# $\left\lfloor x\right\rfloor$ and $[x]$ are the same concept?

The function $$f(x)$$ integer part of $$x$$ is defined as the largest integer less than or equal to $$x$$. Generally I have always use this symbol to indicate the integer part of $$x$$

$$f(x)=[x]$$ instead of

$$f(x)=\left\lfloor x\right\rfloor.$$ Are there some differences or are they represent the same quantity?

• Depending on context / author, $[x]$ may mean the nearest integer (i.e. round instead of floor or ceiling). See e.g. here. – Bungo Sep 27 at 22:34
• This is the same object. Both notations are used. – Mark Sep 27 at 22:34
• If this bothers you, then get ready for binomial coefficients... I've seen no less than 8 variations on notation for that. – JMoravitz Sep 27 at 22:38
• @YiFan $\binom{n}{k},~\binom{k}{n}$ (yes, some places flip which goes on top vs bottom), $~_nC_k,~^nC_k~C^n_k,~C^k_n$ (see previous) $~\text{Binom}(n,k),~C(n,k),~(n)_{k}$ (made more confusing as this pochhammer symbol is also sometimes used for falling factorial instead) – JMoravitz Sep 27 at 22:46
• If $\lfloor x \rfloor$ doesn't mean the largest integer less than or equal then what does it mean? Things mean what we say they mean. But... okay, no, mathematicians don't have some subtle different meaning for it. It (often called the "floor" function) is a function $\lfloor\ \rfloor:\mathbb R \to \mathbb Z$ via $\lfloor x\rfloor= n$ where $n$ is the unique integer so that $n\le n < n+1$. Personally I find it for more clear to read but a real pain to type. – fleablood Sep 27 at 23:29

$$[x]$$ is the older notation for the largest integer that does not exceed $$x$$. Earliest Uses of Various Mathematical Symbols says

Greatest integer function (floor function). Until recently $$[x]$$ has been the standard symbol for the greatest integer function. According to Grinstein (1970), "The use of the bracket notation, which has led some authors to term this the bracket function, stems back to the work of Gauss (1808) in number theory. The function is also referred to by Legendre who used the now obsolete notation $$E(x)$$." The Gauss reference is to Theorematis arithmetici demonstratio nova. Werke Volume: Bd. 2 p. 5.

Recently the symbol $$\lfloor x \rfloor$$ has come into use. It was introduced in 1962 by Kenneth E. Iverson who also coined the name floor function. [...]

$$\lfloor x \rfloor$$ has two advantages: it is completely unambiguous (there is nothing else it is used for) and it has a mnemonic symmetric partner in $$\lceil x \rceil$$, the ceiling function. Examples of use: \begin{align} \lfloor 2 \rfloor &= 2 & \lceil 2 \rceil &= 2 \\ \lfloor 2.5 \rfloor &= 2 & \lceil 2.5 \rceil &= 3 \\ \lfloor -1 \rfloor &= -1 & \lceil -1 \rceil &= -1 \\ \lfloor -1.5 \rfloor &= -2 & \lceil -1.5 \rceil &= -1 \\ \lfloor \pi \rfloor &= 3 & \lceil \pi \rceil &= 4 \end{align}

• Thank you a lot lot for your excellent explanation. Do you can put any example for $\lfloor x \rfloor$ and $\lceil x \rceil$ so I can show my high school students the two quantities so I have everything in one answer? Thank you. – Sebastiano Sep 27 at 22:46
• @Sebastiano Added, will that suffice? – Chappers Sep 27 at 22:50
• Approved and thank you everybody. – Sebastiano Sep 27 at 22:51
• Learn something new …. I thought that $\lfloor\,\cdot\,\rfloor$ was invented by Knuth when he wrote $\TeX$. – Daniel Fischer Oct 1 at 21:05
• Yeah, I guess "most probably did" was where I heard it from. And I approve of what Iverson used the old Gauß brackets for, it's very convenient sometimes. – Daniel Fischer Oct 1 at 21:16