wtf

I haven't followed this thread so I don't know the context of your paste of an excerpt from Keisler's 1977 or so book on NSA. What you wrote here is perfectly true, since both the hyperreals and the standard reals are models of the first-order theory of the reals. But in order to construct the hyperreals, you need a gadget called a non-principal ultrafilter on the natural numbers. Such a thing exists only in the presence of a weak form of the axiom of choice. So the logical principles needed to build the hyperreals exceed those needed to build the standard reals. Secondly, the hyperreals do not satisfy the least upper bound property, because they are non-Archimedean. Third, Keisler's book is not about research, since the hyperreals were first constructed by Hewitt in 1948 and nonstandard analysis was developed by Robinson sometime afterward. Keisler's intent was to write an NSA-based textbook for freshman calculus. It's telling that in the 44 years since then, no other similar books have been written; and calculus is still overwhelmingly taught in the traditional manner based on limits. There are occasional NSA-based calculus courses given, and studies show that by and large, students come away just as confused about NSA-based calculus as they do from traditional calculus. So we see that (1) NSA offers no pedagogical advantages (else more schools would have adopted it since 1977 and more texts would have been written); (2) NSA requires a strictly stronger logical foundation than the standard reals, namely a weak form of the axiom of choice; and (3) the hyperreals lack the fundamental defining property of the standard reals, namely the least upper bound property. As I say I'm not sure what your point is in pasting this excerpt so I can't comment on that. I'm just mentioning some context for NSA that you should know about if you care about NSA or wish to make some point based on it. Some light background reading of interest: https://en.wikipedia.org/wiki/Transfer_principle https://en.wikipedia.org/wiki/Ultrafilter https://math.stackexchange.com/questions/1838272/why-do-we-need-ultrafilter-for-construction-of-hyperreal-numbers https://en.wikipedia.org/wiki/Least-upper-bound_property https://en.wikipedia.org/wiki/Archimedean_property

Darn, I'm busted.

Coursera has a course. It's all video so you can sign up and watch any time you like, there's no schedule as far as I know. https://www.coursera.org/learn/complex-analysis

Nothing personal, I just happened to have run across this exact issue on some other forum a day or two ago so it was fresh in my mind.

I wish to refine this statement because it's a common point of confusion. You have no proof that "At every point on the curve of the function, you can draw a tangent line, such that 1 point ( only ) is common to both," nor do you have a rigorous definition of what a tangent line is. Rather, we have an INTUITION about what a tangent line is. In order to make the notion rigorous, we DEFINE the tangent line at a point to be the straight line passing through that point with slope equal to the derivative at that point, if the derivative exists. That is, the the slope of the tangent line is NOT "equivalent" to the derivative; rather, it's DEFINED that way. The idea is to make precise the intuitive idea of the tangent line at a point. If you think (as students often do) that the derivative is "the same" as the slope of the tangent line, that's a misunderstanding of what's going on. There is no tangent line, formally, until we define it via the derivative. Then (for example) we can make rigorous the intuitively clear observation that the graph of |x| has no tangent line at 0. Otherwise, we could have no proof, since without the derivative we have only an intuitive but not a rigorous notion of tangent line.

I have not given this any detailed thought since weeks ago and, to the minimal extent I've thought about this recently, I agree that you're right. The totient function gives the same answer as the inclusion/exclusion principle for this problem.

I think you're right that inclusion-exclusion and Euler totient give the same result for your problem.

If they give different answers then clearly at least one (or possibly both) are inaccurate. But my statement is just a guess and not based on thinking about the problem much. You're probably right.

Today I learned! Another way is math]\displaystyle e^z = \sum_{n=0}^\infty \frac{z^n}{n!}[/math [math]\displaystyle e^z = \sum_{n=0}^\infty \frac{z^n}{n!}[/math]

Can you help me to understand your point? For example if I write 3 + 4 x 5 = ?, everyone agrees that the anseer is 23; because by convention we multiply first and then add. But that's just a convention. If we all agreed to add first then multiply, the answer would be 35. It's a purely arbitrary convention that could easily be different, as long as everyone agrees on what the convention is. As you know, the Internet is full of order-of-operations puzzles that are the result of poor math education regarding operator precedence. If it makes you happy, 1 isn't a prime because that's the convention. We could make a different convention, regard 1 as prime, and adjust all the theeorems accordingly. It really makes no difference. So what is your core issue or concern? It's just a convention. The reason for the convention can be motivated by higher algebra, but in the end it's still a convention and truly makes no substantive difference. It's similar to why we go on green lights and stop on red. At the dawn of the automobile age we could have adopted the opposite convention. It makes no difference as long as everyone agrees.

I was going to explain that the reason 1 isn't prime is that 1 is a unit in the ring of integers. However, the SciAm article by Evelyn Lamb that you linked already has a thorough discussion of this point. If it doesn't satisfy you then perhaps nothing will; because that's the deepest reason 1 isn't a prime. It's because it's a unit, meaning that it has a multiplicative inverse. A ring is a system of numbers in which you can add and multiply, and there are additive inverses, but not necessarily multiplicative inverses. The integers are a ring because, for example, 5 doesn't have a multiplicative inverse. In the rationals 1/5 is the multiplicative inverse of 5 but 1/5 isn't an integer. In a ring, an element that does happen to have a multiplicative inverse is called a unit. In the integers, 1 and -1 do happen to have multiplicative inverses. In fact in each case they are their own multiplicative inverse. It doesn't make sense to ask if a unit is a prime. for reasons that Ms. Lamb goes into. I suggest re-reading that part of the article. For reasons of simplicity for SciAm readers, she gives the more general definition of prime number but can't use the word ring or the term prime ideal, which is really what's going on here. When she says, "Specifically, one important change was the development of sets of numbers beyond the integers that behave somewhat like integers." she is talking about rings. And the concept that defines what a prime is, is called a prime ideal. https://en.wikipedia.org/wiki/Prime_ideal @studiot, The article mentions Eisenstein's criterion, so that's related. If you consider the multiples of 5 in the integers, namely the set {0, -5, 5, -10, 10, ...}, it has the property that if the product xy of two integers is in that set, then at least one of x and y must be in the set. If you consider the multiples of 6, namely {0, -6, 6, -12, 12, ...} it's possible for the product xy of two integers to be in the set yet neither of x and y is; for example, 3 and 4. Such sets are called ideals. The actual definition is more general and I'm simplifying too, but not as much as the SciAm article did. So the multiples of 5 are a prime ideal and the multiples of 6 aren't. Part of the definition of a prime ideal is that it can't be all of the original ring. But the set of multiples of 1 is all of the integers. So the ideal generated by 1 can't be a prime ideal. But again that's just because of the definition ... which is what you are unhappy about! So perhaps this will still not be satisfactory to you. The other reasons people gave here are good too. Unique factorization doesn't work with 1 as a prime unless you add phrases to qualify it. But really, if you didn't find the SciAm article satisfactory, I don't think there's going to be a better explanation. The real answer is that when we generalize the integers to rings, and we want to define what primes are, we do so in terms of prime ideals. And prime ideals don't include the multiples of 1 because the ideal generated by 1 (or any other unit) is the whole ring. And now we're into abstract algebra. But again, the SciAm article actually explains all that without using the words ring or prime ideal. I think you should just read the article again because it contains the answer you're looking for.

A few weeks ago I had the problem space mapped into my brain for an evening. Too busy to remap it at the moment, so probably won't be able to get back to this. Off the top of my head I don't think the totient function will always give the same answer as inclusion/exclusion, but I could be wrong and probably am. Sorry I can't offer more assistance.

@Tinacity, ps -- Wait DUH! We forgot 11/49. I don't know why we both got confused about this. I solved the problem. The trick is that if n is divisible by one of 2, 3, or 5, so is 60 - n. So the pairs (n, 60-n) where both elements are relatively prime to 2, 3, and 5 are exactly the same as the numbers n with the same property. So the solution is to do inclusion/exclusion on 30 to determine how many numbers are not divisible by 2, 3, or 5; and that's the number of pairs. In the case of 60 there are exactly 8 pairs: 1/59, 7/53, 11/49, 13/47, 17/43, 19/41, 23/37, and 29/31. That's eight. You can now write a program to do inclusion/exclusion on your original number N, or half of 2N if you think of it that way (that is, 2x3x5 = 30, multiply by 2 to get 60, then do inclusion/exclusion on 30). The "sum to 60" is a red herring, an aspect of the problem that adds confusion but doesn't change the problem. The number of pairs that sum to 60 where each element of the pair is not divisible by 2, 3, or 5, is exactly equal to the number of numbers between 1 and 30 not divisible by 2, 3, or 5. And this result generalizes under the conditions of your problem. To do inclusion/exclusion on N = 2*3*5*7*11*13*17*19*23 you take: - The sum of all 8-fold products of N (that is, every combination of 8 factors at a time); - Minus the sum of all 7-fold products; - Plus the sum of all 6-fold products; etc. You then subtract the final sum from N, and that's the number of pairs where both elements are relatively prime. I believe that's it, but if I messed up I hope someone will jump in.

Is this a matter of being concerned that someone will steal your idea? If you post your actual problem perhaps someone can offer some help. For what it's worth, here's what I did with inclusion/exclusion. Suppose we want to know how many numbers from 1 through 30 are not divisible by any of 2, 3, or 5. We calculate how many ARE divisible by at least one of them as follows; 1/2 x 30 = 15 1/3 x 30 = 10 1/5 x 30 = 6 That adds up to 31. Now for the double counts, which must be subtracted: 1/6 x 30 = 5 1/10 x 30 = 3 1/15 x 30 = 2 That adds up to 10, to be subtracted. Now we must ADD back the triple counts: namely, 1/30 x 30 = 1. So we have 31 - 10 + 1 = 22. Therefore there are 30 - 22 = 8 numbers NOT divisible by any of 2, 3, or 5. Indeed we can count them by hand: 1, 7, 11, 13, 17, 19, 23, and 29. Eight as calculated. Now the problem is that we have not accounted for the pairs 29/31, or 23/37, etc., because the larger numbers of the pair are out of our range. So if you figure out how to account for the "sum to 60" aspect of the problem, you'll be able to work this out. Do feel free to give more information about your actual problem, or not, as you see fit. Then again when Hilbert offered to help Einstein with general relativity, Einstein at first welcomed his offer; but then realized that Hilbert was trying to solve the problem first and take credit. So maybe you're right not to give too much away! LOL. ps -- Wait DUH! We forgot 11/49. I don't know why we both got confused about this. I solved the problem. The trick is that if n is divisible by one of 2, 3, or 5, so is 60 - n. So the pairs (n, 60-n) where both elements are relatively prime to 2, 3, and 5 are exactly the same as the numbers n with the same property. So the solution is to do inclusion/exclusion on 30 to determine how many numbers are not divisible by 2, 3, or 5; and that's the number of pairs. In the case of 60 there are exactly 8 pairs: 1/59, 7/53, 11/49, 13/47, 17/43, 19/41, 23/37, and 29/31. That's eight. You can now write a program to do inclusion/exclusion on your original number N, or half of 2N if you think of it that way (that is, 2x3x5 = 30, multiply by 2 to get 60, then do inclusion/exclusion on 30). The "sum to 60" is a red herring, an aspect of the problem that adds confusion but doesn't change the problem. I believe that's it, but if I messed up I hope someone will jump in.

If order matters then you have 14 + 16 and 16 + 14 and likewise for all the other pairs except for 15 + 15, which is it's own reverse. Do you mean order doesn't count? Rather than saying it does? What happened to 1/59? I had a couple of mistakes in my own list, I had 11/49 which shouldn't be there, and I forgot 7/53. So there are 7 such pairs, 14 if order matters. I assume by your example that order DOESN'T matter. Still we have 7 pairs 1/59, 7/53, 13/47, 17/43, 19/41, 23/37, and 29/31. That's seven. I think we have them all. The question comes down to taking 60 and asking, out of the first 30 positive integers, how many are divisible by at least one of 2, 3, or 5. Half are divisible by 2, 1/3 are divisible by 3 but we counted the ones divisible by 6 twice; and 1/5 are divisible by 5 but we counted the 10's and the 15's twice. But then we subtracted the ones divisible by 30 once too much so we have to add it back in. I believe you attack this kind of problem with the inclusion/exclusion formula. In fact the first example here shows how to count how many numbers from 1 to 100 are divisible by at least one of 2, 3, or 5, our exact problem here. https://en.wikipedia.org/wiki/Inclusion–exclusion_principle I jotted down some numbers but I was off-by-one somewhere, which doesn't surprise me.

Can you show your work in detail? I still don't understand the basic question. 2*3*5 = 30. Multiply that by 2 and you get 60. Please explain the rest because I totally do not understand what we are doing here. Where did the -2's come from, that hasn't been part of your exposition. ps -- Ok I totally don't get this. Pairs that sum to 60 and have no divisors among 2,, 3, 5: I get 1,59 11,49 13,47 17, 43 That's already four, eight if you distinguish order, and there are plenty more. So please explain clearly what you are doing. Others are 19, 41 23, 37 29, 31 That's a total of 7, times 2 to account for order as you said earlier, so there are 14 pairs that satisfy your requirement, not 3. Where do these -2's come from? In one example earlier you had 17-2 as a factor but that's not prime. Maybe it's just me but I do not understand what is being calculated. Can you work out a complete example, a simple one? Apologies if I'm being dense and this is obvious to everyone but me. Those of you who wrote programs to solves this problem, what problem are you solving? Am I just missing something that's obvious to everyone else?

Have you tried this out by hand for a simpler case, say N = 2 x 3 x 5 = 30? Then dropping the first and last gives you 3. Can you make your idea come out with that example? Is there something special about the case you're presenting?

Haven't followed the thread but this seems a little ambiguous. Let's take a simpler example, N = 2 x 3 x 5 x 7 = 210. Now you want to consider the pairs of numbers that sum to 210, such as (1, 209), (2,208), ..., (209, 1). Do you care about order? Is (1,209) the same as (209,1) or different? Not a big issue, just a factor of 2, but nice to know what is the intended intepretation. Now "... where neither is divisible by any of the primes which make its product?" was confusing to me. What is "its" in this context? Do you mean that since 209 = 11x19, and neither 11 nor 19 is one of 2, 3, 5, or 7, we count (1,209) and (209,1) as satisfying your condition? And you want to count the number of such pairs? Just want to make sure I'm understanding the question. Apologies to all if this has already been covered in the thread.

The original interpretation is ordinal exponentiation. Here's an overview. If we "keep counting" after 0, 1, 2, 3, ... we can tack on an element after all those, called [math]\omega[/math]. So we have the ordered set 0, 1, 2, 3, 4, ..., [math]\omega[/math]. Now we keep on going: [math]\omega + 1, \omega + 2, ...[/math] and when we're done with all those we've reached [math]\omega + \omega[/math] or [math]\omega 2[/math]. Ordinal multiplication is notated backwards, [math]\omega 2[/math] means 2 copies of [math]\omega[/math], one after another, and NOT [math]\omega[/math] copies of 2, which would actually be equal to [math]\omega[/math]. [Exercise for you: figure out why [math]2 \omega = \omega[/math]. You have to learn about ordinals. This is not an easy exercise until you have worked through the elementary material on what ordinals are]. This example shows that ordinal addition is not commutative. As a historical note, I tracked this backward notation down once. It turns out that Cantor himself went back and forth on whether to call [math]\omega + \omega[/math] [math]\omega 2[/math] or [math]2 \omega[/math]. In the end for whatever reason he went with the notation that I consider backwards. But like everything else in math, you just get used to it. So now we have [math]\omega 2, \omega 3, \omega 4, ...[/math], and at the end of that we tack on [math]\omega \omega[/math] or [math]\omega^2[/math]. Then we have [math]\omega^3, \omega^4 [/math], and we keep on going till we get [math]\omega^\omega[/math]. If we keep on going we get [math]\omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}}[/math], and so forth. If we keep this process going, at the end we tack on a countably infinite power tower of [math]\omega[/math]'s. This infinite power tower is given the name [math]\epsilon_0[/math], and it is the smallest ordinal [math]\epsilon[/math] such that [math]\omega^\epsilon = \epsilon[/math]. Exercise: Can you see why this equation is true of [math]\epsilon_0[/math]? This "tacking on at the end" idea can be formalized by letting the limit ordinal, as it's called, simply be the set of everything that's come before. So if the natural numbers are 0, 1, 2, 3, ..., then we define [math]\omega = \{0, 1, 2, 3, \dots \}[/math]. These are the von Neumann ordinals. https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals. By the way I recommend that you read this entire Wiki page to begin to learn about the ordinal numbers. It's important, and mind boggling, to realize that [math]\epsilon_0[/math] is a countable ordinal; that is, it's a countable set. The countable ordinals are very strange and hard to get one's mind around. It's an amazing theorem that every countable ordinal can be embedded in some subset of the rationals in their usual order. This is the subject of the theorem you are interested in. This is what it's about, not something else. You can be interested in something else if you like, but if you are interested in this particular idea, this is what it's about. https://en.wikipedia.org/wiki/Epsilon_numbers_(mathematics) Interestingly in physics, the notation [math]\epsilon_0[/math] refers to something called the vacuum permitivity, but that has nothing to do with what we're talking about here. I don't expect you to understand all this right now. It's taken me a long time to get even a little bit of understanding of [math]\epsilon_0[/math]. But if you want to learn this material, you have to make an attempt to learn it, starting with understanding what ordinal numbers are. Otherwise you are having a vacuous conversation with yourself.

It's ordinal exponentiation in this case. You need to start with learning about ordinal numbers and then ordinal arithmetic, working up to exponentiation.

Best way to start would be to take the time to understand relativization then.

This refers to ordinal exponentiation, it's not relativization. https://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiation

1 x 1 = 1. That's the multiplication table for the nonzero elements. What's the problem?

0 isn't invertible in the real numbers either, but the reals are a field. 0 never has a multiplicative inverse. Write down the addition and multiplication tables for [math]\mathbb Z_2[/math] and you'll see it's a perfectly good field. Remember that 1 = -1, that's a misunderstanding you had earlier. 1 + 1 = 0 so 1 is its own additive inverse.

Who's rarified now LOL.