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I am trying to figure out how to find the Floor and Ceiling of a nested Multiplication of fractions.

I understand that

$ \left\lceil{\frac{x}{mn}}\right \rceil= \left\lceil{\frac{\lceil{\frac{x}{m}}\rceil}{n}}\right\rceil $

works when x, m, n are all integers, and also for the Floor Function thanks to this great answer by Brian M. Scott but what about something like this:

⌈⌈⌈53 * 1/2⌉ * 9/5⌉ * 4/7⌉

or the inverse

⌊⌊⌊53 * 1/2⌋ * 9/5⌋ *4/7⌋

I have done some noodling and if you simply take the floor of each in a iterative process it sometimes yields different results depending on the order that you choose the fractions to multiply with. I would like to figure what the fixed lower/upper limit would be regardless of the order in which you multiply x by each of the fractions iteratively.

Is there a known method for working this out that is provably true?

Thanks in advance!

user2141965
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1 Answers1

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The property $ \left\lceil{\frac{x}{mn}}\right \rceil= \left\lceil{\frac{\lceil{\frac{x}{m}}\rceil}{n}}\right\rceil $ allows the removal of the inner ceiling operators in an expression when $n$ is a positive integer. In general expressions where the above rule does not apply, the ceilings cannot be eliminated:

\begin{align} &\lceil \lceil \lceil 53 * 1/2 \rceil * 9/5 \rceil * 4/7 \rceil\\ =&\lceil \lceil 27 * 9/5 \rceil * 4/7 \rceil\\ =&\lceil 49 * 4/7 \rceil\\ =&28 \end{align}

Eliminating ceilings arbitrarily is unsound but if you were to do it you would never exceed $28$ since $x \le \lceil x \rceil$ for any $x \in \mathbb{R}$.

Abhishek
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