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

Question

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.

Answer 1

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

Answer 2

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: $$- [ - \pi ] = - [ -3.1415\ldots\, ] = - (-4) = 4$$

Answer 3

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