# What is this notation? f(x)=[x]=the greatest integer less than or equal to x.

I came across the question above. It is a function with domain all real numbers and range all integers. But is this square bracket notation used in general or did the author just use it for this particular case?

I would say that the typical notation for the function you describe (also known as the floor function) is $\lfloor x\rfloor$, but $[x]$ is not uncommon either.

However, I would recommend $\lfloor x\rfloor$, because $[x]$ is sometimes (not that common) used as the "fractional" part of $x$, i.e. $[x] = x-\lfloor x\rfloor$

• To add some history, $[x]$ is the original notation which was introduced by Gauss around 1800, $\lfloor x \rfloor$ is a more modern notation, apparently only going back to the 1960s. – mlk Aug 16 '17 at 7:37
• @mlk I didn't know that! I guess the modern notation is more informative, because it also allows for $\lceil x\rceil$... – 5xum Aug 16 '17 at 7:38
• Yeah I think most people will agree on that. I guess that the old notation will slowly die out. However I assume the old notation was a technical necessity in pre-LaTeX times. One of my professors in the first semester for example used a really old set of lecture notes (I guess from the 80s when he started here), still written on some sort of typewriter. Apparently it had no $\lfloor$ character, so he had to use $[$ instead. – mlk Aug 16 '17 at 7:55
• Yes. $\lceil x \rceil$ and $\lfloor x \rfloor$ were introduced by Kenneth Iverson, inventor of the programming language APL. – Robert Israel Aug 16 '17 at 7:59

Vinogradov uses $[x]$ for the integral part of $x$ and $\{x\}=x-[x]$ for the fractional.

It's often used for the floor function. Another one, which is used almost exclusively for it, is $\lfloor x\rfloor$ (as opposed to the least integer larger or equal than $x$, which would be the ceiling function $\lceil x\rceil$). As a reader, I prefer the latter.

Some programming languages use $\text{floor}(x)$ and $\text{ceil}(x)$.

It's just the floor function, which is nowadays more commonly notated as $\lfloor x \rfloor$. e.g., $$\left\lfloor \frac{-3}{2} \right\rfloor = -2.$$

Look at your computer keyboard. Do you see keys for "[" and "]"? (If you do, most likely they shift to "{" and "}"). I believe typewriters circa 1980 had keys for those square brackets.

But not even today's computers have keys for "$\lfloor$" or "$\rfloor$". Maybe in the days of LCARS you will have completely customizable keyboards, but for now we muddle along as best as we can. Some people just use the square brackets and hope the meaning is clear from the context.