Product of imaginary numbers. A book stated that $\sqrt{-a}•\sqrt{-b} = -\sqrt{ab}$   where, a and b are positive real numbers. I know the proof of the above equation but why isn't this $\sqrt{-a}•\sqrt{-b} = \sqrt{ab}$. It also stated that this equality $\sqrt{a}•\sqrt{b} = \sqrt{ab}$ is true only when either of a and b are positive or zero but is false when a and b are negative, but why? Thank you for your help.
Why is $\sqrt{-x}*\sqrt{-x}=-x?$ Q1 - Why is $\sqrt{-x}*\sqrt{-x}=-x?$  Q2 - I was thinking it would be:  $\sqrt{-x}*\sqrt{-x}=\sqrt{-x*-x}=\sqrt{x^2}$ but apparently not (why not?)  Q3 - What are the formal algebra rules to use? Can I calculate this without using i such as in: $\sqrt{-x}*\sqrt{-x}=i\sqrt{x}*i\sqrt{x}=-\sqrt{x^2}=-x$.
how to prove $437\,$ divides $18!+1$? (NBHM 2012) I was solving some problems and I came across this problem. I didn't understand how to approach this problem. Can we solve this with out actually calculating $18!\,\,?$ How do I get useful SQL Server database performance statistics? We currently have several shared SQL Server servers and instances in our organization. They are going with a different architecture where business units will get their own SQL Server instances. We have SQL Server 2008R2.  In the past, databases were put on servers in a more or less random fashion -- business use and disk space were the main drivers.  I've been tasked with coming up with a plan for allocating our 50+ databases across 3 servers (one instance per server). My criteria for divvying them up is as follows:   Business critical db's should be spread across the servers. i.e. avoid lumping a bunch of business critical db's all one server.   For the remainder of the non-mission critical db's, consider: a. Disk space usage b. I/O usage -- memory, CPU, etc.     1 and 2a are easy for me to figure out. I know how to get data file sizes.   My question is with that last bit - 2b. I do not know how relevant this is, what specific stats to look at, or how to get them.   Frankly, I'm not 100% sure how to phrase this question! I think I basically want to know is, "How 'busy' is this db -- however you define 'busy'?" Or maybe, "How does this db impact the performance of the server?"  I mean, disk space is important -- but it could (in theory) have a ton of rows or binary objects in there that are not written very often, so it has little impact on the performance of the server. I think I would want to know how much impact a db within a date range.  Is there a way to get these kind of general stats? I'm not looking for expensive queries, detailed user stats, or query optimization here. 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Has there been any news from the maintainers of Ubuntu packaging of Ruby as to whether they'll be releasing a package for Ruby 2.0? If so, have they indicated whether they'll be releasing any packages for Ruby 2.0 on 12.04? I asked about it on  and they told me to post it here.  Probably need to add the :active to the :hover meta class or something like that.  Example:      Hammer Time I designed a mockup on my LG monitor. When I view the same design on another monitor that I own, the colors seem drastically different. I reset both of the monitors to their factory defaults, but still the issue persists. I'm wondering if some of my clients might dislike my designs because of this. What should I do?  How can I make sure that what I design is what my client sees?  Even worse still, I've realized that my laptop is showing different colors too. Circular Dependencies? How can I get the list of available databases on a SQL Server instance? 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Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact.  This has been asked before, but all the answers I have seen prove the contrapositive. Realistically, this may be the way to go, but is there way to exhibit an unbounded continuous function (under the assumption $X$ is not compact) without appealing to results beyond introductory real analysis (e.g., the solution I've seen involves the Tietze Extension theorem)?  Because we're working with metric spaces, it's clear that the assumption that $X$ is noncompact (towards proving contrapositive) will lead us to extract a sequence of points in the space with no convergent subsequences. But how can we infer the existence of an unbounded continuous on the space knowing only about some sequence of points in this space (without using anything too advanced)? What is the purpose of JNDI I want to advance in the ranks of Senator Lavigny-Duval. How can I get merit points? 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