OK, obviously I exaggerate a bit. Frankly, though, if you can multiply fractions (which I THINK I learned in 6th grade), you can comprehend the chain rule. In my original posting, it should be apparent that the "dt" terms out (or equate to one, if you prefer). Anyone who, on their first day of calculus class, can't be convinced that the chain rule is true should be sent back for remedial arithmetic instruction.
Note that 'apparently true' is not the same as 'rigorously proven true' - but the differential notation works the way you think it should, provided you understand calculus.
you have obviously not examined the "normal" student in detail Actually, I have; I'm a recovering math teacher.
without really being able to add 17/28 + 87/98 Irrelevant; we're not adding fractions (which is hard) - we're multiplying them (which is easy =) The people you discuss rarely see the inside of a calculus classroom.
As a veteran math teacher, I say you are in for an unpleasant surprise. 90% of what they teach you in education courses (math education and otherwise) is crap. Finish your education degree, and get some teaching under your belt, but be prepared to spend your summers getting an engineering degree.
Second, you've committed the atrocious sin of mixing Leibniz notation with Newton prime notation
(whine) But DAD! This is how ALL the cool calculus textbook writers are doing it these days! (/whine)
Leibniz notation is introduced in all the calc textbooks I've read with the STUPID idea that even though it LOOKS like a fraction, you can't TREAT treat it like a fraction. I suppose this is done to preserve rigor, but it is a STUPID pedagogical technique. End rant.
The functional notation f'(x) is useful, as is the differential notation dy/dx. Mixing them gives you the worst of both worlds.
Just to pick a nit - IIRC, Newton used a dot notation, which this screen is too limited to display. Here's an attempt at explanation. If x were a function, then x with a dot over it would be the first derivative, and x with two dots over it(like an umlaut) would be the second derivative. I don't recall where the 'prime' notation came from.
Anyway, you're hardly qualified to school us in calculus. Now, don't be hard on the boy, just because his teacher is everything we've come to expect from the public education system...
Go sue your calculus teacher for malpractice. Seriously. I wrote THE EXACT SAME THING you wrote, but I used differential notation. You clearly haven't been taught differential notation. You likely wasted time memorizing formulas that should be instantly grokked (and are, with differential notation). I'd bet my next paycheck that you can't explain the purpose of the "dx" in an indefinite integral (and no, it's not to show that "x" is the independent variable in the expression being integrated).
Out of curiousity, did you use calculators in your calculus class?
The chain rule??? Why would anyone have a problem with the chain rule? And what does RPN have to do with it? It's an intuitively obvious rule to anyone who has completed sixth grade:
dx/dy = dx/dt * dt/dy
Since you teach calculus, I'll point out that this is a serious question - what's not to understand?
You make a sound point. I would point out, though, that there is also no legal difference between beating someone to death with a baseball bat in a crowded room and quietly dropping some slow-acting poison into their water line.
They end up dead both ways, but in the latter case, it's a lot less obvious that a murder was committed, and it's certainly harder to prove you did anything wrong.
heh. Wil Wheaton (the mainstream celebrity uber-geek) mentioned that one reason he didn't get into drugs (like other child actors of his era) is that his parents were somewhat hippie-ish, and he associated drug use with them. He wasn't going to be as uncool as his parents.
Your septic tank example...what is really happening is that a legal entity independent of the contractor is assuming legal liability for the septic system. The "homeowner" is not in a position to do this, because the "homeowner" can't repair the damage done by a faulty sewage system. On the other hand, an engineering firm with hundreds of thousands or millions in revenues IS in a position to repair that damage, and in certifying a plan, asssumes that liability. They also stand to lose that revenue forever if they certify a poorly designed system.
Yeah, in theory, all that's involvde is knowing which entry in which table in which book, but in practice there is this whole legal liability structure brought into play. Stuff that in the pipes of the guys at work - ask if they're prepared to deal with the federal, state, and local environmental bureaucrats and pay both the costs and fines associated with the cleanup of a faulty septic system, or if they'd rather pay an engineer $200 to put his mark of approval on it.
ummm....what crime? You're saying that he should have violated the law and NOT sought to increase shareholder value? He was employed by the shareholders and was LEGALLY required to do the sort of thing he did. If you don't like it, then take up the issue with the US Congress. I don't like it either, but he not only has done nothing illegal; he has done what was required of him BY LAW.
BTW, the reason rich people buy 5,000 acres is because they don't want people despoiling it. In the middle of it, instead of constructing a visitor center, they construct a home. If your concern is preservation of the natural beauty, I'd suggest that a rich man's estate, owned and administered by one man for his personal benefit, will have less impact on the environment than a public park, owned by everyone, administered by everyone, for the benefit of everyone.
So? The set of algebraic numbers does not include e. And the set of irrational numbers does not include SQRT(2). And the set of integers does not include 1/2. And the set of whole numbers does not include -1. They are all useful numbers; what is your point?
The only point I can infer from your staement is that you believe the Math definition of the word "real" is the same as the English definition of the word real. I assure you that the difference is just as profound as the word "yo" in Spanish and the word "yo" in English street vernacular.
Should you disagree (and many people do) on the basis that there are no "imaginary" measurements in in real life, then I encourage you to show me an example of any number other than integer. Consider volts (but feel free to substitute amps, seconds, inches, etc.)
SQRT(2) volts? Ha. The best you can demonstrate is 1.4142135623730950488016887242097 volts. I can eliminate irrational measurements merely by looking at my measuring device - it shows only approximate rational values. You can't directly measure an irrational.
Transcendentals (ie non-algebraics) also fall prey to approximation by rationals.
Half a volt? No, that's actually 5 (an integer) decivolts. I can eliminate rational numbers by redefining my unit of measure.
-3 amps? No, that's 3 amps in the other direction. I can eliminate negative numbers by changing direction.
Now I'll try to REALLY blow your mind. Assuming you're with me so far and understand that the only numbers we really NEED in the Real World (as opposed to mathematics) are integers, I encourage you to show me an integer.
5 inches? No, that's just a piece of wood with black markings on it. It's not 5.
5 rocks? No, those are just stones. They're not 5.
That symbol on your monitor? No, that's not 5, that's just a collection of faintly glowing phosphors surrounded by brightly glowing phosphors.
That numeral on your paper? No, that's just a thin trail of clay and graphite. It isn't 5.
In short, numbers are an abstraction (and an AMAZING abstraction - the use of abstract numbers separates us from our prehistoric forebears). All numbers are just as "real" or "imaginary" as all other numbers. The only distinction between the various sets is how long it took us to reach a point where they were useful. All known modern and historical cultures (and some animals) used integers; they are a pre-historical phenomenon. All civilizations that left behind mathematical texts that we can translate could solve quadratic equations; fractions and square roots thus also predate known historical texts. The Greeks proved the existence of irrational numbers, but I don't know how useful they were to them. Zero was used by ancient Mesoamerican cultures. Negative numbers entered widespread use in the middle ages. Complex numbers were discovered around the same time but, as far as I know, no use was found for them until the development of the calculus. (Imaginaries were critical to the solution of 3rd degree equations; though the complex roots were not of practical use, they permitted the derivation of the real roots). They are currently (pardon the pun) indispensable in the study of electrical circuits. The hyperreals were researched early in the 20th century and have been shown useful in providing a rigorous basis for calculus without the need for limits. (What practical use they have outside universities, I don't know, but it seems to me that eliminating two weeks of torture for math undergrads is a useful thing).
Enough already. If you're not sorry to have posted here and elicited this response, then congratulations. You are a math geek.
No, i (heh, that looks funny in HTML) really is a number. As you note, it is the number which, when squared and added to one equals What mentally deficient math teacher told you otherwise?
The deisgners of Star Trek do not get biologists, anthropologists and physicists together to theorize what an alien would look like
Ummm....yes, they did. Besides appearance, they considered biology, culture, and physics. Granted, they needed aliens who could be represented by human actors, but that was an unavoidable compromise. (Consider the Horta, though.) What you say may be true of Enterprise, Voyager, Deep Space Nine, or Star Trek: The Next Generation, but it is most definitely NOT true of Star Trek.
I've been a little gun-shy of blogging about Word for fear of being inundated by what are as far as I can tell a gang of "net thugs" who roam the net making outrageous claims about Microsoft and its behavior
Puh-leeze, Chris, you manage a flagship product for one of the richest monopolists in the country, one that has de facto control of the IT market, and you're afraid of emails from 13-year-old kids?
But, trying to rank them alongside legitimate literature seems mighty presumptuous.
I disagree. "Legitimate literature" is an inclusive category. It encompasses works by Homer, Chaucer, Shakespeare, Dickens, Shelley, and Pound: ancient epics, bawdy medieval tales, plays, novels, and poetry long and short.
They all deal in the written or spoken word, but beyond that they diverge greatly. Epics are the written form of an ancient oral tradition. We perform plays. We read novels. Between poems that are pages long and poems whose titles are 20% of their byte count, there is a qualitative, not merely quantitative, difference. Yet all are considered literature.
I could almost grant your point were it not for the inclusion of Shakespeare in the canon. He wrote PLAYS, for crying out loud. If the set of all literature contains works that feature people walking around and talking in front of paintings (sets) while interacting with objects and each other in front of paintings, then how can you possibly exclude somethings as tame as a story that branches and varies based on the viewer's choices?
Granted, it's new and different, but so is the novel. Only time will tell if interactive fiction is a fad or hobby or an enduring cultural tradition. Meanwhile, let's permit a bit of presumption.
Oh, and if a play/sonnet/novel/epic harbors a deep and worthwhile intellectual point, then a "real" author, rather than belaboring the point, will write a 17-syllable poem. In Chinese.
As Jefferson said, the tree of liberty must occasionally be watered with the blood of patriots and tyrants. Go read your history some time; you'll find that the colonies' beef was not with King George, except insofar as he was the figurehead of the government. The democratically elected Parliament was passing the laws that the colonists opposed. Their opposition was largely based on economic issues, but there were some intrusions by the government into private life, as well (see the amendment regarding the quartering of soldiers.)
Funny, that. A democratically elected body, but one that the revolutionaries felt did not represent them. A series of laws designed to increase the wealth of the already wealthy. Intrusions by the government into private life. Does this sound like ANY government you've heard of recently?
The difference between then and now, of course, is that there is no ocean separating the government and the governed, so I predict no revolution in the USA.
Slashdot Mirror
Heh.
OK, obviously I exaggerate a bit. Frankly, though, if you can multiply fractions (which I THINK I learned in 6th grade), you can comprehend the chain rule. In my original posting, it should be apparent that the "dt" terms out (or equate to one, if you prefer). Anyone who, on their first day of calculus class, can't be convinced that the chain rule is true should be sent back for remedial arithmetic instruction.
Note that 'apparently true' is not the same as 'rigorously proven true' - but the differential notation works the way you think it should, provided you understand calculus.
you have obviously not examined the "normal" student in detail
Actually, I have; I'm a recovering math teacher.
without really being able to add 17/28 + 87/98
Irrelevant; we're not adding fractions (which is hard) - we're multiplying them (which is easy =) The people you discuss rarely see the inside of a calculus classroom.
As a veteran math teacher, I say you are in for an unpleasant surprise. 90% of what they teach you in education courses (math education and otherwise) is crap. Finish your education degree, and get some teaching under your belt, but be prepared to spend your summers getting an engineering degree.
Second, you've committed the atrocious sin of mixing Leibniz notation with Newton prime notation
(whine)
But DAD! This is how ALL the cool calculus textbook writers are doing it these days!
(/whine)
Leibniz notation is introduced in all the calc textbooks I've read with the STUPID idea that even though it LOOKS like a fraction, you can't TREAT treat it like a fraction. I suppose this is done to preserve rigor, but it is a STUPID pedagogical technique. End rant.
The functional notation f'(x) is useful, as is the differential notation dy/dx. Mixing them gives you the worst of both worlds.
Just to pick a nit - IIRC, Newton used a dot notation, which this screen is too limited to display. Here's an attempt at explanation. If x were a function, then x with a dot over it would be the first derivative, and x with two dots over it(like an umlaut) would be the second derivative. I don't recall where the 'prime' notation came from.
Anyway, you're hardly qualified to school us in calculus.
Now, don't be hard on the boy, just because his teacher is everything we've come to expect from the public education system...
Go sue your calculus teacher for malpractice. Seriously. I wrote THE EXACT SAME THING you wrote, but I used differential notation. You clearly haven't been taught differential notation. You likely wasted time memorizing formulas that should be instantly grokked (and are, with differential notation). I'd bet my next paycheck that you can't explain the purpose of the "dx" in an indefinite integral (and no, it's not to show that "x" is the independent variable in the expression being integrated).
Out of curiousity, did you use calculators in your calculus class?
The chain rule??? Why would anyone have a problem with the chain rule? And what does RPN have to do with it? It's an intuitively obvious rule to anyone who has completed sixth grade:
dx/dy = dx/dt * dt/dy
Since you teach calculus, I'll point out that this is a serious question - what's not to understand?
"black holes do not exist"
Are you quibbling over the definition of "to exist"? Or do you genuinely believe that light can transit every point in space?
"I see no legal difference"
You make a sound point. I would point out, though, that there is also no legal difference between beating someone to death with a baseball bat in a crowded room and quietly dropping some slow-acting poison into their water line.
They end up dead both ways, but in the latter case, it's a lot less obvious that a murder was committed, and it's certainly harder to prove you did anything wrong.
heh. Wil Wheaton (the mainstream celebrity uber-geek) mentioned that one reason he didn't get into drugs (like other child actors of his era) is that his parents were somewhat hippie-ish, and he associated drug use with them. He wasn't going to be as uncool as his parents.
Druggie parents: My anti-drug.
Ooh, ooh, GREAT point! Excellent debunking of the parent poster's straw man!
(move along, no interesting content here, just meta-meta-meta moderation)
Your septic tank example...what is really happening is that a legal entity independent of the contractor is assuming legal liability for the septic system. The "homeowner" is not in a position to do this, because the "homeowner" can't repair the damage done by a faulty sewage system. On the other hand, an engineering firm with hundreds of thousands or millions in revenues IS in a position to repair that damage, and in certifying a plan, asssumes that liability. They also stand to lose that revenue forever if they certify a poorly designed system.
Yeah, in theory, all that's involvde is knowing which entry in which table in which book, but in practice there is this whole legal liability structure brought into play. Stuff that in the pipes of the guys at work - ask if they're prepared to deal with the federal, state, and local environmental bureaucrats and pay both the costs and fines associated with the cleanup of a faulty septic system, or if they'd rather pay an engineer $200 to put his mark of approval on it.
It was published 4 years after 1984
Why not just say it was published in 1988?
Grinning, ducking, and running...
ummm....what crime? You're saying that he should have violated the law and NOT sought to increase shareholder value? He was employed by the shareholders and was LEGALLY required to do the sort of thing he did. If you don't like it, then take up the issue with the US Congress. I don't like it either, but he not only has done nothing illegal; he has done what was required of him BY LAW.
BTW, the reason rich people buy 5,000 acres is because they don't want people despoiling it. In the middle of it, instead of constructing a visitor center, they construct a home. If your concern is preservation of the natural beauty, I'd suggest that a rich man's estate, owned and administered by one man for his personal benefit, will have less impact on the environment than a public park, owned by everyone, administered by everyone, for the benefit of everyone.
Linda who literally busts her butt to run everything
Really? In the course of administering her father's estate, she's broken her pelvis or torn her gluteals? Wow. That's dedication.
So? The set of algebraic numbers does not include e. And the set of irrational numbers does not include SQRT(2). And the set of integers does not include 1/2. And the set of whole numbers does not include -1. They are all useful numbers; what is your point?
The only point I can infer from your staement is that you believe the Math definition of the word "real" is the same as the English definition of the word real. I assure you that the difference is just as profound as the word "yo" in Spanish and the word "yo" in English street vernacular.
Should you disagree (and many people do) on the basis that there are no "imaginary" measurements in in real life, then I encourage you to show me an example of any number other than integer. Consider volts (but feel free to substitute amps, seconds, inches, etc.)
SQRT(2) volts? Ha. The best you can demonstrate is 1.4142135623730950488016887242097 volts. I can eliminate irrational measurements merely by looking at my measuring device - it shows only approximate rational values. You can't directly measure an irrational.
Transcendentals (ie non-algebraics) also fall prey to approximation by rationals.
Half a volt? No, that's actually 5 (an integer) decivolts. I can eliminate rational numbers by redefining my unit of measure.
-3 amps? No, that's 3 amps in the other direction. I can eliminate negative numbers by changing direction.
Now I'll try to REALLY blow your mind. Assuming you're with me so far and understand that the only numbers we really NEED in the Real World (as opposed to mathematics) are integers, I encourage you to show me an integer.
5 inches? No, that's just a piece of wood with black markings on it. It's not 5.
5 rocks? No, those are just stones. They're not 5.
That symbol on your monitor? No, that's not 5, that's just a collection of faintly glowing phosphors surrounded by brightly glowing phosphors.
That numeral on your paper? No, that's just a thin trail of clay and graphite. It isn't 5.
In short, numbers are an abstraction (and an AMAZING abstraction - the use of abstract numbers separates us from our prehistoric forebears). All numbers are just as "real" or "imaginary" as all other numbers. The only distinction between the various sets is how long it took us to reach a point where they were useful. All known modern and historical cultures (and some animals) used integers; they are a pre-historical phenomenon. All civilizations that left behind mathematical texts that we can translate could solve quadratic equations; fractions and square roots thus also predate known historical texts. The Greeks proved the existence of irrational numbers, but I don't know how useful they were to them. Zero was used by ancient Mesoamerican cultures. Negative numbers entered widespread use in the middle ages. Complex numbers were discovered around the same time but, as far as I know, no use was found for them until the development of the calculus. (Imaginaries were critical to the solution of 3rd degree equations; though the complex roots were not of practical use, they permitted the derivation of the real roots). They are currently (pardon the pun) indispensable in the study of electrical circuits. The hyperreals were researched early in the 20th century and have been shown useful in providing a rigorous basis for calculus without the need for limits. (What practical use they have outside universities, I don't know, but it seems to me that eliminating two weeks of torture for math undergrads is a useful thing).
Enough already. If you're not sorry to have posted here and elicited this response, then congratulations. You are a math geek.
Cheers.
No, i (heh, that looks funny in HTML) really is a number. As you note, it is the number which, when squared and added to one equals What mentally deficient math teacher told you otherwise?
The deisgners of Star Trek do not get biologists, anthropologists and physicists together to theorize what an alien would look like
Ummm....yes, they did. Besides appearance, they considered biology, culture, and physics. Granted, they needed aliens who could be represented by human actors, but that was an unavoidable compromise. (Consider the Horta, though.) What you say may be true of Enterprise, Voyager, Deep Space Nine, or Star Trek: The Next Generation, but it is most definitely NOT true of Star Trek.
where's the incentive to purchase their stock?
It's the "Greater Fool" investment strategy:
No matter how much I pay for this stock, a greater fool than I will pay even more.
The parent poster makes the most important point that I've seen in this discussion.
I've been a little gun-shy of blogging about Word for fear of being inundated by what are as far as I can tell a gang of "net thugs" who roam the net making outrageous claims about Microsoft and its behavior
Puh-leeze, Chris, you manage a flagship product for one of the richest monopolists in the country, one that has de facto control of the IT market, and you're afraid of emails from 13-year-old kids?
Try to at least ACT like a man.
Sure, but how many of them can be carried out from 10,000 miles away by a total stranger armed with nothing more than a keyboard?
And ELF transmitter? Perhaps they spent TOO MUCH time playing D&D on the naval base in Wisconsin...
Heh. Plus one informative.
But, trying to rank them alongside legitimate literature seems mighty presumptuous.
I disagree. "Legitimate literature" is an inclusive category. It encompasses works by Homer, Chaucer, Shakespeare, Dickens, Shelley, and Pound: ancient epics, bawdy medieval tales, plays, novels, and poetry long and short.
They all deal in the written or spoken word, but beyond that they diverge greatly. Epics are the written form of an ancient oral tradition. We perform plays. We read novels. Between poems that are pages long and poems whose titles are 20% of their byte count, there is a qualitative, not merely quantitative, difference. Yet all are considered literature.
I could almost grant your point were it not for the inclusion of Shakespeare in the canon. He wrote PLAYS, for crying out loud. If the set of all literature contains works that feature people walking around and talking in front of paintings (sets) while interacting with objects and each other in front of paintings, then how can you possibly exclude somethings as tame as a story that branches and varies based on the viewer's choices?
Granted, it's new and different, but so is the novel. Only time will tell if interactive fiction is a fad or hobby or an enduring cultural tradition. Meanwhile, let's permit a bit of presumption.
Oh, and if a play/sonnet/novel/epic harbors a deep and worthwhile intellectual point, then a "real" author, rather than belaboring the point, will write a 17-syllable poem. In Chinese.
Yes, they will. All you have to do is to provide a working model.
So your theory is that programs are MORE secure when they have LESS security features?
I suppose that having passwords on user accounts is silly, too, because some rogue program could log keystrokes and post them to the web.
" The only way to fix this situation"
As Jefferson said, the tree of liberty must occasionally be watered with the blood of patriots and tyrants. Go read your history some time; you'll find that the colonies' beef was not with King George, except insofar as he was the figurehead of the government. The democratically elected Parliament was passing the laws that the colonists opposed. Their opposition was largely based on economic issues, but there were some intrusions by the government into private life, as well (see the amendment regarding the quartering of soldiers.)
Funny, that. A democratically elected body, but one that the revolutionaries felt did not represent them. A series of laws designed to increase the wealth of the already wealthy. Intrusions by the government into private life. Does this sound like ANY government you've heard of recently?
The difference between then and now, of course, is that there is no ocean separating the government and the governed, so I predict no revolution in the USA.