Taken from the Rhode Island Mathletes Documentation from 1986 (I encountered this problem while scanning through older documents from a contest I'm currently preparing for)

if $[x]$ is the largest integer not bigger than $x$, what is $[-2.4] + [2.4]$

I'm not sure what this question is asking at all, for a start, how can $[x]$ be less than $x$? And what do the brackets mean at all?

  • 1
    $\begingroup$ The definition is given to you: "$[x]$ is the largest integer not bigger than $x$." You may know this as "the result after rounding down $x$ to the nearest integer." We do have $[x]=x$ if $x$ is an integer, but in general it might be that $[x] < x$. $\endgroup$ – angryavian Oct 26 '17 at 2:28
  • $\begingroup$ The meaning of the brackets is explained in the line you quoted: $[x[$ means the largest integer not bigger than $x.$ $\endgroup$ – bof Oct 26 '17 at 2:28
  • $\begingroup$ This is a common notation. See en.wikipedia.org/wiki/Floor_and_ceiling_functions#Notation $\endgroup$ – Cornman Oct 26 '17 at 2:29
  • $\begingroup$ @bof So [x] basically means to floor the value of $x$? $\endgroup$ – Travis Oct 26 '17 at 2:31
  • $\begingroup$ "How can $[x]$ be less than $x$?" If $x$ is not an integer, then $[x]$ has to be less than $x.$ It can't be equal to $x$ because it's an integer and $x$ is not an integer; it can't be bigger than $x$ because it's "not bigger than $x;$ the only possibility left is that it's less than $x.$ For example, if $x=3.5$ then $[x]=3\lt3.5=x.$ $\endgroup$ – bof Oct 26 '17 at 2:32

This is the floor function, which takes any number and "rounds down" to the closest integer that's smaller than or equal to $x$. The ceiling function is very similar, but "rounds up" to the closest integer greater than or equal to $x$. Note that if $x$ is whole, $\lfloor x \rfloor$ = $x$, not $x-1$.


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