prove that x is a real number then $\lfloor −x \rfloor = −\lceil x\rceil$ and $\lceil −x \rceil = −\lfloor x \rfloor$. If I were to put a real number say like $1$, wouldn't it be right? But if I put a value like $1.5$ it would be incorrect? I don't think this is the way I should be proving it though so could anyone help me out.
[Math] Floor and Ceiling function
discrete mathematics
Best Answer
HINT: As you noted, it’s easy to see that the identities hold when $x$ is an integer. Suppose that $x$ is not an integer; then there is an integer $n$ such that $n<x<n+1$. Multiplying this by $-1$, we see that $-(n+1)<-x<-n$. What are $\lfloor x\rfloor$, $\lceil x\rceil$, $\lfloor-x\rfloor$, and $\lceil-x\rceil$ in terms of $n$? Do the identities hold?
If you think in visual or physical terms, it might be helpful to visualize multiplying the number line by $-1$ as rotating it $180$° about the point $0$, so that the numbers $x$ and $-x$ change places. If $x$ is trapped between two integers, it will still be trapped between their negatives after the rotation, but the order of the three numbers will have been reversed.