 0$ confusion.

Well, that just isn't true (although 0.9999.... is the only exception).

Well.... You have to keep in mind 0.9999.... is an integer even if it doesn't look like one.

Keep in mind math results are based on what thing are; not what they look like. 0.999.... is the only exception to 0.xyz.... < 1. And it's because 0.999.... $\not <$ 1 it doesn't follow that [0.9999... ] < 1.

Believe me. If I could apologize for the confusion, I would.

Answer 2

Let's recall the definition. $\lfloor x\rfloor$ is the greatest integer $n$ satisfying $n\leq x$. But now we have $1\leq 0.999\dots$, and no integer greater than $1$ doesn't satisfy $n\leq 0.999\dots$, so, from the definition, $\lfloor 0.999\dots\rfloor=1$.

What probably confuses you is the mentioned fact that $\lfloor 0.xyz\dots\rfloor=0$ for any $xyz\dots$. The problem is that this is wrong. It certainly doesn't follow from the definition above, though it might follow from some informal definitions (of the sort "remove everything after decimal point"). Indeed, there is precisely one counterexample to $\lfloor 0.xyz\dots\rfloor=0$, namely $0.xyz\dots=0.999\dots$.