You can't derive existence of multiplicative inverse from existence of multiplicative unity, you have to assume existence of multiplicative inverse.
Right. I'm not deriving that.
And there is no identity rule for division
Formally speaking, your right. But for a precalculus audience x/1 = x is taught as a 'property of division'. I perhaps misspoke calling it a 'rule'.
Huh? You've just stated two formulations that mean the same exact thing.
No. They are NOT quite the exact same thing. The identy x*1 = x is defined for all x in the set of Real numbers. The 2nd form is not.
x*1 = x is defined for x=0
x/x = 1 is not defined for x=0
Just as a graph of y=1, is not quite the same as a graph of y = x/x. The latter has a discontinuity at 0, the former is continuous.
When you have 0/0, you hit two "obvious" but contradictory rules in basic algebra:
Rule one: anything multiplied by zero is zero
Rule two: anything divided by itself is one
Ugh no, just no.
"Rule one: anything multiplied by zero is zero"
Yes, this is called, amongst other things, the zero property of multiplication. However 0/0 is not a multiplication and the rule is not relevant, and there is no conflict.
Secondly your "rule two" is not actually rule of algebra. There is no rule x/x = 1.
There is an identity rule for division: anything divided by one is itself (x/1 = x) but there is no rule that says x/x = 1
You can derive "rule two" from the identity rule for multiplication x*1 = x --> x/x = 1
However, that transformation always stipulates that x 0 because division by zero is undefined.
Mathematicians have no issue determine which rule has precedence, because neither rule applies to 0/0.
There is no conflict. Division by zero is specifically "undefined".
Consider the equation; x/x.
http://www.wolframalpha.com/in...
The graph of the function is a horizontal line at y=1, with a discontinuity at 0. (if x=0, x/x=0/0) So 0/0 should be 1 right? Because everywhere else on the graph x/x = 1??
http://www.wolframalpha.com/in...
Now consider the equation 2x/x.
http://www.wolframalpha.com/in...
As x approaches 0 (lim x->0) from either the left or right the limit of the equation is 2. A graph of the function is horizontal line at y=2, with a discontinuity at 0. But every where else 2x/x = 2. So shouldn't 2(0)/0 = 0/0 = 2? So 0/0 should be 2 right?
http://www.wolframalpha.com/in...
Neither. Its not defined.
Now consider the equation 1/x.
http://www.wolframalpha.com/in...
As x approaches 0 from the left it goes to negative infinity. As x approaches 0 from the right it goes to positive infinity. This graph doesn't even suggest a value for 0/0? Is it + infinity? Or - infinity?
I can write a function that makes 0/0 look like it should be anything I want.
0/0 is undefined. It doesn't violate any rules of algebra. It's a rule of algebra that division by 0 is undefined.
I don't disagree with you. But one role of Windows servers you sort of glossed over is simply: "managing windows desktops" (active directory, group policy, etc, etc). If you are running more than a handful of windows desktops you've probably got a windows server.
And if you've already got a bunch of windows desktops plus a windows server, it doesn't really make sense to spin up one *nix box in that environment unless you really need something that only *nix can do. It generally makes more sense to either use the existing windows server, or drop in another one.
unless, of course, you're attempting to run a windows program
I'm not arguing with you that *nix is a better server overall. But windows is a better server if your working with windows apps, serving windows apps, or managing windows desktops.
With 90%+ of the desktop market; most people are running windows applications. They are using remote desktop into servers running windows applications. They are using local windows applications that talk to server windows applications. etc.And the whole thing is managed by another windows server running active directory, and their backups and antivirus and wsus
Windows and Windows servers is not a niche market. And its not going to be a niche market any time soon. Even if its in "decline" relative to the ascension of mobile apps...
If you are dividing by zero in the first place you are doing it wrong. You should never be throwing divide by zero errors; because doing so means you've already failed.
You should know you have a valid denominator before you attempt the division.
In other words, ensure your denominator is non-zero before you divide rather than trapping divide by zero errors, because if your code is asking a computer to divide by zero you've already fucked up somewhere else.
This seems more like an acknowledgement that ios and android are where the majority of new development jobs are right now than anything else.
Does that mean C# or
True but it seems kind of ridiculous to choose an entire platform based on some minor differences in syntax.
Wait, Is that why I should stop using c# and use Java? Or the other way around? I mean, your statement seems to have this built in assumption that java is the default choice; and not-java needs to justify itself relative to java. I really see no reason that Java is the default choice.
If anything Java's big corporate sponsor is worse than Microsoft.
But otherwise, it's a smaller ecosystem with fewer libraries, fewer developers, fewer choices, fewer large scale deployments to learn from.
I dunno; the windows ecosystem is pretty darn big after all. c# bindings for pretty much everything exist. There are definitely lots of situations where Java makes more sense... but its hardly as one sided as you make out.
When you're in public, anyone can recognize you whether it be man or machine. Anyone can take pictures of you and what you are doing. There is no concept of "privacy" when you are in public.
Yes. Actually there is.
If i sit down at a table with a restuarant with my friends,
I fully expect that the restaurant isn't going to record my conversation and put it online, sell it, or do anything else.
I fully anticipate that the waiter or other patrons might overhear snippets of our conversation. But if I catch them actually making a concerted effort to listen and record it; my entire party will be extremely pissed off.
Why? Because there is a concept of "privacy" when you are in public. Nearly all of us in society implicitly abide by this informal social contract.
When an entity moves from incidentally intruding on our privacy in public spaces to methodically and deliberately intruding and we NOTICE it, they get dealt with.
The current trend for corporations is to do it, but to do it beneath our notice. Everything from facial recognition cameras to advertising cookies... all do things that we'd vigorously object to if they weren't beneath our notice. If we noticed google employees snapping our pictures, following us around, taking notes, sitting behind us at our desks writing down everything we searched for and did, they'd be out on their ass in very short order.
But the cameras and the cookies are beneath our notice; that doesn't make it ethical or right. And the legal system SHOULD catch up to our expectations here.
We still have the BSDs that would have gotten some additional attention.
Maybe. Big maybe. I think there is a 'community' to GPL code that the BSDs wouldn't have necessarily "inherited" had GNU/Linux not happened. I think people were attracted to the GPL and its ideology who wouldn't have simply contributed to BSD if that were the only thing.
The BSDs might also have had a lot more trouble gaining commercial / enterprise viability without the GPL GNU/Linux out front. The BSDs are much more susceptible to embrace/extend than GPL is; and the BSDs are at greater risk of becoming closed source / proprietary than GPL code.
I don't doubt that the BSDs would have existed as student projects at universities etc, but I'm not convinced they'd be as strong as they are today without the GPL and GNU/Linux in the world; let alone stronger, which is what you suggest. I contend they might even be weaker.
In fact, ANYBODY who has earned significantly more money than Linus Travois
Once is a typo... but twice needs correcting: Torvalds not Travois.
in the last 20 years by definition has had more economic impact.
I disagree. How on earth do you rationalize defining ones economic impact in terms of simply earning more money than someone?
I define it as how much economic activity can trace back to the actions of that person. And I credit Linus writing Linux and releasing it under the GPL as having absolutely immense impact on business and enterprise around the world. I credit Stallman with that economic impact as well; for releasing the GPL.
Further, I think those events were relatively unique; I don't think that something else equivalent would have happened if Stallman and Torvalds hadn't taken those specific steps. For example, Red Hat doesn't exist; vMware doesn't exist; Xen doesn't exist... Would we have virtualization ? Sure but it might just be proprietary; and it might all belong to IBM and Sun. Assuming windows still happens, and we roll aorund to 2012 with HyperV... that comes out but its competing with IBM and Sun
Meanwhile in another facet... SCO v IBM never happens... that right there is several lifetimes of economic activity for normal people...
He built Linux off the work of people and institutions that came before him or were even contemporaries.
Of course. To a point.
He also wasn't the only one working on a free or low cost "unix like" operating system, but his system is the one that took off.
GPL + Linux is why it took off. I give Stallman a lot credit; perhaps even more than Linus; because the GPL was a big deal. Still Linus chose to use it.
If there had been no Linus and no Linux, there would surely have been something else.
Surely? I'm not remotely convinced. GNU Hurd might have gotten off the ground if it had received the attention that went to linux... or it might still be a toy project in a university somewhere chasing ideological perfection rather than the practical.
It really is the unique blend of ideology + practicality that made GNU/Linux special and I'm not convinced it was inevitably going to happen.
Rick Perry and Chris Christy both managed states with large economies.
But they didn't create those economies. They were there when they got there. All they did was steer them for a few years. If Rick Perry hadn't run and been elected Governor, someone else would have been.
Most of the economy churns along entirely with or without him.
The rest of the elected officials, and the bureaucracy would have been the same. Therefore most of the policy decisions would have mostly been exactly the same if someone else had been elected.
At best Rick Perry put his personal touches on a few things here and there; and has a few signature bits that are - his influence. How much economic impact did those personal touches actually have? They're mostly just little rivulets and swirls on top of the larger economy that moves with its own momentum.
Ditto for Christie
If you're claiming that Torvalds had more economic impact than Greenspan, Bernanke or Yellen, you, quite frankly are out of touch.
You are missing my point.
Torvalds and his Linux project planted a seed that otherwise might not have existed at all. The ramifications of that seed have been felt the world over.
Greenspan et al; someone was going to be sitting in his chair, doing his job. If it wasn't Greenspan, it would have been someone else. If it had been someone else it would have been someone else working within the same political landscape, with the same goals, and they would have been picked and vetted for the job by the same committees. Then they'd be faced with the same problems, and they'd have the same advisory committees working for them; they'd still just one voice on the federal reserve board; the rest of the board would have been the same; they'd be presented the same options, solutions, and recommendations.
Did Greenspan break the mold? Did he go in a bold and unexpected new direction? Or did he simply do, more or less, exactly what any of the other eligible candidate for his job would have done had they been appointed instead?
Would another chairman of the board caught the subprime crisis in 2007 before it hit? I mean there were ample committees, advisers, and other board members... was Greenspan personally a singular force of will in suppressing them all? Or was it a systemic failure of the entire bureaucracy there? I contend it was the latter.
If Greenspan had taken the year off for health reasons, and ets pick someone else... lets say then vice chair Donald Kohn sat as interim chair that year would the actions of the Fed in 2006 been significantly different? Would HE have seen the subprime mortgage collapse sitting as interim chair instead of merely being a board member? What would he have done differently? What makes you think anything at all would have been different?
If it doesn't really matter who is actually sitting in the chair, then how personal impact can you claim they had?
There's no question that the ramifications of the policy actions were felt the world over. BUT how much was *his* policy decisions vs the momentum of committee group-think set against nearly inexorable economic turbulence that reflected not just current policy but the accumulation of decades of momentum? The fact that he was even selected as the chair in itself is a self-fulfilling prophecy -- the fed got exactly exactly the leadership that it selected. Had Greenspan had dramatically different views Greenspan would not have been selected for the chair in the first place.
Most of those are only mildly influential in local (American spheres). Seriously... what did Rick Perry or Chris Christy ever do that affected the lives of people in Turkey? In Austria? In Australia?
Even Bernake, Greenspan, sure some fairly wild gyrations in the stockmarket but its debatable how much impact they personally REALLY had on it; and billions of people are really only tangentially affected by it.
Hell, I'm a home owner in north america with mortgage; and although its surely impacted my mortgage rate and home equity values and it really hasn't affected my life much.
As in would it have been much different with anyone else at the wheel? Or was it, to channel Asimov, predictable by 'pschohistory'; as a likely outcome of a systemic flaw; and all but inevitable; given the banking infrastructure, political landscapes, international dependencies, and so on...
All I ask is a chance to prove that money can't make me happy.
