# How does $-[-\pi]$ equal 4?

For Christmas I got a math watch and for 4 it was $-[-\pi]$. I know that $\pi$ does not equal 4 so how does $-[-\pi]$ equal 4? Thank you.

• The symbol $[ \cdot ]$ means to take the floor. It's not a general bracket. The floor of $-\pi$ is $-4$ (since it is the largest integer smaller than $-\pi$) and then taking the negative of it gives you $4$. – Cameron Williams Dec 27 '15 at 2:45
• what do that $[.]$ stand for? – mrs Dec 27 '15 at 2:46
• I think the $\cdot$ was meant as a place holder. $\lfloor x \rfloor$ means the largest integer less than or equal to x. – fleablood Dec 27 '15 at 3:02
• Sounds like it should have just been $\lceil \pi \rceil$. That's gimmicky enough. – Em. Dec 27 '15 at 3:08
• "should"? "gimmicky enough"? the entire point of the watch is to be convoluted so "$-\lfloor \pi \rfloor$ is *much better than ceiling pi because it's much less direct. – fleablood Dec 27 '15 at 3:32

$[x]$ is the floor of $x$, the largest integer less than or equal to $x$. For example $[4.6]=4$, $[7]=7$ and $[-67.4]=-68$.

So $-[-\pi]=-(-4)=4$

• @CameronWilliams Thanks :) – user223391 Dec 27 '15 at 2:48
• Thank you! So is [x] just called the floor? or is there another name for it? – 3141 Dec 27 '15 at 2:52
• Isn't $[x]$ often the nearest integer? – TokenToucan Dec 27 '15 at 2:55
• @ArchisWelankar That wouldn't make sense here. – user223391 Dec 27 '15 at 3:33
• @user236182: Not "the nearest integer to $n$", but rather "$n$ rounded up to the nearest integer". There is no such thing as "the nearest integer" since, for example, $0$ and $1$ are equally near to $\frac12$. – MPW Dec 27 '15 at 4:14

Well... $[ x ]$ gives the largest integer $\leq x$. That is, $[ x ] = \sup\{z : z \in \mathbb{Z}, z\leq x\}$ if one likes to complicate things. Anyway, it's rather straightforward to prove your claim: $$- [ - \pi ] = - [ -3.1415\ldots\, ] = - (-4) = 4$$

Hint: By definition, $x=[x]+\{x\}.$ By convention, $\{x\}\ge0$.