If we have a number line of numbers between say -3 and 3 (-3  -2  -1  0  1  2  3) and we multiply by a  factor $k$ then the positive values will become $k, 2k, 3k$ respectively with 0 remaining zero. If the same is done to the negative values we have $(k)(-1) , (k)(-2), (k)(-3)$ which are $-k, -2k, -3k$ respectively since the negative implies they are opposite sides of the $0$.  What I fail to understand is if I multiply by $(-1)$ to all the terms, the positives will become negative (by the additive inverse property of numbers). But how do the values of $(-1)(-1), (-2)(-1), (-3)(-1)$ become $1, 2, 3$? What is the intuition behind this?  I am doing this without knowing that the product of two negatives is positive.
Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.  I went ahead and gave them a proof by contradiction like so:  Assume $(-x) \cdot (-y) = -xy$  Then divide both sides by $(-x)$ and you get $(-y) = y$  Since we have a contradiction, then our first assumption must be incorrect.  I'm guessing I did something wrong here. Since the conclusion of $(-x) \cdot (-y) = (xy)$ is hard to derive from what I wrote.  Is there a better way to explain this?  Is my proof incorrect?  Also, what would be an intuitive way to explain the negation concept, if there is one?
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This seems well targeted to appeal to the Stack Overflow / Server Fault / Super User audience, but way off the mark if the goal is to "make the Internet a better place to get expert answers to your questions."    There are experts who aren't interested in or focused on technology oriented topics.  Little aliens are unlikely to be something that someone interested in organizing a site on gardening would gravitate too.  No, I'm not suggesting that no one interested in gardening would be bother to look at a site with little aliens, but won't it be likely that the sort of mass appeal to to a very broad cross section of the internet will be lacking? Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k&gt;0$ is constant. Show that $f$ is a polynomial of degree at most $n$. Open Attachment on Create of Child Record Finding the maximum likelihood estimators for this shifted exponential PDF? I'm trying to align a legend inside a table caption, so that the colons are vertically aligned. I can do it with the align ambient outside of the caption, but if I use the same code inside the caption the pdf won't compile (I use LuaTeX to compile). I'm using the package caption.  EDIT: i want something like this:    for which i've written this code:  \begin{align} \begin{split} a : {} &amp; the~first~letter~of~the~alphabet \\ b : {} &amp; the~second~one \\ and : {} &amp; so~on \end{split} \end{align}   There might be a more elegant way... but i don't know it.  Anyway, I can't do it inside the caption, that's why i'm stuck. save as a video as .mp4 Why do objects sometimes fall on their own? 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