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.
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2The 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 02:45
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what do that $[.]$ stand for? – Mikasa Dec 27 '15 at 02:46
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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 03:02
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Sounds like it should have just been $\lceil \pi \rceil$. That's gimmicky enough. – Em. Dec 27 '15 at 03:08
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"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 03:32
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@fleablood "Should" in the sense that $[x]$ is not as "universal" as $\lfloor x\rfloor$. And to what extent it should be convoluted I guess is a matter of opinion. – Em. Dec 27 '15 at 05:12
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I have a suspicion it was $\lfloor \pi \rfloor$. If it isn't supposed to be convoluted it'd say "4". The entire point is something quasi-convoluted. – fleablood Dec 27 '15 at 05:21
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Cool watch. I want a mathy watch that uses complex values from the unit circle on the complex plane to tell me time. ;D – Simply Beautiful Art Dec 27 '15 at 16:55
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$[x]$ 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 $-[-\pi]=-(-4)=4$
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Thank you! So is [x] just called the floor? or is there another name for it? – 3141 Dec 27 '15 at 02:52
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@CuddlyCuttlefish Yes it can often be used in that context. The notation is unfortunately not very uniform. – Cameron Williams Dec 27 '15 at 02:59
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@CuddlyCuttlefish The nearest integer to $n$ is $\lfloor n+0.5\rfloor$. – user236182 Dec 27 '15 at 03:10
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2@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 04:14
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@MPW Numbers of the form $\overline{k.5}$ with $k$ integer is the only exception where "the nearest integer" depends on your convention. In all other cases "nearest integer" is unambiguous and given by $\lfloor n+0.5\rfloor$. I should've mentioned that. There's a Wikipedia page on "nearest integer". – user236182 Dec 27 '15 at 04:19
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@user236182: I understand. My point is that there are lots of rounding rules to handle the problematic points. Round up, round down, round up or down according to whether the $1+ \lfloor |n|\rfloor$th decimal digit of $\pi$ is odd or even... – MPW Dec 27 '15 at 04:42
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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: $$- [ - \pi ] = - [ -3.1415\ldots\, ] = - (-4) = 4$$
MathIsNice1729
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