There definitely would be multiple representations of numbers, allowing infinite decimal places and arbitrary integer "digits". You could write pi as 10 or as 3.(stuff that never ends), which is presumably what you meant to type. You could perhaps limit your decimal places to some set of integers, but that seems pretty artificial. Though, perhaps {0, 1} would be sufficient.... Nope. Even 0.111111... would max out at 1/(1-(1/pi)) ~= 1.47 pi. However, 3 would seem to work from an analogous analysis.
you still have to do an amount of work proportional to 60 trillion
Actually, Wikipedia lists the runtime of computing the nth digit of pi as O((n log n)^3) using the original BBP algorithm. Apparently this has been improved O(n^2). I'm not sure what the runtime of algorithms which compute the first n digits of pi are.
I doubt it, though I do like that series. I can't quite remember generalizations of it, though.... That is, what is zeta(m) for m a positive integer? Wikipedia lists a formula valid at positive even integers here, wherein \zeta(2n) = \frac{(-1)^{n+1} B_{2n}(2 \pi)^{2n}}{2(2n)!}, so for instance 1 + 1/2^4 + 1/3^4 +... = pi^4 / 90, and in general 1 + 1/2^(2n) + 1/3^(2n) +... converges to r*pi^(2n) for some rational number r that can be found quite easily. I presume Euler's method (the one which first proved your identity) can be extended to prove this case. I don't remember a characterization for the odd integers though.... Isn't it wonderful that the formula I listed uses \tau = 2 \pi? I'm convinced; I don't recall seeing any good evidence for using \pi over \tau other than for historical reasons. I certainly use it in my own work. The Apery's constant (that is, \zeta(3)) Wikipedia page lists some formulas, but nothing even remotely as elegant as the above even integer characterization. I suppose there isn't one currently known:(.
Well there's a couple of BBP-type formulas for pi^2 that MathWorld lists here that might even convert to spigot algorithms (I'm not in the mood to check whether or not the usual BBP-type pi formula's spigot algorithm applies to these cases, where there's a fraction out front). Hmm, that's very jargon filled. What I meant is that it may be possible to compute arbitrary digits of pi^2 without computing the previous digits. This is indeed possible with pi. The thing here, though, is that all of the first 40 trillion digits were computed. TFA is just written by someone not steeped in math enough to know to be careful about equating such computations. The natural log of 2, for instance, admits a very simple spigot algorithm through manipulation of its basic Maclaurin series, as Wikipedia will tell you.
In part I meant it was strange that the same figure was given in three different units with wildly different precision each time--that suggested someone might have rounded arbitrarily to the nearest mile after a conversion. I also wondered if mile-scale figures were even possible to measure, but the magnitude of the Pioneer anomaly is a very good point, and suggests that mile-scale measurements are possible. (I've never heard of a "Voyager anomaly". A brief search for the phrase only revealed a few forum posts by, well, idiots, and an apparent typo where "Pioneer anomaly" was meant.)
Yes, our smallness is astonishing. The relative difference in scale going from pencils to atoms is dwarfed by the relative difference in scale going from light-years to pencils. Very roughly, going from the size of a galaxy to the size of a person is comparable to going from the size of a person to the atomic scale twice. On the scale of galaxies, each person is an atom's atom.
The source of the 40,000 figure is here, which is at least hosted on jpl.nasa.gov. I was hoping an author careful enough to include star movement was careful enough to include whatever other relevant effects may exist. You mentioned momentum transfer. Off the top of my head, there may also be electrostatic effects, gravitational fields, or a particle field drifting in some direction which may or may not modify the calculation significantly.
Relevant Wikipedia text for Voyager 1, the farther and faster of the two (taken from here):
As of April 21, 2011, Voyager 1 was about 116.825 AU, or about 10,843,294,886 miles or about 0.00183 of a light-year from the Sun. [...] Voyager 1's current relative velocity to the sun is 17.061 km/s, or 61,452 kilometres per hour (38,185 mph). This calculates as 3.599 AU per year, about 10% faster than Voyager 2. At this velocity, 73,600 years would pass before reaching the nearest star, Proxima Centauri, were the spacecraft traveling in the direction of that star. [...]
Voyager 1 is not heading towards any particular star, but in about 40,000 years it will pass within 1.6 light years of the star AC+79 3888 in the constellation Camelopardalis. That star is generally moving towards our Solar System at about 119 kilometers per second.
You might want to take that with a grain of salt, though, since there's an incorrect calculation in part of the paragraph I didn't quote and 11 significant figures is very suspiciously precise in the miles figure. Also, the 73,600 figure doesn't agree with my own calculations in the hundreds place. But I imagine what I quoted is pretty close to the truth.
Not in quantum mechanics.... At least, not in the sense you mean. There an expectation value for, say, energy might be conserved, but it's not really the same thing. That is, your post seems to take the classical view that things "are as they are, as measured by the following variables that take on definite values: mass, energy, momentum,..." modified by relativity saying "and they might turn into energy and geometry is weird so you really need to measure those things carefully, keeping track of your reference frame and the curvature of spacetime". That's basically a hidden variable theory which cannot explain quantum effects as proven by the Bell inequalities.
In a nutshell--quantum is well and truly screwy. The only definite conclusions are probabilistic ones. It only seems like you have a definite momentum (mass*velocity) because the quantum probability curve for your momentum means you'll only notice fluctuations from repeated measurements after many decimal points. Similarly your position is in a sense constantly fluctuating minutely, giving you a wavelength just like the light hitting your eyes has. (Of course I'm vastly oversimplifying. If your really interested go find a textbook. Wikipedia articles somehow don't do justice to topics like these.)
School districts certainly vary hugely. My high school, for instance, had a 30-40% dropout rate. To their credit, they actually managed to do a decent job of giving lots of AP classes to advanced students, with generally good--sometimes great--teachers. Compared to some of my college friends who went to prep schools, though, it was a joke. I remember having to teach my chemistry teacher logarithms after she tried to teach us (incorrect) voodoo on related rates. My strong dislike for chemistry is probably partly due to a couple of awful high school chemistry teachers.
I looked at the Titan test and wondered if the OEIS were allowable, since it's an online encyclopedia. I threw the integer sequences into it and it had all of them. Two of those integer sequences and the other non-integer sequence pretty much require remembering random math facts, just like the first verbal analogy. A trained mathematician would do quite well on the math section as well, in part because of their training. I agree with the GP that IQ tests do not have the level of resolution to totally order people on intelligence.
Presumably, Erno Rubik was the first to solve it. He had real difficulty unscrambling it, and seems to have worked out the usual algorithmic approach after around a month of trying:
He twisted and twisted, and the colors only got more scrambled. It was like ''staring at a piece of writing written in a secret code. But for me, it was a code I myself had invented! Yet I could not read it. This was such an extraordinary situation that I simply could not accept it.''
Rubik was in even worse shape than his disciples. He didn't knowif the problem could be solved. Perhaps there was only one sure way to get back to the start: by exactly retracing every step he had taken. Rubik couldn't hope to do that. Randomly twisting the Cube would eventually produce the ordered state, but he suspected that the laws of probability were against this occurring in his lifetime. (It has since been calculated that if every person on earth randomly twisted a Cube once every second, about once every three centuries one Cube would return to its original state.)
Rubik had only an intuition that there must be a method. He started out by aligning the eight corner cubies correctly, and he discovered certain sequences of moves for rearranging just a few cubies at a time. One sequence of four twists, for instance, would temporarily scramble the cube, exchange the positions of three cubies, and then restore the rest of the Cube to its previous state. Other sequences took twelve twists--with chaotic results if he lost track of what he was doing halfway through. But Rubik persevered in his room for more than a month and emerged in the summer to show his mother a pristine Cube.
Taken from here. People who do it blind folded most likely memorize the algorithms and apply them to an accurate mental model while their hands move the physical version. I once knew someone who could do it blindfolded, though I never asked how. I always assumed like everyone I've ever known who could do it that he used the algorithmic approach. Do you have examples of people who don't use this approach, or were you just guessing?
Anyone can learn something if they really want to.
I have to disagree. For instance, my mother wouldn't be able to master the classification of finite simple groups before she dies (roughly speaking, that's thousands of pages of advanced math). If you're instead talking about some sort of idealized human who lives forever but isn't ravaged by age, then maybe I'd agree. In any case your point was that there's a correlation between speed of learning and intelligence, and that many people could understand many, many things if they were willing to put in possibly enormous amounts of time, which is certainly true.
It seems like TFA's author might have made the same mistake, or their wording is extremely poor. They say
The software does this by breaking a file to be hidden into a number of fragments and placing the individual pieces in clusters scattered around the hard drive. [...] The method that Khan and his colleagues developed avoids this problem by hiding small pieces of a sensitive file various random places on a hard drive. [...] as the sensitive files are not actually hidden but rather dispersed in pieces.
The file is broken into bits and placed in the arrangement of clusters--these bits are not literally written to the hard drive.
I believe he did earth-moon calculations, and universal gravitation predicted an otherwise unobserved planet (Neptune?). Maybe Newton, in his calculations involving the sun, did always assume the sun was so much more massive than other planets that it was alright to assume it was at the center of mass of the solar system. That's a practical consideration that's different from what I understand you were saying in your first post, which was that Newton did not even theorize universal gravitation as applying to the solar system, instead fixing the sun at the center for some reason and making it necessarily immune to other body's gravitational pulls.
When I was taught special relativity, it was built up pretty much axiomatically. "Warped" geometry ended up falling out quite naturally from first principles (speed of light constant in all inertial reference frames, principle of relativity, continuous space, etc.). If they had instead just told me "apply this length contraction to space" I would have thought "well, that's completely random. Why would I do that?" Once in a while the evolutionary growth of science/math isn't the best pedagogically. A great example is quaternions, which were invented before vector analysis.
I still don't know what you mean about "other" SR/GR/QM geometries. Putting Newtonian mechanics on other geometries is clever, though.
That's not the same situation as an app or Apple having access to location history. It would single me out from the other millions of customers. If the iPhone were used by, say, 20 people and I were one of them, then I would care.
I don't care if Apple knows where I've been, or if other apps can read this log. Is there some non-paranoid reason I should? Out of millions of customers, is Apple really going to care whatsoever if I went to the bank and then rented some porn? If some unscrupulous government agency wants your current location, triangulation should work on any cell phone anyway, as would logging GPS data.
If I really were scared of the government knowing where I've been, I wouldn't trust Apple's good-naturedness to protect me anyway--that would be moronic. TFA suggests a private detective could catch a cheating spouse with this location data, but that's pretty far over on the paranoid spectrum. So again, why should I care, given that I'm neither cheating nor afraid of the government knowing where I've been?
What do you mean by the "addition law of vectors"? I've also never heard that Newton conceived universal gravitation as not necessarily universal and only applying in some limits. Newton actually said of Hooke's hypotheses on gravity that "without my Demonstrations, to which Mr Hook is yet a stranger, it cannot be beleived by a judicious Philosopher to be any where accurate"--Hooke's claims weren't sufficiently tested, either physically or mathematically, though Newton put his own similar/equivalent theory through its paces as best he could at the time. Newton did not confine himself to objects with significantly different masses, as you suggest: "I deduced that the forces which keep the planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve...". It's very common to consider Newton's laws as they apply to a set of objects of roughly the same mass. See the 2-, 3-, or n-body problems.
I don't know what you mean when you talk about GR/SR geometries. SR is flat Minkowski space, and GR uses a pseudo-Riemannian manifold with metric given by the Einstein equations. One might be able to apply Newtonian gravity on, say, a sphere, with geodesic distance replacing regular distance in gravity computations. That seems a bit strange, though; I've never heard of it applied to anything besides standard Euclidean space.
Slashdot Mirror
There definitely would be multiple representations of numbers, allowing infinite decimal places and arbitrary integer "digits". You could write pi as 10 or as 3.(stuff that never ends), which is presumably what you meant to type. You could perhaps limit your decimal places to some set of integers, but that seems pretty artificial. Though, perhaps {0, 1} would be sufficient.... Nope. Even 0.111111... would max out at 1/(1-(1/pi)) ~= 1.47 pi. However, 3 would seem to work from an analogous analysis.
you still have to do an amount of work proportional to 60 trillion
Actually, Wikipedia lists the runtime of computing the nth digit of pi as O((n log n)^3) using the original BBP algorithm. Apparently this has been improved O(n^2). I'm not sure what the runtime of algorithms which compute the first n digits of pi are.
Perhaps you shouldn't tell her about tau....
I doubt it, though I do like that series. I can't quite remember generalizations of it, though.... That is, what is zeta(m) for m a positive integer? Wikipedia lists a formula valid at positive even integers here, wherein \zeta(2n) = \frac{(-1)^{n+1} B_{2n}(2 \pi)^{2n}}{2(2n)!}, so for instance 1 + 1/2^4 + 1/3^4 + ... = pi^4 / 90, and in general 1 + 1/2^(2n) + 1/3^(2n) + ... converges to r*pi^(2n) for some rational number r that can be found quite easily. I presume Euler's method (the one which first proved your identity) can be extended to prove this case. I don't remember a characterization for the odd integers though.... Isn't it wonderful that the formula I listed uses \tau = 2 \pi? I'm convinced; I don't recall seeing any good evidence for using \pi over \tau other than for historical reasons. I certainly use it in my own work. The Apery's constant (that is, \zeta(3)) Wikipedia page lists some formulas, but nothing even remotely as elegant as the above even integer characterization. I suppose there isn't one currently known :(.
Well there's a couple of BBP-type formulas for pi^2 that MathWorld lists here that might even convert to spigot algorithms (I'm not in the mood to check whether or not the usual BBP-type pi formula's spigot algorithm applies to these cases, where there's a fraction out front). Hmm, that's very jargon filled. What I meant is that it may be possible to compute arbitrary digits of pi^2 without computing the previous digits. This is indeed possible with pi. The thing here, though, is that all of the first 40 trillion digits were computed. TFA is just written by someone not steeped in math enough to know to be careful about equating such computations. The natural log of 2, for instance, admits a very simple spigot algorithm through manipulation of its basic Maclaurin series, as Wikipedia will tell you.
Well, presumably they compare strings of digits. But yeah, the article is less than clear.
In part I meant it was strange that the same figure was given in three different units with wildly different precision each time--that suggested someone might have rounded arbitrarily to the nearest mile after a conversion. I also wondered if mile-scale figures were even possible to measure, but the magnitude of the Pioneer anomaly is a very good point, and suggests that mile-scale measurements are possible. (I've never heard of a "Voyager anomaly". A brief search for the phrase only revealed a few forum posts by, well, idiots, and an apparent typo where "Pioneer anomaly" was meant.)
Yes, our smallness is astonishing. The relative difference in scale going from pencils to atoms is dwarfed by the relative difference in scale going from light-years to pencils. Very roughly, going from the size of a galaxy to the size of a person is comparable to going from the size of a person to the atomic scale twice. On the scale of galaxies, each person is an atom's atom.
The source of the 40,000 figure is here, which is at least hosted on jpl.nasa.gov. I was hoping an author careful enough to include star movement was careful enough to include whatever other relevant effects may exist. You mentioned momentum transfer. Off the top of my head, there may also be electrostatic effects, gravitational fields, or a particle field drifting in some direction which may or may not modify the calculation significantly.
As of April 21, 2011, Voyager 1 was about 116.825 AU, or about 10,843,294,886 miles or about 0.00183 of a light-year from the Sun. [...] Voyager 1's current relative velocity to the sun is 17.061 km/s, or 61,452 kilometres per hour (38,185 mph). This calculates as 3.599 AU per year, about 10% faster than Voyager 2. At this velocity, 73,600 years would pass before reaching the nearest star, Proxima Centauri, were the spacecraft traveling in the direction of that star. [...]
Voyager 1 is not heading towards any particular star, but in about 40,000 years it will pass within 1.6 light years of the star AC+79 3888 in the constellation Camelopardalis. That star is generally moving towards our Solar System at about 119 kilometers per second.
You might want to take that with a grain of salt, though, since there's an incorrect calculation in part of the paragraph I didn't quote and 11 significant figures is very suspiciously precise in the miles figure. Also, the 73,600 figure doesn't agree with my own calculations in the hundreds place. But I imagine what I quoted is pretty close to the truth.
Does it work like this yet?
Not in quantum mechanics.... At least, not in the sense you mean. There an expectation value for, say, energy might be conserved, but it's not really the same thing. That is, your post seems to take the classical view that things "are as they are, as measured by the following variables that take on definite values: mass, energy, momentum, ..." modified by relativity saying "and they might turn into energy and geometry is weird so you really need to measure those things carefully, keeping track of your reference frame and the curvature of spacetime". That's basically a hidden variable theory which cannot explain quantum effects as proven by the Bell inequalities.
In a nutshell--quantum is well and truly screwy. The only definite conclusions are probabilistic ones. It only seems like you have a definite momentum (mass*velocity) because the quantum probability curve for your momentum means you'll only notice fluctuations from repeated measurements after many decimal points. Similarly your position is in a sense constantly fluctuating minutely, giving you a wavelength just like the light hitting your eyes has. (Of course I'm vastly oversimplifying. If your really interested go find a textbook. Wikipedia articles somehow don't do justice to topics like these.)
Yeah, I agree. I can't come up with any variant I really like, though.
We need a word for this kind of article.
How about... "padded". It's both a pun and descriptive inasmuch as a useless comparison was added to the article, probably to get more views.
School districts certainly vary hugely. My high school, for instance, had a 30-40% dropout rate. To their credit, they actually managed to do a decent job of giving lots of AP classes to advanced students, with generally good--sometimes great--teachers. Compared to some of my college friends who went to prep schools, though, it was a joke. I remember having to teach my chemistry teacher logarithms after she tried to teach us (incorrect) voodoo on related rates. My strong dislike for chemistry is probably partly due to a couple of awful high school chemistry teachers.
Unfortunately for the joke, the infinity is negative :(.
I looked at the Titan test and wondered if the OEIS were allowable, since it's an online encyclopedia. I threw the integer sequences into it and it had all of them. Two of those integer sequences and the other non-integer sequence pretty much require remembering random math facts, just like the first verbal analogy. A trained mathematician would do quite well on the math section as well, in part because of their training. I agree with the GP that IQ tests do not have the level of resolution to totally order people on intelligence.
He twisted and twisted, and the colors only got more scrambled. It was like ''staring at a piece of writing written in a secret code. But for me, it was a code I myself had invented! Yet I could not read it. This was such an extraordinary situation that I simply could not accept it.''
Rubik was in even worse shape than his disciples. He didn't knowif the problem could be solved. Perhaps there was only one sure way to get back to the start: by exactly retracing every step he had taken. Rubik couldn't hope to do that. Randomly twisting the Cube would eventually produce the ordered state, but he suspected that the laws of probability were against this occurring in his lifetime. (It has since been calculated that if every person on earth randomly twisted a Cube once every second, about once every three centuries one Cube would return to its original state.)
Rubik had only an intuition that there must be a method. He started out by aligning the eight corner cubies correctly, and he discovered certain sequences of moves for rearranging just a few cubies at a time. One sequence of four twists, for instance, would temporarily scramble the cube, exchange the positions of three cubies, and then restore the rest of the Cube to its previous state. Other sequences took twelve twists--with chaotic results if he lost track of what he was doing halfway through. But Rubik persevered in his room for more than a month and emerged in the summer to show his mother a pristine Cube.
Taken from here. People who do it blind folded most likely memorize the algorithms and apply them to an accurate mental model while their hands move the physical version. I once knew someone who could do it blindfolded, though I never asked how. I always assumed like everyone I've ever known who could do it that he used the algorithmic approach. Do you have examples of people who don't use this approach, or were you just guessing?
Anyone can learn something if they really want to.
I have to disagree. For instance, my mother wouldn't be able to master the classification of finite simple groups before she dies (roughly speaking, that's thousands of pages of advanced math). If you're instead talking about some sort of idealized human who lives forever but isn't ravaged by age, then maybe I'd agree. In any case your point was that there's a correlation between speed of learning and intelligence, and that many people could understand many, many things if they were willing to put in possibly enormous amounts of time, which is certainly true.
It seems like TFA's author might have made the same mistake, or their wording is extremely poor. They say
The software does this by breaking a file to be hidden into a number of fragments and placing the individual pieces in clusters scattered around the hard drive. [...] The method that Khan and his colleagues developed avoids this problem by hiding small pieces of a sensitive file various random places on a hard drive. [...] as the sensitive files are not actually hidden but rather dispersed in pieces.
The file is broken into bits and placed in the arrangement of clusters--these bits are not literally written to the hard drive.
I believe he did earth-moon calculations, and universal gravitation predicted an otherwise unobserved planet (Neptune?). Maybe Newton, in his calculations involving the sun, did always assume the sun was so much more massive than other planets that it was alright to assume it was at the center of mass of the solar system. That's a practical consideration that's different from what I understand you were saying in your first post, which was that Newton did not even theorize universal gravitation as applying to the solar system, instead fixing the sun at the center for some reason and making it necessarily immune to other body's gravitational pulls.
When I was taught special relativity, it was built up pretty much axiomatically. "Warped" geometry ended up falling out quite naturally from first principles (speed of light constant in all inertial reference frames, principle of relativity, continuous space, etc.). If they had instead just told me "apply this length contraction to space" I would have thought "well, that's completely random. Why would I do that?" Once in a while the evolutionary growth of science/math isn't the best pedagogically. A great example is quaternions, which were invented before vector analysis.
I still don't know what you mean about "other" SR/GR/QM geometries. Putting Newtonian mechanics on other geometries is clever, though.
That's not the same situation as an app or Apple having access to location history. It would single me out from the other millions of customers. If the iPhone were used by, say, 20 people and I were one of them, then I would care.
I don't care if Apple knows where I've been, or if other apps can read this log. Is there some non-paranoid reason I should? Out of millions of customers, is Apple really going to care whatsoever if I went to the bank and then rented some porn? If some unscrupulous government agency wants your current location, triangulation should work on any cell phone anyway, as would logging GPS data.
If I really were scared of the government knowing where I've been, I wouldn't trust Apple's good-naturedness to protect me anyway--that would be moronic. TFA suggests a private detective could catch a cheating spouse with this location data, but that's pretty far over on the paranoid spectrum. So again, why should I care, given that I'm neither cheating nor afraid of the government knowing where I've been?
but they are less evil than...say, Hitler.
Hey, Godwin's law.
What do you mean by the "addition law of vectors"? I've also never heard that Newton conceived universal gravitation as not necessarily universal and only applying in some limits. Newton actually said of Hooke's hypotheses on gravity that "without my Demonstrations, to which Mr Hook is yet a stranger, it cannot be beleived by a judicious Philosopher to be any where accurate"--Hooke's claims weren't sufficiently tested, either physically or mathematically, though Newton put his own similar/equivalent theory through its paces as best he could at the time. Newton did not confine himself to objects with significantly different masses, as you suggest: "I deduced that the forces which keep the planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve...". It's very common to consider Newton's laws as they apply to a set of objects of roughly the same mass. See the 2-, 3-, or n-body problems.
I don't know what you mean when you talk about GR/SR geometries. SR is flat Minkowski space, and GR uses a pseudo-Riemannian manifold with metric given by the Einstein equations. One might be able to apply Newtonian gravity on, say, a sphere, with geodesic distance replacing regular distance in gravity computations. That seems a bit strange, though; I've never heard of it applied to anything besides standard Euclidean space.