Re: Working with the set of real numbers (was: Finding size of Variable)
Following up on my own post. On Wed, 05 Mar 2014 07:52:01 +, Steven D'Aprano wrote: On Tue, 04 Mar 2014 23:25:37 -0500, Roy Smith wrote: I stopped paying attention to mathematicians when they tried to convince me that the sum of all natural numbers is -1/12. [...] In effect, the author Mark Carrol-Chu in the GoodMath blog above wants to make the claim that the divergent sum is not equal to ζ(-1), but everywhere you find that divergent sum in your calculations you can rub it out and replace it with ζ(-1), which is -1/12. In other words, he's accepting that the divergent sum behaves *as if* it were equal to -1/12, he just doesn't want to say that it *is* equal to -1/12. Is this a mere semantic trick, or a difference of deep and fundamental importance? Mark C-C thinks it's an important difference. Mathematicians who actually work on this stuff all the time think he's making a semantic trick to avoid facing up to the fact that sums of infinite sequences don't always behave like sums of finite sequences. Here's another mathematician who is even more explicit about what she's complaining about: http://blogs.scientificamerican.com/roots-of-unity/2014/01/20/is-the-sum-of-positive-integers-negative/ [quote] There is a meaningful way to associate the number -1/12 to the series 1+2+3+4…, but in my opinion, it is misleading to call it the sum of the series. [end quote] Evelyn Lamb's objection isn't about the mathematics that leads to the conclusion that the sum of natural numbers is equivalent to -1/12. That's conclusion is pretty much bulletproof. Her objection is over the use of the word equals to describe that association. Or possibly the use of the word sum to describe what we're doing when we replace the infinite series with -1/12. Whatever it is that we're doing, it doesn't seem to have the same behavioural properties as summing finitely many finite numbers. So perhaps she is right, and we shouldn't call the sum of a divergent series a sum? -- Steven -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
Mathematics? The Flexible String Representation is a very nice example of a mathematical absurdity. jmf PS Do not even think to expect to contradict me. Hint: sheet of paper and pencil. -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On 5 March 2014 07:52, Steven D'Aprano st...@pearwood.info wrote: On Tue, 04 Mar 2014 23:25:37 -0500, Roy Smith wrote: I stopped paying attention to mathematicians when they tried to convince me that the sum of all natural numbers is -1/12. I'm pretty sure they did not. Possibly a physicist may have tried to tell you that, but most mathematicians consider physicists to be lousy mathematicians, and the mere fact that they're results seem to actually work in practice is an embarrassment for the entire universe. A mathematician would probably have said that the sum of all natural numbers is divergent and therefore there is no finite answer. Why the dig at physicists? I think most physicists would be able to tell you that the sum of all natural numbers is not -1/12. In fact most people with very little background in mathematics can tell you that. The argument that the sum of all natural numbers comes to -1/12 is just some kind of hoax. I don't think *anyone* seriously believes it. Well, that is, apart from mathematicians like Euler and Ramanujan. When people like them tell you something, you better pay attention. Really? Euler didn't even know about absolutely convergent series (the point in question) and would quite happily combine infinite series to obtain a formula. snip Normally mathematicians will tell you that divergent series don't have a total. That's because often the total you get can vary depending on how you add them up. The classic example is summing the infinite series: 1 - 1 + 1 - 1 + 1 - ... There is a distinction between absolute convergence and convergence. Rearranging the order of the terms in the above infinite sum is invalid because the series is not absolutely convergent. For this particular series there is no sense in which its sum converges on an answer but there are other series that cannot be rearranged while still being convergent: http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Alternating_harmonic_series Personally I think it's reasonable to just say that the sum of the natural numbers is infinite rather than messing around with terms like undefined, divergent, or existence. There is a clear difference between a series (or any limit) that fails to converge asymptotically and another that just goes to +-infinity. The difference is usually also relevant to any practical application of this kind of maths. Oscar -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Wed, 05 Mar 2014 12:21:37 +, Oscar Benjamin wrote: On 5 March 2014 07:52, Steven D'Aprano st...@pearwood.info wrote: On Tue, 04 Mar 2014 23:25:37 -0500, Roy Smith wrote: I stopped paying attention to mathematicians when they tried to convince me that the sum of all natural numbers is -1/12. I'm pretty sure they did not. Possibly a physicist may have tried to tell you that, but most mathematicians consider physicists to be lousy mathematicians, and the mere fact that they're results seem to actually work in practice is an embarrassment for the entire universe. A mathematician would probably have said that the sum of all natural numbers is divergent and therefore there is no finite answer. Why the dig at physicists? There is considerable professional rivalry between the branches of science. Physicists tend to look at themselves as the paragon of scientific hardness, and look down at mere chemists, who look down at biologists. (Which is ironic really, since the actual difficulty in doing good science is in the opposite order. Hundreds of years ago, using quite primitive techniques, people were able to predict the path of comets accurately. I'd like to see them predict the path of a house fly.) According to this greedy reductionist viewpoint, since all living creatures are made up of chemicals, biology is just a subset of chemistry, and since chemicals are made up of atoms, chemistry is likewise just a subset of physics. Physics is the fundamental science, at least according to the physicists, and Real Soon Now they'll have a Theory Of Everything, something small enough to print on a tee-shirt, which will explain everything. At least in principle. Theoretical physicists who work on the deep, fundamental questions of Space and Time tend to be the worst for this reductionist streak. They have a tendency to think of themselves as elites in an elite field of science. Mathematicians, possibly out of professional jealousy, like to look down at physics as mere applied maths. They also get annoyed that physicists often aren't as vigorous with their maths as they should be. The controversy over renormalisation in Quantum Electrodynamics (QED) is a good example. When you use QED to try to calculate the strength of the electron's electric field, you end up trying to sum a lot of infinities. Basically, the interaction of the electron's charge with it's own electric field gets larger the more closely you look. The sum of all those interactions is a divergent series. So the physicists basically cancelled out all the infinities, and lo and behold just like magic what's left over gives you the right answer. Richard Feynman even described it as hocus-pocus. The mathematicians *hated* this, and possibly still do, because it looks like cheating. It's certainly not vigorous, at least it wasn't back in the 1940s. The mathematicians were appalled, and loudly said You can't do that! and the physicists basically said Oh yeah, watch us! and ignored them, and then the Universe had the terribly bad manners to side with the physicists. QED has turned out to be *astonishingly* accurate, the most accurate physical theory of all time. The hocus-pocus worked. I think most physicists would be able to tell you that the sum of all natural numbers is not -1/12. In fact most people with very little background in mathematics can tell you that. Ah, but there's the rub. People with *very little* background in mathematics will tell you that. People with *a very deep and solid* background in mathematics will tell you different, particularly if their background is complex analysis. (That's *complex numbers*, not complicated -- although it is complicated too.) The argument that the sum of all natural numbers comes to -1/12 is just some kind of hoax. I don't think *anyone* seriously believes it. You would be wrong. I suggest you read the links I gave earlier. Even the mathematicians who complain about describing this using the word equals don't try to dispute the fact that you can identify the sum of natural numbers with ζ(-1), or that ζ(-1) = -1/12. They simply dispute that we should describe this association as equals. What nobody believes is that the sum of natural numbers is a convergent series that sums to -1/12, because it is provably not. In other words, this is not an argument about the maths. Everyone who looks at the maths has to admit that it is sound. It's an argument about the words we use to describe this. Is it legitimate to say that the infinite sum *equals* -1/12? Or only that the series has the value -1/12? Or that we can associate (talk about a sloppy, non-vigorous term!) the series with -1/12? Well, that is, apart from mathematicians like Euler and Ramanujan. When people like them tell you something, you better pay attention. Really? Euler didn't even know about absolutely convergent series (the point in question) and would quite happily
Re: Working with the set of real numbers (was: Finding size of Variable)
On Thu, Mar 6, 2014 at 4:43 AM, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: Physics is the fundamental science, at least according to the physicists, and Real Soon Now they'll have a Theory Of Everything, something small enough to print on a tee-shirt, which will explain everything. At least in principle. Everything is, except what isn't. That's my theory, and I'm sticking to it! ChrisA -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Wed, Mar 5, 2014 at 9:43 AM, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: At one time, Euler summed an infinite series and got -1, from which he concluded that -1 was (in some sense) larger than infinity. I don't know what justification he gave, but the way I think of it is to take the number line from -∞ to +∞ and then bend it back upon itself so that there is a single infinity, rather like the projective plane only in a single dimension. If you start at zero and move towards increasingly large numbers, then like Buzz Lightyear you can go to infinity and beyond: 0 - 1 - 10 - 1 - ... ∞ - ... -1 - -10 - -1 - 0 This makes me think that maybe the universe is using ones or two complement math (is there a negative zero?)... Chris -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On 2014-03-05, Chris Kaynor ckay...@zindagigames.com wrote: On Wed, Mar 5, 2014 at 9:43 AM, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: At one time, Euler summed an infinite series and got -1, from which he concluded that -1 was (in some sense) larger than infinity. I don't know what justification he gave, but the way I think of it is to take the number line from -∞ to +∞ and then bend it back upon itself so that there is a single infinity, rather like the projective plane only in a single dimension. If you start at zero and move towards increasingly large numbers, then like Buzz Lightyear you can go to infinity and beyond: 0 - 1 - 10 - 1 - ... ∞ - ... -1 - -10 - -1 - 0 This makes me think that maybe the universe is using ones or two complement math (is there a negative zero?)... If the Universe (like most all Python implementations) is using IEEE-754 floating point, there is. -- Grant Edwards grant.b.edwardsYow! This PIZZA symbolizes at my COMPLETE EMOTIONAL gmail.comRECOVERY!! -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On 5 March 2014 17:43, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: On Wed, 05 Mar 2014 12:21:37 +, Oscar Benjamin wrote: The argument that the sum of all natural numbers comes to -1/12 is just some kind of hoax. I don't think *anyone* seriously believes it. You would be wrong. I suggest you read the links I gave earlier. Even the mathematicians who complain about describing this using the word equals don't try to dispute the fact that you can identify the sum of natural numbers with ζ(-1), or that ζ(-1) = -1/12. They simply dispute that we should describe this association as equals. What nobody believes is that the sum of natural numbers is a convergent series that sums to -1/12, because it is provably not. In other words, this is not an argument about the maths. Everyone who looks at the maths has to admit that it is sound. It's an argument about the words we use to describe this. Is it legitimate to say that the infinite sum *equals* -1/12? Or only that the series has the value -1/12? Or that we can associate (talk about a sloppy, non-vigorous term!) the series with -1/12? This is the point. You can identify numbers with many different things. It does not mean to say that the thing is equal to that number. I can associate the number 2 with my bike since it has 2 wheels. That doesn't mean that the bike is equal to 2. So the problem with saying that the sum of the natural numbers equals -1/12 is precisely as you say with the word equals because they're not equal! If you restate the conclusion in more accurate (but technical and less accessible) way that the analytic continuation of a related set of convergent series has the value -1/12 at the value that would correspond to this divergent series then it becomes less mysterious. Do I really have to associate the finite negative value found in the analytic continuation with the sum of the series that is provably greater than any finite number? snip At one time, Euler summed an infinite series and got -1, from which he concluded that -1 was (in some sense) larger than infinity. I don't know what justification he gave, but the way I think of it is to take the number line from -∞ to +∞ and then bend it back upon itself so that there is a single infinity, rather like the projective plane only in a single dimension. If you start at zero and move towards increasingly large numbers, then like Buzz Lightyear you can go to infinity and beyond: 0 - 1 - 10 - 1 - ... ∞ - ... -1 - -10 - -1 - 0 In this sense, -1/12 is larger than infinity. There are many examples that appear to show wrapping round from +infinity to -infinity e.g. the tan function. The thing is that it is not really physical (or meaningful in any direct sense). So for example I might consider the forces on a particle, apply Newton's 2nd law and arrive at a differential equation for the acceleration of the particle, solve the equation and find that the position of the particle at time t is given by tan(t). This would seem to imply that as t increases toward pi/2 the particle heads off infinity miles West but at the exact time pi/2 it wraps around to reappear at infinity miles East and starts heading back toward its starting point. The truth is less interesting: the solution tan(t) becomes invalid at pi/2 and mathematics can tell us nothing about what happens after that even if all the physics we used was exactly true. Now of course this is an ad hoc sloppy argument, but I'm not a professional mathematician. However I can tell you that it's pretty close to what the professional mathematicians and physicists do with negative absolute temperatures, and that is rigorous. http://en.wikipedia.org/wiki/Negative_temperature The key point from that page is the sentence A definition of temperature can be based on the relationship It is clear that temperature is a theoretical abstraction. We have intuitive understandings of what it means but in order for the current body of thermodynamic theory to be consistent it is necessary to sometimes give negative values to the temperature. There's nothing unintuitive about negative temperatures if you understand the usual thermodynamic definitions of temperature. Personally I think it's reasonable to just say that the sum of the natural numbers is infinite rather than messing around with terms like undefined, divergent, or existence. There is a clear difference between a series (or any limit) that fails to converge asymptotically and another that just goes to +-infinity. The difference is usually also relevant to any practical application of this kind of maths. And this is where you get it exactly backwards. The *practical application* comes from physics, where they do exactly what you argue against: they associate ζ(-1) with the sum of the natural numbers (see, I too can avoid the word equals too), and *it works*. I don't know all the details of what they do there and whether or not
Re: Working with the set of real numbers (was: Finding size of Variable)
In article 53176225$0$29987$c3e8da3$54964...@news.astraweb.com, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: Physics is the fundamental science, at least according to the physicists, and Real Soon Now they'll have a Theory Of Everything, something small enough to print on a tee-shirt, which will explain everything. At least in principle. A mathematician, a chemist, and a physicist are arguing the nature of prime numbers. The chemist says, All odd numbers are prime. Look, I can prove it. Three is prime. Five is prime. Seven is prime. The mathematician says, That's nonsense. Nine is not prime. The physicist looks at him and says, H, you may be right, but eleven is prime, and thirteen is prime. It appears that within the limits of experimental error, all odd number are indeed prime! -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Wed, 05 Mar 2014 21:31:51 -0500, Roy Smith wrote: In article 53176225$0$29987$c3e8da3$54964...@news.astraweb.com, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: Physics is the fundamental science, at least according to the physicists, and Real Soon Now they'll have a Theory Of Everything, something small enough to print on a tee-shirt, which will explain everything. At least in principle. A mathematician, a chemist, and a physicist are arguing the nature of prime numbers. The chemist says, All odd numbers are prime. Look, I can prove it. Three is prime. Five is prime. Seven is prime. The mathematician says, That's nonsense. Nine is not prime. The physicist looks at him and says, H, you may be right, but eleven is prime, and thirteen is prime. It appears that within the limits of experimental error, all odd number are indeed prime! They ask a computer programmer to adjudicate who is right, so he writes a program to print out all the primes: 1 is prime 1 is prime 1 is prime 1 is prime 1 is prime ... -- Steven D'Aprano http://import-that.dreamwidth.org/ -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Thu, Mar 6, 2014 at 2:06 PM, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: They ask a computer programmer to adjudicate who is right, so he writes a program to print out all the primes: 1 is prime 1 is prime 1 is prime 1 is prime 1 is prime And he claimed that he was correct, because he had - as is known to be true in reality - a countably infinite number of primes. ChrisA -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On 2014-03-06, Roy Smith r...@panix.com wrote: In article 53176225$0$29987$c3e8da3$54964...@news.astraweb.com, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: Physics is the fundamental science, at least according to the physicists, and Real Soon Now they'll have a Theory Of Everything, something small enough to print on a tee-shirt, which will explain everything. At least in principle. A mathematician, a chemist, and a physicist are arguing the nature of prime numbers. The chemist says, All odd numbers are prime. Look, I can prove it. Three is prime. Five is prime. Seven is prime. The mathematician says, That's nonsense. Nine is not prime. The physicist looks at him and says, H, you may be right, but eleven is prime, and thirteen is prime. It appears that within the limits of experimental error, all odd number are indeed prime! Assuming spherical odd numbers in a vacuum on a frictionless surface, of course. -- Grant -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
In article 5317e640$0$29985$c3e8da3$54964...@news.astraweb.com, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: On Wed, 05 Mar 2014 21:31:51 -0500, Roy Smith wrote: In article 53176225$0$29987$c3e8da3$54964...@news.astraweb.com, Steven D'Aprano steve+comp.lang.pyt...@pearwood.info wrote: Physics is the fundamental science, at least according to the physicists, and Real Soon Now they'll have a Theory Of Everything, something small enough to print on a tee-shirt, which will explain everything. At least in principle. A mathematician, a chemist, and a physicist are arguing the nature of prime numbers. The chemist says, All odd numbers are prime. Look, I can prove it. Three is prime. Five is prime. Seven is prime. The mathematician says, That's nonsense. Nine is not prime. The physicist looks at him and says, H, you may be right, but eleven is prime, and thirteen is prime. It appears that within the limits of experimental error, all odd number are indeed prime! They ask a computer programmer to adjudicate who is right, so he writes a program to print out all the primes: 1 is prime 1 is prime 1 is prime 1 is prime 1 is prime ... So, a mathematician, a biologist, and a physicist are watching a house. The physicist says, It appears to be empty. Sometime later, a man and a woman go into the house. Shortly after that, the man and the woman come back out, with a child. The biologist says, They must have reproduced. The mathematician says, If one more person goes into the house, it'll be empty again. -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Mon, Mar 3, 2014 at 11:35 PM, Chris Angelico ros...@gmail.com wrote: In constant space, that will produce the sum of two infinite sequences of digits. (And it's constant time, too, except when it gets a stream of nines. Adding three thirds together will produce an infinite loop as it waits to see if there'll be anything that triggers an infinite cascade of carries.) Now, if there's a way to do that for square rooting a number, then the CF notation has a distinct benefit over the decimal expansion used here. As far as I know, there's no simple way, in constant space and/or time, to progressively yield more digits of a number's square root, working in decimal. The code for that looks like this: def cf_sqrt(n): Yield the terms of the square root of n as a continued fraction. m = 0 d = 1 a = a0 = floor_sqrt(n) while True: yield a next_m = d * a - m next_d = (n - next_m * next_m) // d if next_d == 0: break next_a = (a0 + next_m) // next_d m, d, a = next_m, next_d, next_a def floor_sqrt(n): Return the integer part of the square root of n. n = int(n) if n == 0: return 0 lower = 2 ** int(math.log(n, 2) // 2) upper = lower * 2 while upper - lower 1: mid = (upper + lower) // 2 if n mid * mid: upper = mid else: lower = mid return lower The floor_sqrt function is merely doing a simple binary search and could probably be optimized, but then it's only called once during initialization anyway. The meat of the loop, as you can see, is just a constant amount of integer arithmetic. If it were desired to halt once the continued fraction starts to repeat, that would just be a matter of checking whether the triple (m, d, a) has been seen already. Going back to your example of adding generated digits though, I don't know how to add two continued fractions together without evaluating them. -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Tue, Mar 4, 2014 at 4:19 AM, Ian Kelly ian.g.ke...@gmail.com wrote: def cf_sqrt(n): Yield the terms of the square root of n as a continued fraction. m = 0 d = 1 a = a0 = floor_sqrt(n) while True: yield a next_m = d * a - m next_d = (n - next_m * next_m) // d if next_d == 0: break next_a = (a0 + next_m) // next_d m, d, a = next_m, next_d, next_a Sorry, all that next business is totally unnecessary. More simply: def cf_sqrt(n): Yield the terms of the square root of n as a continued fraction. m = 0 d = 1 a = a0 = floor_sqrt(n) while True: yield a m = d * a - m d = (n - m * m) // d if d == 0: break a = (a0 + m) // d -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
In article mailman.7702.1393932047.18130.python-l...@python.org, Ian Kelly ian.g.ke...@gmail.com wrote: On Mon, Mar 3, 2014 at 11:35 PM, Chris Angelico ros...@gmail.com wrote: In constant space, that will produce the sum of two infinite sequences of digits. (And it's constant time, too, except when it gets a stream of nines. Adding three thirds together will produce an infinite loop as it waits to see if there'll be anything that triggers an infinite cascade of carries.) Now, if there's a way to do that for square rooting a number, then the CF notation has a distinct benefit over the decimal expansion used here. As far as I know, there's no simple way, in constant space and/or time, to progressively yield more digits of a number's square root, working in decimal. The code for that looks like this: def cf_sqrt(n): Yield the terms of the square root of n as a continued fraction. m = 0 d = 1 a = a0 = floor_sqrt(n) while True: yield a next_m = d * a - m next_d = (n - next_m * next_m) // d if next_d == 0: break next_a = (a0 + next_m) // next_d m, d, a = next_m, next_d, next_a def floor_sqrt(n): Return the integer part of the square root of n. n = int(n) if n == 0: return 0 lower = 2 ** int(math.log(n, 2) // 2) upper = lower * 2 while upper - lower 1: mid = (upper + lower) // 2 if n mid * mid: upper = mid else: lower = mid return lower The floor_sqrt function is merely doing a simple binary search and could probably be optimized, but then it's only called once during initialization anyway. The meat of the loop, as you can see, is just a constant amount of integer arithmetic. If it were desired to halt once the continued fraction starts to repeat, that would just be a matter of checking whether the triple (m, d, a) has been seen already. Going back to your example of adding generated digits though, I don't know how to add two continued fractions together without evaluating them. That is highly non-trivial indeed. See the gosper.txt reference I gave in another post. Groetjes Albert -- Albert van der Horst, UTRECHT,THE NETHERLANDS Economic growth -- being exponential -- ultimately falters. albert@spearc.xs4all.nl =n http://home.hccnet.nl/a.w.m.van.der.horst -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
In article mailman.7687.1393902132.18130.python-l...@python.org, Chris Angelico ros...@gmail.com wrote: On Tue, Mar 4, 2014 at 1:45 PM, Albert van der Horst alb...@spenarnc.xs4all.nl wrote: No, the Python built-in float type works with a subset of real numbers: To be more precise: a subset of the rational numbers, those with a denominator that is a power of two. And no more than N bits (53 in a 64-bit float) in the numerator, and the denominator between the limits of the exponent. (Unless it's subnormal. That adds another set of small numbers.) It's a pretty tight set of restrictions, and yet good enough for so many purposes. But it's a far cry from all real numbers. Even allowing for continued fractions adds only some more; I don't think you can represent surds that way. Adding cf's adds all computable numbers in infinite precision. However that is not even a drop in the ocean, as the computable numbers have measure zero. A cf object yielding its coefficients amounts to a program that generates an infinite amount of data (in infinite time), so it is not very surprising it can represent any computable number. Pretty humbling really. ChrisA Groetjes Albert -- Albert van der Horst, UTRECHT,THE NETHERLANDS Economic growth -- being exponential -- ultimately falters. albert@spearc.xs4all.nl =n http://home.hccnet.nl/a.w.m.van.der.horst -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Wed, 05 Mar 2014 02:15:14 +, Albert van der Horst wrote: Adding cf's adds all computable numbers in infinite precision. However that is not even a drop in the ocean, as the computable numbers have measure zero. On the other hand, it's not really clear that the non-computable numbers are useful or necessary for anything. They exist as mathematical abstractions, but they'll never be the result of any calculation or measurement that anyone might do. -- Steven -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Wednesday, March 5, 2014 9:11:13 AM UTC+5:30, Steven D'Aprano wrote: On Wed, 05 Mar 2014 02:15:14 +, Albert van der Horst wrote: Adding cf's adds all computable numbers in infinite precision. However that is not even a drop in the ocean, as the computable numbers have measure zero. On the other hand, it's not really clear that the non-computable numbers are useful or necessary for anything. They exist as mathematical abstractions, but they'll never be the result of any calculation or measurement that anyone might do. There are even more extreme versions of this amounting to roughly this view: Any infinity supposedly 'larger' than the natural numbers is a nonsensical notion. See eg http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory and Weyl/Polya bet (pg 10 of http://research.microsoft.com/en-us/um/people/gurevich/Opera/123.pdf ) I cannot find the exact quote so from memory Weyl says something to this effect: Cantor's diagonalization PROOF is not in question. Its CONCLUSION very much is. The classical/platonic mathematician (subject to wooly thinking) concludes that the real numbers are a superset of the integers The constructvist mathematician (who supposedly thinks clearly) only concludes the obvious, viz that real numbers cannot be enumerated To go from 'cannot be enumerated' to 'is a proper superset of' requires the assumption of 'completed infinities' and that is not math but theology -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
In article c39d5b44-6c7b-40d1-bbb5-791a36af6...@googlegroups.com, Rustom Mody rustompm...@gmail.com wrote: I cannot find the exact quote so from memory Weyl says something to this effect: Cantor's diagonalization PROOF is not in question. Its CONCLUSION very much is. The classical/platonic mathematician (subject to wooly thinking) concludes that the real numbers are a superset of the integers The constructvist mathematician (who supposedly thinks clearly) only concludes the obvious, viz that real numbers cannot be enumerated To go from 'cannot be enumerated' to 'is a proper superset of' requires the assumption of 'completed infinities' and that is not math but theology I stopped paying attention to mathematicians when they tried to convince me that the sum of all natural numbers is -1/12. Sure, you can manipulate the symbols in a way which is consistent with some set of rules that we believe govern the legal manipulation of symbols, but it just plain doesn't make sense. -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Tue, 04 Mar 2014 23:25:37 -0500, Roy Smith wrote: I stopped paying attention to mathematicians when they tried to convince me that the sum of all natural numbers is -1/12. I'm pretty sure they did not. Possibly a physicist may have tried to tell you that, but most mathematicians consider physicists to be lousy mathematicians, and the mere fact that they're results seem to actually work in practice is an embarrassment for the entire universe. A mathematician would probably have said that the sum of all natural numbers is divergent and therefore there is no finite answer. Well, that is, apart from mathematicians like Euler and Ramanujan. When people like them tell you something, you better pay attention. We have an intuitive understanding of the properties of addition. You can't add 1000 positive whole numbers and get a negative fraction, that's obvious. But that intuition only applies to *finite* sums. They don't even apply to infinite *convergent* series, and they're *easy*. Remember Zeno's Paradoxes? People doubted that the convergent series: 1/2 + 1/4 + 1/8 + 1/16 + ... added up to 1 for the longest time, even though they could see with their own eyes that it had to. Until they worked out what *infinite* sums actually meant, their intuitions were completely wrong. This is a good lesson for us all. The sum of all the natural numbers is a divergent infinite series, so we shouldn't expect that our intuitions hold. We can't add it up as if it were a convergent series, because it's not convergent. Nobody disputes that. But perhaps there's another way? Normally mathematicians will tell you that divergent series don't have a total. That's because often the total you get can vary depending on how you add them up. The classic example is summing the infinite series: 1 - 1 + 1 - 1 + 1 - ... Depending on how you group them, you can get: (1 - 1) + (1 - 1) + (1 - 1) ... = 0 + 0 + 0 + ... = 0 or you can get: 1 - (1 - 1 + 1 - 1 + ... ) = 1 - (1 - 1) - (1 - 1) - ... ) = 1 - 0 - 0 - 0 ... = 1 Or you can do a neat little trick where we define the sum as x: x = 1 - 1 + 1 - 1 + 1 - ... x = 1 - (1 - 1 + 1 - 1 + ... ) x = 1 - x 2x = 1 x = 1/2 So at first glance, summing a divergent series is like dividing by zero. You get contradictory results, at least in this case. But that's not necessarily always the case. You do have to be careful when summing divergent series, but that doesn't always mean you can't do it and get a meaningful answer. Sometimes you can, sometimes you can't, it depends on the specific series. With the sum of the natural numbers, rather than getting three different results from three different methods, mathematicians keep getting the same -1/12 result using various methods. That's a good hint that there is something logically sound going on here, even if it seems unintuitive. Remember Zeno's Paradoxes? Our intuitions about equality and plus and sums of numbers don't apply to infinite series. We should be at least open to the possibility that while all the *finite* sums: 1 + 2 1 + 2 + 3 1 + 2 + 3 + 4 ... and so on sum to positive whole numbers, that doesn't mean that the *infinite* sum has to total to a positive whole number. Maybe that's not how addition works. I don't know about you, but I've never personally added up an infinite number of every-increasing quantities to see what the result is. Maybe it is a negative fraction. (I'd say try it and see, but I don't have an infinite amount of time to spend on it.) And in fact that's exactly what seems to be case here. Mathematicians can demonstrate an identity (that is, equality) between the divergent sum of the natural numbers with the zeta function ζ(-1), and *that* can be worked out independently, and equals -1/12. So there are a bunch of different ways to show that the divergent sum adds up to -1/12, some of them are more vigorous than others. The zeta function method is about as vigorous as they come. The addition of an infinite number of things behaves differently than the addition of finite numbers of things. More here: http://scitation.aip.org/content/aip/magazine/physicstoday/news/10.1063/PT.5.8029 http://math.ucr.edu/home/baez/week126.html http://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_%E2%8B%AF and even here: http://scientopia.org/blogs/goodmath/2014/01/20/oy-veh-power-series-analytic-continuations-and-riemann-zeta/ where a mathematician tries *really hard* to discredit the idea that the sum equals -1/12, but ends up proving that it does. So he simply plays a linguistic slight of hand and claims that despite the series and the zeta function being equal, they're not *actually* equal. In effect, the author Mark Carrol-Chu in the GoodMath blog above wants to make the claim that the divergent sum is not equal to ζ(-1), but everywhere you find that divergent sum in your calculations you can rub it out and replace it with ζ(-1), which is -1/12. In other words, he's
Re: Working with the set of real numbers (was: Finding size of Variable)
In article mailman.6735.1392194885.18130.python-l...@python.org, Chris Angelico ros...@gmail.com wrote: On Wed, Feb 12, 2014 at 7:17 PM, Ben Finney ben+pyt...@benfinney.id.au wrote: Chris Angelico ros...@gmail.com writes: I have yet to find any computer that works with the set of real numbers in any way. Never mind optimization, they simply cannot work with real numbers. Not *any* computer? Not in *any* way? The Python built-in âfloatâ type âworks with the set of real numbersâ, in a way. No, the Python built-in float type works with a subset of real numbers: To be more precise: a subset of the rational numbers, those with a denominator that is a power of two. float(pi) Traceback (most recent call last): File pyshell#1, line 1, in module float(pi) ValueError: could not convert string to float: 'pi' float(Ï) Traceback (most recent call last): File pyshell#2, line 1, in module float(Ï) ValueError: could not convert string to float: 'Ï' Same goes for fractions.Fraction and [c]decimal.Decimal. All of them are restricted to some subset of rational numbers, not all reals. The URL:http://docs.python.org/2/library/numbers.html#numbers.Real ABC defines behaviours for types implementing the set of real numbers. What specific behaviour would, for you, qualify as âworks with the set of real numbers in any wayâ? Being able to represent surds, pi, e, etc, for a start. It'd theoretically be possible with an algebraic notation (eg by carrying through some representation like 2*pi rather than 6.28), but otherwise, irrationals can't be represented with finite storage and a digit-based system. An interesting possibility is working with rules that generate the continued fraction sequence of a real number. Say yield() gives the next coefficient (or the next hex digit). It was generally believed that summing two numbers in their cf representation was totally impractical because it required conversion to a rational number. OTOH if we consider a cf as an ongoing progress, the situation is much better. Summing would be a process that yields coefficients of the sum, and you could just stop when you've enough precision. Fascinating stuff. It is described in a self contained, type writer style document gosper.txt that is found on the web in several places e.g. http://home.strw.leidenuniv.nl/~gurkan/gosper.pdf I have a gosper.txt, don't know from where. It really is a cookbook, one could built a python implementation from there, without being overly math savvy. I'd love to hear if some one does it. ( in principle a coefficient of a cf can overflow machine precision, that has never been observed in the wild. A considerable percentage of the coefficients for a random number are ones or otherwise small. The golden ratio has all ones.) ChrisA Groetjes Albert -- Albert van der Horst, UTRECHT,THE NETHERLANDS Economic growth -- being exponential -- ultimately falters. albert@spearc.xs4all.nl =n http://home.hccnet.nl/a.w.m.van.der.horst -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Tue, Mar 4, 2014 at 1:45 PM, Albert van der Horst alb...@spenarnc.xs4all.nl wrote: No, the Python built-in float type works with a subset of real numbers: To be more precise: a subset of the rational numbers, those with a denominator that is a power of two. And no more than N bits (53 in a 64-bit float) in the numerator, and the denominator between the limits of the exponent. (Unless it's subnormal. That adds another set of small numbers.) It's a pretty tight set of restrictions, and yet good enough for so many purposes. But it's a far cry from all real numbers. Even allowing for continued fractions adds only some more; I don't think you can represent surds that way. ChrisA -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Tuesday, March 4, 2014 8:32:01 AM UTC+5:30, Chris Angelico wrote: On Tue, Mar 4, 2014 at 1:45 PM, Albert van der Horst wrote: No, the Python built-in float type works with a subset of real numbers: To be more precise: a subset of the rational numbers, those with a denominator that is a power of two. And no more than N bits (53 in a 64-bit float) in the numerator, and the denominator between the limits of the exponent. (Unless it's subnormal. That adds another set of small numbers.) It's a pretty tight set of restrictions, and yet good enough for so many purposes. But it's a far cry from all real numbers. Even allowing for continued fractions adds only some more; I don't think you can represent surds that way. See http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#sqrts -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Tue, Mar 4, 2014 at 2:13 PM, Rustom Mody rustompm...@gmail.com wrote: But it's a far cry from all real numbers. Even allowing for continued fractions adds only some more; I don't think you can represent surds that way. See http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#sqrts That's neat, didn't know that. Is there an efficient way to figure out, for any integer N, what its sqrt's CF sequence is? And what about the square roots of non-integers - can you represent √π that way? I suspect, though I can't prove, that there will be numbers that can't be represented even with an infinite series - or at least numbers whose series can't be easily calculated. ChrisA -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Tuesday, March 4, 2014 9:16:25 AM UTC+5:30, Chris Angelico wrote: On Tue, Mar 4, 2014 at 2:13 PM, Rustom Mody wrote: But it's a far cry from all real numbers. Even allowing for continued fractions adds only some more; I don't think you can represent surds that way. See http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#sqrts That's neat, didn't know that. Is there an efficient way to figure out, for any integer N, what its sqrt's CF sequence is? And what about the square roots of non-integers - can you represent √π that way? I suspect, though I can't prove, that there will be numbers that can't be represented even with an infinite series - or at least numbers whose series can't be easily calculated. You are now asking questions that are really (real-ly?) outside my capacities. What I know (which may be quite off the mark :-) ) Just as all real numbers almost by definition have a decimal form (may be infinite eg 1/3 becomes 0.3...) all real numbers likewise have a CF form For some mathematical (aka arcane) reasons the CF form is actually better. Furthermore: 1. Transcendental numbers like e and pi have non-repeating infinite CF forms 2. Algebraic numbers (aka surds) have repeating maybe finite(?) forms 3. For some numbers its not known whether they are transcendental or not (vague recollection pi^sqrt(pi) is one such) 4 Since e^ipi is very much an integer, above question is surprisingly non-trivial -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Tue, 04 Mar 2014 14:46:25 +1100, Chris Angelico wrote: That's neat, didn't know that. Is there an efficient way to figure out, for any integer N, what its sqrt's CF sequence is? And what about the square roots of non-integers - can you represent √π that way? I suspect, though I can't prove, that there will be numbers that can't be represented even with an infinite series - or at least numbers whose series can't be easily calculated. Every rational number can be written as a continued fraction with a finite number of terms[1]. Every irrational number can be written as a continued fraction with an infinite number of terms, just as every irrational number can be written as a decimal number with an infinite number of digits. Most of them (to be precise: an uncountably infinite number of them) will have no simple or obvious pattern. [1] To be pedantic, written as *two* continued fractions, one ending with the term 1, and one with one less term which isn't 1. That is: [a; b, c, d, ..., z, 1] == [a; b, c, d, ..., z+1] Any *finite* CF ending with one can be simplified to use one fewer term. Infinite CFs of course don't have a last term. -- Steven -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Tue, Mar 4, 2014 at 4:53 PM, Steven D'Aprano st...@pearwood.info wrote: On Tue, 04 Mar 2014 14:46:25 +1100, Chris Angelico wrote: That's neat, didn't know that. Is there an efficient way to figure out, for any integer N, what its sqrt's CF sequence is? And what about the square roots of non-integers - can you represent √π that way? I suspect, though I can't prove, that there will be numbers that can't be represented even with an infinite series - or at least numbers whose series can't be easily calculated. Every irrational number can be written as a continued fraction with an infinite number of terms, just as every irrational number can be written as a decimal number with an infinite number of digits. It's easy enough to have that kind of expansion, I'm wondering if it's possible to identify it directly. To render the decimal expansion of a square root by the cut-and-try method, you effectively keep dividing until you find that you're close enough; that means you (a) have to keep the entire number around for each step, and (b) need to do a few steps to find that the digits aren't changing. But if you can take a CF (finite or infinite) and do an O(n) transformation on it to produce that number's square root, then you have an effective means of representing square roots. Suppose I make a generator function that represents a fraction: def one_third(): while True: yield 3 def one_seventh(): while True: yield 1; yield 4; yield 2; yield 8; yield 5; yield 7 I could then make a generator that returns the sum of those two: def add_without_carry(x, y): whiile True: yield next(x)+next(y) Okay, that's broken for nearly any case, but with a bit more sophistication: def add(x, y): prev=None nines=0 while True: xx,yy=next(x),next(y) tot=xx+yy if tot==9: nines+=1 continue if tot9: if prev is None: raise OverflowError(exceeds 1.0) yield prev+1 tot-=10 for _ in range(nines): yield 0 nines=0 else: if prev is not None: yield prev prev=tot def show(n): return ''.join(str(_) for _ in itertools.islice(n,20)) show(add(one_third(),one_seventh())) '47619047619047619047' show(add(add(add(one_seventh(),one_seventh()),add(one_seventh(),one_seventh())),add(one_seventh(),one_seventh( '85714285714285714285' In constant space, that will produce the sum of two infinite sequences of digits. (And it's constant time, too, except when it gets a stream of nines. Adding three thirds together will produce an infinite loop as it waits to see if there'll be anything that triggers an infinite cascade of carries.) Now, if there's a way to do that for square rooting a number, then the CF notation has a distinct benefit over the decimal expansion used here. As far as I know, there's no simple way, in constant space and/or time, to progressively yield more digits of a number's square root, working in decimal. ChrisA -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
Integers are integers. (1) Characters are characters. (2) (1) is a unique natural set. (2) is an artificial construct working with 3 sets (unicode). jmf -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On Wed, Feb 12, 2014 at 7:17 PM, Ben Finney ben+pyt...@benfinney.id.au wrote: Chris Angelico ros...@gmail.com writes: I have yet to find any computer that works with the set of real numbers in any way. Never mind optimization, they simply cannot work with real numbers. Not *any* computer? Not in *any* way? The Python built-in ‘float’ type “works with the set of real numbers”, in a way. No, the Python built-in float type works with a subset of real numbers: float(pi) Traceback (most recent call last): File pyshell#1, line 1, in module float(pi) ValueError: could not convert string to float: 'pi' float(π) Traceback (most recent call last): File pyshell#2, line 1, in module float(π) ValueError: could not convert string to float: 'π' Same goes for fractions.Fraction and [c]decimal.Decimal. All of them are restricted to some subset of rational numbers, not all reals. The URL:http://docs.python.org/2/library/numbers.html#numbers.Real ABC defines behaviours for types implementing the set of real numbers. What specific behaviour would, for you, qualify as “works with the set of real numbers in any way”? Being able to represent surds, pi, e, etc, for a start. It'd theoretically be possible with an algebraic notation (eg by carrying through some representation like 2*pi rather than 6.28), but otherwise, irrationals can't be represented with finite storage and a digit-based system. ChrisA -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
Le mercredi 12 février 2014 09:35:38 UTC+1, wxjm...@gmail.com a écrit : Integers are integers. (1) Characters are characters. (2) (1) is a unique natural set. (2) is an artificial construct working with 3 sets (unicode). jmf Addendum: One should not confuse unicode and the implementation of unicode. jmf -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
Chris Angelico writes: On Wed, Feb 12, 2014 at 7:17 PM, Ben Finney wrote: What specific behaviour would, for you, qualify as “works with the set of real numbers in any way”? Being able to represent surds, pi, e, etc, for a start. It'd theoretically be possible with an algebraic notation (eg by carrying through some representation like 2*pi rather than 6.28), but otherwise, irrationals can't be represented with finite storage and a digit-based system. I've seen papers on exact computable reals that would, in effect, generate more precision when needed for some operation. It wasn't symbolic like 2pi, more like 6.28... with a promise to delve into the ellipsis, and some notable operations not supported. Equality testing was missing, I think, and I think it could not be known in general whether such a number is positive, zero or negative, so even approximate printing in the usual digit notation would not be possible. (Interval arithmetic, I hear, has a similar problem about not knowing the sign of a number.) In stark contrast, exact rationals work nicely, up to efficiency considerations. -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
On 2014-02-12, Ben Finney ben+pyt...@benfinney.id.au wrote: Chris Angelico ros...@gmail.com writes: I have yet to find any computer that works with the set of real numbers in any way. Never mind optimization, they simply cannot work with real numbers. Not *any* computer? Not in *any* way? The Python built-in float type works with the set of real numbers, in a way. The only people who think that are people who don't actualy _use_ floating point types on computers. What specific behaviour would, for you, qualify as works with the set of real numbers in any way There's a whole laundry list of things (some of them rather nasty and difficult) you have to worry about when using FP that simply don't apply to real numbers. -- Grant Edwards grant.b.edwardsYow! HUGH BEAUMONT died at in 1982!! gmail.com -- https://mail.python.org/mailman/listinfo/python-list
Re: Working with the set of real numbers (was: Finding size of Variable)
Grant Edwards wrote: Not *any* computer? Not in *any* way? The Python built-in float type works with the set of real numbers, in a way. The only people who think that are people who don't actualy _use_ floating point types on computers. FPU parsing the IEEE spec, or?. I didn't quite parse what *you* wrote. To paraphrase: #include math.h there are FP_NORMAL and FP_SUBNORMAL people in the world; those who understand IEEE 754 and those who don't. .. --gv -- https://mail.python.org/mailman/listinfo/python-list
