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Everything posted by uncool

  1. Unfortunately, some of the concepts involved in the definition require that complexity. Ignoring the philosophical issues, yes. Correct. I mean that "infinity" is not a single concept by itself, so it doesn't really make sense to say "past infinity" in the same way as, e.g., "past 2". The closest thing I can say is that for any cardinal, there is a larger cardinal.
  2. Before I answer some of these questions, I'd like to note that it may be easier if you learn how the hyperreals are defined formally. However, the formal definition is somewhat complex and based in advanced set theory (namely ultrafilters and the existence of free ultrafilters on the natural numbers). Yes. In fact, every real number is also a hyperreal number (if you have an exacting philosophy of math, the statement might be more accurately stated as "Every real number has a hyperreal counterpart", but I'm going to ignore philosophical issues for now). Slightly more formally, we really consider the hyperreals as an extension of the real numbers - the set of hyperreals includes the real numbers, and then some. You can create "hyper-hyperreals" through a similar formal process to the way the hyperreals are constructed. However, 2*R isn't special - it is "merely" hyperreal, in the same way that 0 is "merely" real. You can do any formal algebraic operation on the hyperreals that you could do on the reals - add them, subtract them, multiply them, etc. Formally, they are an ordered field just like the real numbers. I strongly suspect that you misunderstood something here; the ordinals, cardinals, and hyperreals are all in some way or another generalizations of the idea of infinity. All of them have some idea of "greater than infinity". If I had to give an answer, I'd say that they are like quantities in that they can be added, subtracted, multiplied, divided, etc. However, I'd more say that ordinals, cardinals, and hyperreals are simply attempts to extend different collections of properties of finite things - in the case of ordinals, ordering; in the case of cardinals, counting; in the case of hyperreals, the arithmetic.
  3. 1) Wouldn't 1-1 be 1|2? 2) What do you mean by 1-1-1-1? Is the dash meant to be a minus sign?
  4. If there's no "fail condition", then yes, it is a self-fulfilling prophecy. What fractions do you expect? How small is "relatively small" for numerator and denominator?
  5. Using the example I gave (1.4444387260): Always start with p0 = 0, q0 = 1, p1 = 1, q1 = 0. a1 = 1, p2 = a1*p1 + p0 = 1, q2 = a1*q1 + q0 = 1, convergent = 1/1 a2 = 2, p3 = a2*p2 + p1 = 3, q3 = a2*q2 + q1 = 2, convergent = 3/2 a3 = 3, p4 = a3*p3 + p2 = 10, q4 = a3*q3 + q2 = 7, convergent = 10/7 a4 = 1, p5 = a4*p4 + p3 = 13, q5 = a4*q4 + q3 = 9, convergent = 13/9. a5 = 2158, cutoff reached. Alternatively, at the end, it could be: a5 = 2158, p6 = a5*p5 + p4 = 28064, q6 = a5*q5 + q4 = 19429. 19429 is too large, use last convergent, return 13/9.
  6. Hrm. That's less able to handle small deviations than I expected - 101/97 differs from 101/97.00001 by about 0.0000001, or about 1/10 the precision I thought it would be able to handle. Also, if you want your algorithm to finish quickly, it might be a good idea to stop it when the denominator gets too large (so you don't get these absurd 10^185 answers). The numerator and denominator can be calculated at each step (as described at https://en.wikipedia.org/wiki/Continued_fraction#Infinite_continued_fractions_and_convergents , storing 2 convergents at a time), which keeps the algorithm quick.
  7. Inverting it only works if the number is a natural number plus the reciprocal of a natural number. Take the 13/9 example I gave above. The generalization that works (see if you have something close to an integer; if not, remove the integer part and try again) is continued fractions, as I described.
  8. So in short, what you are given is a number between 1 and 2, up to about 16 decimal places. You suspect that it is a rational number with relatively small numerator and denominator, and want to find those quickly. My suggestion would be to use continued fractions with a "cut-off" of a large number. For example: Let's say the number we want to find is 13/9 = 1.4444444...., but we have some source of error and instead get x0 = 1.4444387260. Choose a cutoff of 100. Continued fractions would work as follows: a0 = 1, x1 = 1/(x0 - 1) = 2.25002895 a1 = 2, x2 = 1/(x1 - 2) = 3.99953685363 a2 = 3, x3 = 1/(x2 - 3) = 1.00046336097 a3 = 1, x4 = 1/(x3 - 1) = 2158.1446534 2158 is larger than the cutoff, so we stop and assume our number is: 1 + 1/(2 + 1/(3 + 1/1)) = 1 + 1/(2 + 1/4) = 1 + 4/9 = 13/9. This all assumes that your "noise" is relatively small (to be precise, if I'm not mistaken it should be a small fraction of the square of the reciprocal of the largest denominator you allow). If the noise is larger than that, then there will be multiple fractions that "fit".
  9. What, exactly, are you "given"? Just a partial decimal expansion of the fraction?
  10. It's not, and that doesn't answer the question I'm asking. The point is that a fisherman might use this method because of the lack of easier methods. If there are two completely separate shores, determining the shore is as easy as finding which side of a divide you are on. It's only hard because of the continuity between them.
  11. The quote doesn't read that way to me at all; if they were separate shores, why would a fisherman need to look at pebbles to determine which shore he was on? Also, look at a map - Chesil Beach and Bridport are part of one continuous coastline.
  12. For the sixth position: the number there is always divisible by 3, as it can be written in the form 6n + 3 for some integer n. For the distributions of 010000 and 000100 being equivalent: https://en.wikipedia.org/wiki/Prime_number_theorem#Prime_number_theorem_for_arithmetic_progressions
  13. I don't see the word "vector', which is the relevant part of the statement I am asking about, in those posts. You are the one who first used it in the context of whether the Lorentz factor is one. I'll ask in a slightly different way: what point were you trying to make with the statement "Besides that the Lorentz factor is not a vector."?
  14. No, you did. You are the one who said "Besides that the Lorentz factor is not a vector." I'm asking you, who said it, how that statement is relevant.
  15. The Lorentz factor is always taken to be positive. How does the Lorentz factor being a scalar affect this argument?
  16. Whether 0^0 = 1 is a matter of convention, and depends on the context. For example, if the power is held constant and the base variable (as in the case of Taylor series), the convention is 0^0 = 1. If the base is constant and the power variable (but positive), the convention is 0^0 = 0.
  17. I found several more by googling <bijection 0 1 1 infinity> (without the braces), so...
  18. Infinity is usually not thought of as a number; though there are some cases where you can think of it as a number, those cases treat infinity in different ways, meaning that to answer your question, I'd have to ask what you are trying to do with these "numbers".
  19. From what I've seen lawyers saying, that's not what the bill says. Specifically, the relevant portion of the bill says: It says they can't be penalized based on religious content. Not that they can't be penalized for getting the question wrong, or for not answering the question in the relevant way.
  20. sexadecimal. Hexadecimal = base 16; sexadecimal = base 60. Also, time is measured in very mixed base; 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, 365.25odd days in a year, and (mostly) decimal from that point on.
  21. So your dispute isn't with subatomic physics, but with cosmology (and cosmogony) alone?
  22. So is your "theory" a new theory which disagrees with current physics, or is it an explanation of some outcome of current physics?
  23. Do you have any evidence to back up this "theory"?
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