I'm afraid this is going to get buried in all the responses about your proof of 0 = 0 x 0, but...
Aside from quantum physics, I like to explain the yearning for "i" as follows: (1) Start with your arguments for why we might want zero. (2) In order solve problems like 6 + z = 4, we want the negative numbers. For example, z might be $-2 if we were doing accounting. (3) Now, we also want to be able to split an orange or a dollar among, say, 5 people. That requires solving 5 * z = 1. Thus, we need fractions, or the rational numbers. (4) We start thinking about other multiplicative equations, such as z * z = 9/4 where we get z=3/2. That's fine, but z * z = 2 has no rational solution. It seems weird we should be able to solve this only for certain values of the RHZ, and we end up getting ourselves to the real numbers. (5) But wait! The real numbers only get us solutions for nonnegative values of the RHS. What if we want to solve z * z = -2 ? Bingo, we need irrational numbers.
So basically, you can think of irrational numbers as being a way of "completing" our usual notion of numbers so as to have solutions to equations that "feel" as though they ought to have them.
That said, I suspect most people consider quantum mechanics a better justification.
IAAMSICA [I am a mathematician specializing in complex analysis]
Bonus Proof
Theorem:
A ham sandwich is better than eternal happiness.
Proof:
- A ham sandwich is better than nothing.
- Nothing is better than eternal happiness
- Therefore, a ham sandwich is better than eternal happiness. QED.
OpenBSD encrypts the swap space by default, specifically to avoid these problems. I would hazard a guess somebody has hacked Linux to do the same, but I haven't seen it.
Of course, if you have so much RAM that you never swap, this is less of an issue.
Plus: a car? No one should have to watch for that on a sidewalk. A parent can easily keep a 3 year old off the street without holding hands or carrying. But somehow I doubt you would know that.
Anyone should have the reasonable expectation that anything moving on the sidewalk is capable of avoiding injurious collisions with a human of whatever size. For example, someone engrossed in a map should not be walking at a brisk pace.
Segways and bicycles operated at medium+ speeds or by unskilled riders belong on the street, where they shoulder more of their own risk.
This perpetrator sounds like the classic 20something gadget-phile guy who knows nothing about children. He blames a 3-year-old for suddenly changing direction??!!
Of course, SF is -- for various reasons like expense and convenience -- relatively empty of children, so the lack of experience at least is predictable. I suppose there are a lot of equivalents to this guy around.
Actually, since I use the lakefront path for transportation I am kind of dreading these. Bike Chicago already rent sociables (side-by-side pedal powered vehicles) for the non-cyclists along the lakefront, and of course lack the rear-end-toughness issue. Tourists wandering all over the path and taking up several feet of space in these things are a real problem.
The Segways will not be as wide, but I expect that the clueless tourists using them will not ride single file, but rather wander all over, risking collisions with the rest of us. I would prefer they stay on the Grant Park and Museum Campus sidewalks.
It's odd -- the last Chicago/Segway news I recall hearing was that the city was not going to allow these things. That was way back when Kamen had the political machine running to convince state legislatures that these devices were sidewalk-aapropriate. Which, if the governors are set to 5mph, I concede they probably are. At 12mph I think they should be on the street.
I would not characterize it as "a lot". I ride my bicycle on city streets and the lakefront daily, and though I see bicycle cops all the time I have yet to see a Segway.
A buggy codec can't take down every program, even if it does not use QuickTime.
True, though from personal experience I can tell you that a buggy codec can make your machine pretty much unusable. There was a version of DivX that was buggy on dual-processor G4's after a QuickTime upgrade. This made it completely impossible to run Finder, and hence to launch any other apps.
That sucked. Thank goodness for a second machine and a running ssh daemon. I finally got backtraces and figured out the problem.
Though all the speculation about US backwardness and hostility toward foreign systems is certainly entertaining, there's a benign historical explanation that I find compelling. It goes something like this:
Recall that industrial mass production is essentially a 20th century invention, and that by the 1940's it still had not really spread beyond the U.S. and Europe. In World War II, most European industrial capacity was destroyed at one point or another, providing a clean slate to rethink standards for every industry, and to adopt logical standards with no switchover cost.
After WWII, Europe wisely went to the metric system. Developing countries wisely adopted it as well. But the U.S., with its factories intact (and now back to making cars and vacuum cleaners) was saddled (and remains cursed with) with tremendous switching costs. The expense in lost customers and supplier confusion is too great for a company in most industries to unilaterally change. And agreements to change all at once are very hard to achieve.
Empirical evidence:
Newer US major industries (e.g. semiconductors) usually work in metric
(As noted elsewhere) US science is in metric; because switchover costs are lower scientists could switch almost right away.
Well-meaning attempts to effect a switch have been ignored by industry (because of the cost)
US industries with a big international component are often metric (bicycle manufacture)
I suppose the conclusion to draw is that the US is unlikely to switch until either something destroys its industrial factories, or the "old" unswitched industries become so dwarfed by new metric ones that it is actually cheaper for them to change.
Reminds me of the college kids who like to play with radiation warning labels
My father-in-law is an M.D at a local hospital. He brings home biohazard ziploc bags for my mother-in-law to use. She often sends us home with tasty leftovers in those bags. They tend to freak out our houseguests.
Can I suggest Tomato Torrent instead? Excellent program. Caveat: the author says it's open source, but the links are broken and appear to be out of date anyhow.
I'm not terribly knowledgeable about linguistics, but I do know that Korean is written in a bona fide alphabet that separates vowel and consonant symbols (contrast to the two syllabaries {katakana and hiragana} used by Japanese). It's invention is attributed to a king many hundreds of years ago.
Academic Korean writing (I've been told) still uses lots of Chinese characters. I get the impression it's a style/showoff thing.
Like English, with germanic-derived and latin-derived vocabularies combined ["hate" versus "detest"], Korean has "native" Korean words and chinese derived words.
You are considered an eloquent speaker if you use a lot of "chinese character words" -- words derived from the Chinese that in previous decades might have been written with the Chinese characters, but these days are probably spelled out like almost everything else. This is somewhat analogous to English, where the sentences comprised of latin-derived words are usully considered more erudite.
They can get away with such heavy restrictions because you are in high school (minors for students, little or no faculty research, etc.). A college would have quite a bit more trouble, methinks.
For those who want a quick excerpt of the science:
Gravitational microlensing uses a distant star (or other massive object) to bend light the same way as a lens would. If that star is perfectly aligned with an even more distant star (from our perspective) then the lens will call the more distant star to brighten, at least for as long as the alignment lasts.
The brightening comes with a spike (from "caustics" which are like irregularities in the lens), as the alignment gets good and them bad again. If you see a second, smaller spike, or an unusual extra image, that's evidence of a planet.
I'm not sure how you distinguish planets from weird caustics.
Note: this technique is good for detecting planets with long-period orbits, whereas the doppler-shift techniques are lousy for that, because they only work if the planet's revolution period is small (like in days).
I love the magnets, too. I also used to use the platter as cool-looking drink coasters. But it eventually occured to me that I had no idea how hazardous (or not) the platter materials were w.r.t human ingestion.
I checked EPA sites to see if there were special disposal rules for hard drives, and didn't really see anything. But I gave up the coasters anyway.
Does anybody have positive information that they are safe?
I've since entered the USA on a K-1 fiancee visa, and ultimately it's easier to relax when your residency depends on your marriage being successful and not your tech company
Indeed. Think of how much you'll expand your skillset... at giving head!
If you can evacuate the tunnel, and run maglev, then if you dig your tunnel correctly you would need no extra power to make it run: with a gentle slope going down and then back up you could let the train run under the influence of gravity, with zero departure speed and zero arrival speed. I wonder how long the trip would take.
Well, if we ignore friction then you don't actually have to worry about how you dig your tunnel, so long as the gravitational potential (roughly: height) of your destination matches that of your departure.
Ignoring the rotation of the earth and nonuniformity of its gravitational field, your trip would be fastest if the path were cycloidal, as this solves the brachistochrone problem. In this simple view, the problem is basically just finding a satellite orbit connecting the two points.
Accounting for the rotation of the earth is probably just a matter of setting up your path so that the earth rotates "just right" underneath it. However, the gravitational field will change quite a bit on your ideal path, since (roughly speaking) only those parts of the earth closer to the center than you are have any gravitational effect. So you've got to solve the brachistochrone in a varying gravitational field.
I think you'd be hard-pressed to find some software or technology works on Linux but doesn't work on Mac OS X
Here's one: try setting up an OSX box as an OpenAFS server (plus a Kerboros authentication server for extra points) for your local network, from scratch. As far as Google and I can tell, no one has ever done it.
I eventually went with Debian for this, and even that was hard enough. I greatly prefer OSX though, so I never touch the Debian box except for OpenAFS and setting up my LEAF router.
Actually, it appears you misunderstand what StillNeedMoreCoffee was saying, perhaps because you don't quite understand Fourier series and transforms.
Although engineers commonly think of a Fourier transform (or series) as a switch into the "frequency domain", the mathematics of that switch is intimately related to circles.
In particular, we take any function (e.g. planetary positions) and decompose it into a sereis of sines and cosines (or exp(ikx) terms if you like). Note that those sines and cosines are side lengths of triangles inscribed in -- you guessed it -- imaginary circles of varying radii.
Voila...the grandparent is insightful. I'm not sure I agree with the implication that Ptolemy discovered any general principle of decomposition of arbitrary functions by basis functions rather than an ad hoc kludge. But hey, it's an interesting perspective.
I'm a big PostgreSQL fan, but when it came time to do a big project, I ended up with SQLite (and I'm very happy I did).
Basically, the problem was to take a bunch of huge text files, parse them into a set of inputs for a big simulator written in ANSI C, run the simulation, save the results, and postprocess them.
I wrote the parsing in a mixture of Python and SQL (using PySQLite), had the simulator write the results back out the the SQLite database (which was amazingly easy and bug-free), and then did postprocessing again in Python and SQL.
It was WAY easier than my previous attempts to do similar things using embedded Python and/or PostgreSQL. I still love PostgreSQL, but not for this project.
Not true. I teach at the Illinois Institute of Technology. Most of my students are from the subcontinent, and have even less money to spend than the typical American college student.
I'm well aware of that. I teach from my own book-in-progress, which they get in the form of free photocopies.
I must admit, though, to being tempted to start charging for any student whose cellphone goes off in class!
I posted about this on the Apple forums, but it's worth repeating here, perhaps in a little more detail. I'm running a Dual G4/800, so it looks to be a DP problem.
If you examine the stack traces that the crash catcher asks to send back to Apple you will see that all the non-working iApps (iTunes, iPhoto, Safari, Mozilla, etc.) crash at initQuickTimeFoo()+44. Sorry I don't recall the exact function.
That is consistent with the "fix" that I found...namely to grab/System/Library/Frameworks/QuickTime.framework off a machine that had not yet upgraded from QT6.4, and replace the one on the Dual G4 with it. interestingly even iTunes 4.2 appears to still be working fine with the older QT. I'm sure there's a crash lurking somewhere though.
Obviously if one has already done an Archive and Reinstall, this problem should be avoidable just by never upgrading QT to 6.5 until Apple fixes this.
For good measure I also replaced some other QT components, like the plugins, but I'm not certain that they mattered. They were
I'm not sure what makes me special, but this went very badly for me. After the install, no app the loads the QT framework would start -- they all crashed at Init()+44.
That meant no Finder, no iPhoto, no iTunes, no Safari, no Mozilla, etc, etc. They all crashed on startup.
I solved the problem by rolling my own downgrade -- I found a machine that still had 6.4 and made a tarball of/System/Library/Frameworks/QuickTime
which I then installed and rebooted. So far everything seems OK.
Well, in the UK (if my understanding based on the Slashdot analysis is correct), they can claim you have some encrypted data in that jpeg -- even if you don't -- and jail you until you give up the keys.
Slashdot Mirror
I'm afraid this is going to get buried in all the responses about your proof of 0 = 0 x 0, but...
Aside from quantum physics, I like to explain the yearning for "i" as follows:
(1) Start with your arguments for why we might want zero.
(2) In order solve problems like 6 + z = 4, we want the negative numbers. For example, z might be $-2 if we were doing accounting.
(3) Now, we also want to be able to split an orange or a dollar among, say, 5 people. That requires solving 5 * z = 1. Thus, we need fractions, or the rational numbers.
(4) We start thinking about other multiplicative equations, such as z * z = 9/4 where we get z=3/2. That's fine, but z * z = 2 has no rational solution. It seems weird we should be able to solve this only for certain values of the RHZ, and we end up getting ourselves to the real numbers.
(5) But wait! The real numbers only get us solutions for nonnegative values of the RHS. What if we want to solve z * z = -2 ? Bingo, we need irrational numbers.
So basically, you can think of irrational numbers as being a way of "completing" our usual notion of numbers so as to have solutions to equations that "feel" as though they ought to have them.
That said, I suspect most people consider quantum mechanics a better justification.
IAAMSICA [I am a mathematician specializing in complex analysis]
Bonus Proof
Theorem:
A ham sandwich is better than eternal happiness.
Proof:
- A ham sandwich is better than nothing.
- Nothing is better than eternal happiness
- Therefore, a ham sandwich is better than eternal happiness. QED.
OpenBSD encrypts the swap space by default, specifically to avoid these problems. I would hazard a guess somebody has hacked Linux to do the same, but I haven't seen it.
Of course, if you have so much RAM that you never swap, this is less of an issue.
Spoken like a true childless objectivist.
Plus: a car? No one should have to watch for that on a sidewalk. A parent can easily keep a 3 year old off the street without holding hands or carrying. But somehow I doubt you would know that.
Anyone should have the reasonable expectation that anything moving on the sidewalk is capable of avoiding injurious collisions with a human of whatever size. For example, someone engrossed in a map should not be walking at a brisk pace.
Segways and bicycles operated at medium+ speeds or by unskilled riders belong on the street, where they shoulder more of their own risk.
This perpetrator sounds like the classic 20something gadget-phile guy who knows nothing about children. He blames a 3-year-old for suddenly changing direction??!!
Of course, SF is -- for various reasons like expense and convenience -- relatively empty of children, so the lack of experience at least is predictable. I suppose there are a lot of equivalents to this guy around.
Actually, since I use the lakefront path for transportation I am kind of dreading these. Bike Chicago already rent sociables (side-by-side pedal powered vehicles) for the non-cyclists along the lakefront, and of course lack the rear-end-toughness issue. Tourists wandering all over the path and taking up several feet of space in these things are a real problem.
The Segways will not be as wide, but I expect that the clueless tourists using them will not ride single file, but rather wander all over, risking collisions with the rest of us. I would prefer they stay on the Grant Park and Museum Campus sidewalks.
It's odd -- the last Chicago/Segway news I recall hearing was that the city was not going to allow these things. That was way back when Kamen had the political machine running to convince state legislatures that these devices were sidewalk-aapropriate. Which, if the governors are set to 5mph, I concede they probably are. At 12mph I think they should be on the street.
I would not characterize it as "a lot". I ride my bicycle on city streets and the lakefront daily, and though I see bicycle cops all the time I have yet to see a Segway.
A buggy codec can't take down every program, even if it does not use QuickTime.
True, though from personal experience I can tell you that a buggy codec can make your machine pretty much unusable. There was a version of DivX that was buggy on dual-processor G4's after a QuickTime upgrade. This made it completely impossible to run Finder, and hence to launch any other apps.
That sucked. Thank goodness for a second machine and a running ssh daemon. I finally got backtraces and figured out the problem.
Recall that industrial mass production is essentially a 20th century invention, and that by the 1940's it still had not really spread beyond the U.S. and Europe. In World War II, most European industrial capacity was destroyed at one point or another, providing a clean slate to rethink standards for every industry, and to adopt logical standards with no switchover cost.
After WWII, Europe wisely went to the metric system. Developing countries wisely adopted it as well. But the U.S., with its factories intact (and now back to making cars and vacuum cleaners) was saddled (and remains cursed with) with tremendous switching costs. The expense in lost customers and supplier confusion is too great for a company in most industries to unilaterally change. And agreements to change all at once are very hard to achieve.
Empirical evidence:
Newer US major industries (e.g. semiconductors) usually work in metric
(As noted elsewhere) US science is in metric; because switchover costs are lower scientists could switch almost right away.
Well-meaning attempts to effect a switch have been ignored by industry (because of the cost)
US industries with a big international component are often metric (bicycle manufacture)
I suppose the conclusion to draw is that the US is unlikely to switch until either something destroys its industrial factories, or the "old" unswitched industries become so dwarfed by new metric ones that it is actually cheaper for them to change.
Reminds me of the college kids who like to play with radiation warning labels
My father-in-law is an M.D at a local hospital. He brings home biohazard ziploc bags for my mother-in-law to use. She often sends us home with tasty leftovers in those bags. They tend to freak out our houseguests.
Can I suggest Tomato Torrent instead? Excellent program. Caveat: the author says it's open source, but the links are broken and appear to be out of date anyhow.
I'm not terribly knowledgeable about linguistics, but I do know that Korean is written in a bona fide alphabet that separates vowel and consonant symbols (contrast to the two syllabaries {katakana and hiragana} used by Japanese). It's invention is attributed to a king many hundreds of years ago.
Academic Korean writing (I've been told) still uses lots of Chinese characters. I get the impression it's a style/showoff thing.
Like English, with germanic-derived and latin-derived vocabularies combined ["hate" versus "detest"], Korean has "native" Korean words and chinese derived words.
You are considered an eloquent speaker if you use a lot of "chinese character words" -- words derived from the Chinese that in previous decades might have been written with the Chinese characters, but these days are probably spelled out like almost everything else. This is somewhat analogous to English, where the sentences comprised of latin-derived words are usully considered more erudite.
They can get away with such heavy restrictions because you are in high school (minors for students, little or no faculty research, etc.). A college would have quite a bit more trouble, methinks.
Enjoy college this fall!
For those who want a quick excerpt of the science:
Gravitational microlensing uses a distant star (or other massive object) to bend light the same way as a lens would. If that star is perfectly aligned with an even more distant star (from our perspective) then the lens will call the more distant star to brighten, at least for as long as the alignment lasts.
The brightening comes with a spike (from "caustics" which are like irregularities in the lens), as the alignment gets good and them bad again. If you see a second, smaller spike, or an unusual extra image, that's evidence of a planet.
I'm not sure how you distinguish planets from weird caustics.
Note: this technique is good for detecting planets with long-period orbits, whereas the doppler-shift techniques are lousy for that, because they only work if the planet's revolution period is small (like in days).
I love the magnets, too. I also used to use the platter as cool-looking drink coasters. But it eventually occured to me that I had no idea how hazardous (or not) the platter materials were w.r.t human ingestion.
I checked EPA sites to see if there were special disposal rules for hard drives, and didn't really see anything. But I gave up the coasters anyway.
Does anybody have positive information that they are safe?
I've since entered the USA on a K-1 fiancee visa, and ultimately it's easier to relax when your residency depends on your marriage being successful and not your tech company
... at giving head!
Indeed. Think of how much you'll expand your skillset
If you can evacuate the tunnel, and run maglev, then if you dig your tunnel correctly you would need no extra power to make it run: with a gentle slope going down and then back up you could let the train run under the influence of gravity, with zero departure speed and zero arrival speed. I wonder how long the trip would take.
Well, if we ignore friction then you don't actually have to worry about how you dig your tunnel, so long as the gravitational potential (roughly: height) of your destination matches that of your departure.
Ignoring the rotation of the earth and nonuniformity of its gravitational field, your trip would be fastest if the path were cycloidal, as this solves the brachistochrone problem. In this simple view, the problem is basically just finding a satellite orbit connecting the two points.
Accounting for the rotation of the earth is probably just a matter of setting up your path so that the earth rotates "just right" underneath it. However, the gravitational field will change quite a bit on your ideal path, since (roughly speaking) only those parts of the earth closer to the center than you are have any gravitational effect. So you've got to solve the brachistochrone in a varying gravitational field.
Maybe I'll solve that one sometime.
I think you'd be hard-pressed to find some software or technology works on Linux but doesn't work on Mac OS X
Here's one: try setting up an OSX box as an OpenAFS server (plus a Kerboros authentication server for extra points) for your local network, from scratch. As far as Google and I can tell, no one has ever done it.
I eventually went with Debian for this, and even that was hard enough. I greatly prefer OSX though, so I never touch the Debian box except for OpenAFS and setting up my LEAF router.
Actually, it appears you misunderstand what StillNeedMoreCoffee was saying, perhaps because you don't quite understand Fourier series and transforms.
Although engineers commonly think of a Fourier transform (or series) as a switch into the "frequency domain", the mathematics of that switch is intimately related to circles.
In particular, we take any function (e.g. planetary positions) and decompose it into a sereis of sines and cosines (or exp(ikx) terms if you like). Note that those sines and cosines are side lengths of triangles inscribed in -- you guessed it -- imaginary circles of varying radii.
Voila...the grandparent is insightful. I'm not sure I agree with the implication that Ptolemy discovered any general principle of decomposition of arbitrary functions by basis functions rather than an ad hoc kludge. But hey, it's an interesting perspective.
I'm a big PostgreSQL fan, but when it came time to do a big project, I ended up with SQLite (and I'm very happy I did).
Basically, the problem was to take a bunch of huge text files, parse them into a set of inputs for a big simulator written in ANSI C, run the simulation, save the results, and postprocess them.
I wrote the parsing in a mixture of Python and SQL (using PySQLite), had the simulator write the results back out the the SQLite database (which was amazingly easy and bug-free), and then did postprocessing again in Python and SQL.
It was WAY easier than my previous attempts to do similar things using embedded Python and/or PostgreSQL. I still love PostgreSQL, but not for this project.
Not true. I teach at the Illinois Institute of Technology. Most of my students are from the subcontinent, and have even less money to spend than the typical American college student.
I'm well aware of that. I teach from my own book-in-progress, which they get in the form of free photocopies.
I must admit, though, to being tempted to start charging for any student whose cellphone goes off in class!
I would certainly love to swap this for the power supply in my Dual G4/800. Does anyone know how likely that is?
Dan et al.
/System/Library/Frameworks/QuickTime.framework off a machine that had not yet upgraded from QT6.4, and replace the one on the Dual G4 with it. interestingly even iTunes 4.2 appears to still be working fine with the older QT. I'm sure there's a crash lurking somewhere though.
/Library/Internet Plug-Ins
/System/Library/QuickTime*
/Library/QuickTime
I posted about this on the Apple forums, but it's worth repeating here, perhaps in a little more detail. I'm running a Dual G4/800, so it looks to be a DP problem.
If you examine the stack traces that the crash catcher asks to send back to Apple you will see that all the non-working iApps (iTunes, iPhoto, Safari, Mozilla, etc.) crash at initQuickTimeFoo()+44. Sorry I don't recall the exact function.
That is consistent with the "fix" that I found...namely to grab
Obviously if one has already done an Archive and Reinstall, this problem should be avoidable just by never upgrading QT to 6.5 until Apple fixes this.
For good measure I also replaced some other QT components, like the plugins, but I'm not certain that they mattered. They were
- Brian K Boonstra
I'm not sure what makes me special, but this went very badly for me. After the install, no app the loads the QT framework would start -- they all crashed at Init()+44.
/System/Library/Frameworks/QuickTime
That meant no Finder, no iPhoto, no iTunes, no Safari, no Mozilla, etc, etc. They all crashed on startup.
I solved the problem by rolling my own downgrade -- I found a machine that still had 6.4 and made a tarball of
which I then installed and rebooted. So far everything seems OK.
Best...comment...ever!
Well, in the UK (if my understanding based on the Slashdot analysis is correct), they can claim you have some encrypted data in that jpeg -- even if you don't -- and jail you until you give up the keys.
Ugly.