uncool

It may be worth noting, however, that that idea of infinitesimals is not the same as the hyperreal idea of infinitesimals.

And?

I assume you mean the equation for the derivative. The denominator is specifically assumed to be infinitesimal, and 0 is not infinitesimal.

"Nonstandard analysis deals primarily with the pair [real numbers, hyperreal numbers], where the hyperreals are an ordered field extension of the reals, and contain infinitesimals, in addition to the reals." In addition to the reals.

Once again: The fact that the set of hyperreal numbers contains infinite numbers doesn't mean it can't contain finite numbers. Not all hyperreal numbers are infinite. Infinitude is not a "requirement" for all hyperreal numbers. There are finite hyperreal numbers. Some hyperreal numbers are finite.

The fact that the set of hyperreal numbers contains infinite numbers doesn't mean it can't contain finite numbers. Read that definition again: the hyperreals are an extension of the real numbers. The set of hyperreal numbers includes the set of real numbers. The statement "Such numbers are infinite" refers specifically to "numbers greater than anything of the form", not to all hyperreal numbers.

It is a hyperreal number. It's also a real number. Every real number is a hyperreal number (more formally: there is a natural map embedding the real numbers in the hyperreal numbers).

That's not how infinitesimals work. 1/pi is not infinitesimal.

... No, I'm not colorblind. No, I'm not arguing that "anything even has a color". https://en.wikipedia.org/wiki/Lyman_series (specifically, the n = 3 case) It sounds like you thinking of something akin to the Rutherford model of the atom, with electrons physically circling the nucleus; even the standard version of that is a century out of date. Further, "a frequency that is a multiple of a full wave wavelength" doesn't make sense - frequency and wavelength don't even have the same units.

In other words, by completely changing what I said. My argument is not that there is no such thing as color. Electrons don't "have" colors. That would indicate that there is only one color that can be seen emitted by an electron. But, for example (and using a crude approximation) an electron in the 3rd shell can emit photons corresponding to two colors: one for when it drops to the first shell, and one when it drops to the second shell (without the approximation, there are many more colors). The color comes from the interaction of the electron and the nucleons around which it orbits.

...no, that's not what I said. Color is related to photon frequency.

...irrational numbers are not infinitesimals. They are real numbers.

...no matter what idea of "discovery of electrons" you mean, it wasn't done by Newton, and generally was done nearly two centuries after him. A reasonable choice: "In 1897, the British physicist J. J. Thomson, with his colleagues John S. Townsend and H. A. Wilson, performed experiments indicating that cathode rays really were unique particles, rather than waves, atoms or molecules as was believed earlier.[5] Thomson made good estimates of both the charge e and the mass m, finding that cathode ray particles, which he called "corpuscles," had perhaps one thousandth of the mass of the least massive ion known: hydrogen.[5] He showed that their charge-to-mass ratio, e/m, was independent of cathode material. He further showed that the negatively charged particles produced by radioactive materials, by heated materials and by illuminated materials were universal.[5][35] The name electron was adopted for these particles by the scientific community, mainly due to the advocation by G. F. Fitzgerald, J. Larmor, and H. A. Lorenz.[36]:273" https://en.wikipedia.org/wiki/Electron#Discovery_of_free_electrons_outside_matter Further, it's not that electrons have a color. Yes, light is emitted by the photons, but that doesn't mean the electrons themselves have that color. If any particle can be said to have a color (in the "you see it" sense, not the QCD sense), it is the photon.

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.

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.

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?

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?

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.

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.

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.

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".

What, exactly, are you "given"? Just a partial decimal expansion of the fraction?

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.

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.

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