John

Why not? Crows can order things which are cardinally equivelant to certain constructions of simple arithmetic, but that in no way means that they know what numbers are. Numbers are an abstract construction.

 

Comparing cardinality doesn't even require numbers; it merely requires matching.

This is why I mentioned that we have to be careful with what we're asking here. My point is that, regardless of living observers, quantity varies, and as such, numbers exist, as numbers are equivalent to variations in quantity. Consider two rocks. If no life existed in the universe, then two rocks would still be two rocks, more than one rock, fewer than three rocks. The term we use to describe the number of rocks is a construct, but the fact remains that there are two rocks.

I suppose this depends on what we mean by "numbers." Clearly the symbols themselves are constructs, in the sense that we've taken particular shapes to associate with particular quantities. But as for the concept of quantity itself, I wouldn't call that a human construct.

As far as I know, as far as we know, objects within the universe exist regardless of whether life is around to observe them. And by virtue of the fact that objects can move, combine, break apart, etc., the quantity of some arbitrary "single" object within some sufficiently small region of space varies. Numbers describe variations in quantity, and therefore, since quantity exists and can vary, then numbers exist.

Don't get me wrong, I think he term "expert" or "moderator" implies they have a high likelihood of being well versed in a variety of sciences or highly versed in a particular field, but still, opinions are not science. I'm not saying the site is overall bad, I just want there to be a site where everything is really strict and formal as a place people can turn to when they can't get confirmed answers, because as you know both mass media and the internet can have a tenancy to manipulate data, and if this site can't be that site, someone else or I will have to get one designed. This site does a good job for science beginners, but there's many topics I see where if people start talking about too complex of mathematics, no one even attempts to post because no one is experienced enough to confidently answer.

I don't think anyone would argue against these points. The purpose of this site is simply to be a forum (appropriately enough) for science enthusiasts to discuss what they love, and for anyone else to join the conversation as well. The community has gotten quite large, and so some structure is in place; but while SFN is (hopefully) informative and worthwhile, I'm not sure it intends to be what you're envisioning here.

If you do attempt to start the sort of website you're talking about, then I wish you the best of luck. You may even find that some of the most active members of this forum would enjoy such a site as well.

Edit: Just as a note, I should say I've brought up your last point before, though briefly, and I'm sure others have as well. No one wants to feel left out, and we certainly don't want people to feel left out either. How best to go about avoiding that is a pretty big topic in and of itself.

Anyone is free to say anything they like within any set of establish rules, but I suppose if the site wants to risk credibility and efficiency for opinion and appeal I can't stop that.

Indeed. I suppose it could be argued that "Resident Expert" places a certain expectation on the content and quality of these users' posts that is easy to betray, even if it's inadvertent. However, again, I trust the moderators and administrators to handle that if it comes up, and to take appropriate action if it becomes a real problem. The mere possibility isn't enough to damage the credibility of the site, though, so unless the problem is shown to exist, I'm not sure dramatic action is warranted.

Two other things to consider, of course, are a) there is a certain expectation placed on the reader to distinguish between opinion and fact, which doesn't strike me as a bad thing; and b) there is some level of peer review here, such that even a respected member of the community can be called out on statements that seem incorrect. Granted, certain members may be held in such high regard that others are reluctant to speak out against them, but I believe our members are, on average, honest enough to overcome that reluctance if necessary.

A few thoughts:

First, and maybe I'm being a bit silly here, a website with a requirement that new experts be vetted by existing experts may run into a bootstrapping problem. Putting into place a seed population of experts may help with that, of course, but then some or all of that seed population may need to be removed later for the sake of consistency.

Second (and forgive me if I'm wrong, as I've not been active for most of SFN's life), it's my understanding that Resident Expert status is an honor rather than an occupation. The system stems from a desire to recognize those who have demonstrated knowledge with informative and helpful posts, not from a desire to fill those pesky vacant "Resident Expert" positions the admins just refuse to list on Monster. Of course, experts are given some limited powers of moderation, but that strikes me as being a perk more than an obligation.

Third, this place is a science forum, not a free university. Experts, and anyone else, are free to say what they like, within the established rules, based on their best understanding of the topics at hand. Even in a university, it's not as if professors refuse to provide insight or share opinions and speculations during lectures. This is one of the primary advantages of having a knowledgeable person available to assist with learning. As mentioned elsewhere in this thread, a textbook (or, dare I say, a well-cited Wikipedia article) is sufficient if dry information is all that's desired. In any case, I'd trust the powers that be to remove an expert whose posts fell dramatically in quality over time.

Fourth, there are members who, despite having not yet been recognized as Resident Experts, generally make useful posts on a variety of topics. Perhaps more Experts would be nice, but they aren't the only sources of good information here.

Fifth, this thread seems to be getting needlessly hostile.

SamBridge, in light of all that's been said, do you still hold the same views you did in your original post? And if not, what still bothers you about the way SFN is set up currently?

You're welcome.

Don't sweat that last part. Given the kinds of problems you're apparently working on now, it'll probably be covered before too long, and you'll probably find it's a pretty handy operation.

Correct on both counts.

Just to be sure I'm not insane, I also wrote a couple of small programs to find both answers manually, and they agree.

I don't know if you've covered combinations yet, but the reason we multiply by three is, more formally, we want to find the number of ways the middle three digits can include exactly two zeroes, which is equal to [math]9\times{{3}\choose{2}}[/math] (the parenthetical part is read "3 choose 2" and equals [math]\frac{3!}{2!(3-2)!} = 3[/math]). So we choose the first and last digits from the odd digits, giving us [math]5\times5[/math], then multiply that by the [math]9\times{{3}\choose{2}}[/math] mentioned before to arrive at [math]5\times5\times9\times3 = 675[/math].

For number 2, start with the greatest restriction, i.e. the first and last digits must be odd. But remember that the first and last digits can't be equal. Once those are chosen, then look at the middle three digits, keeping in mind that they can't match each other and they also can't match the first or last digit.

For number 3, remember that there are multiple ways to have exactly two zeroes. Your answer only accounts for one of those ways. For instance (though this may be giving too much away), your answer includes the number 34005, but not 30405 or 30045.

I'm taking it to essentially mean changes in where certain organisms live, how large an area they inhabit, etc.

As an example, our climate's current warming trend has resulted in some species appearing in places they've never lived before as they move further away from the equator. Therefore, the geographical extent of these species' habitats has changed over time.

Your own words in two successive posts.

 

 

 

 

A general categorical answer brooking no exceptions.

 

 

 

 

An exception.

You are of course correct there, and I worried after I posted it that someone would respond as you did. Perhaps I should have erased the edit. However, surely the fact that I myself had previously referenced exceptions should have indicated there was an understood "with the exception of the cases mentioned previously." In the context of the OP's post, my claim stands. There are certain areas in which division by zero can be defined (see this for an example in complex analysis, and note that, as with any case in which division by zero is defined, there are caveats), but the OP is tossing infinity into simple arithmetic expressions where it isn't valid. A defined value for division by zero, in the way the OP intended to use it, leads to absurdity. It pains me that I have to spell out what should have been the obvious interpretation, but, Internet forum.

So how should I take this insulting response?

 

Surely the correct response to someone who has a misconception is to say something along the lines of

 

"Division by zero is not defined in the circumstance you are proposing, but can be achieved in certain specialised cases."

 

with or without further amplification.

This is exactly what my posts have claimed. As for how to take the "insulting" response, I would suggest in future posting "you're wrong, and here's why" rather than "you're wrong and you're misleading someone" without any explanation.

I do wish folks would stop being so categoric (and wrong) because it is preventing the OP moving on to understanding what can be evaluated (without calculus at his level).

If you would like to show me how my statement is wrong, please feel free. The fact that there are certain contexts in which infinity and division by zero are treated differently has been covered elsewhere and referenced here. However, the OP is proposing simple mathematical statements that are absurd. Those are what I was addressing.

Your examples near the beginning of the thread do not apply here, and I wish you wouldn't troll while referencing them as some evidence that division by zero is defined.

Okay, I'm only here to learn. Please point out where I went wrong in my mathematics.

In the specific post I quoted, you started with the claim that

[math]\frac{\infty}{1} = 0.000...001 = 0[/math],

which doesn't make much sense, firstly because "a decimal, then infinite zeros with a one at the end" isn't valid, and secondly because [math]\frac{\infty}{1}[/math] would equal [math]\infty[/math] anyway. I'll assume you meant to claim that

[math]\frac{1}{\infty} = 0[/math],

which still isn't valid, but let's assume it is. Then we can manipulate this expression to show that

[math]1 = 0(\infty)[/math].

That's all well and good, but let's take a real number [math]n[/math] and do the following:

[math]1(n) = 0(n)(\infty)[/math].

This means

[math]n = 0(\infty) = 1[/math],

but clearly not all real numbers are equal to 1.

Your mathematical error lies with treating infinity as just another number (when it isn't). You can't just toss infinity into an equation and pretend it's just some really big real number. Now, it is true that, approaching zero from the right,

[math]\lim_{x\to0^{+}}\frac{1}{x} = \infty[/math],

but that doesn't mean [math]\frac{1}{0}[/math] is infinity (especially since, approaching from the left, the limit is [math]-\infty[/math]). It means that as x approaches (but never reaches!) 0, the value of [math]\frac{1}{x}[/math] increases without bound. It may seem like a subtle difference, but it is very important.

Edit: To answer your more general question, division by zero is not defined because any definition we give it results in logical absurdity like the above.

As a note, 0.999... = 1.

As for the rest, as others have noted, division by zero isn't defined and standards for handling computer arithmetic are based on the limitations of computer hardware and don't overrule mathematical theory.

Dividing by zero is not defined.

 

Yes it is according to my simple calculation.

 

infinity/ 1 = infinitesimal or 0.000recurring 1

So the inverse of that is infinity

 

Proof of infinitesimals being rounded can be proved by this.

 

x = 0.999recurring

 

10x = 9.999recurring

10x - x = 9x

9x = 9

9x/ 9 = 1

x = 1

 

 

 

It's something I have worked out myself but it has been confirmed. Apparently it was actually some of the early work of Sir Issac Newton!

You can't treat infinity as just another number, because it isn't. As discussed in a recent thread, there are instances in which infinity is added to the real number line, but even then infinity isn't a number and has to be treated carefully.

Relevant Wikis:

http://en.wikipedia....ivision_by_zero

http://en.wikipedia.org/wiki/.999...

For your first assumption, a distribution of 32 and 18 in the second population leads to much nicer numbers than (32 or 42) and 8, so you're probably correct there. Given the wording of the question, your other assumptions are probably fair too. I also don't see any errors in your calculations or reasoning, except that

Because each of the blue-eyed individuals has 2 copies of the b allele' date=' and because the total number of people on the planet is 50 + 50 = 100, the frequency of b homozygotes must be 2(18) / 100 = 36 / 100 = 0.36.[/quote']

is a bit unclear to my eyes. If you look at the frequency of homozygous recessive individuals with respect to the total number of people, you get .18, not .36. However, I assume this isn't what you meant, since it doesn't mesh with your later addition of 100% B frequency on the first ship with the lower B frequency on the second ship.

Unless I'm blind and/or crazy, your final answer is correct.

I don't post much. When I first joined, I was something like 18, and at that time I was a very quiet person in real life and a bit more outgoing online in general.

Since then, I've become somewhat more sociable in real life and less sociable online, so I'm probably about the same here as I am in person in that regard. I will say I try to be a bit more careful with what I post on SFN nowadays, though, probably because I realize that in this community there are many people who know quite a bit more about things than I do (that wasn't necessarily the case ten years ago) while in real life I'm generally regarded as a fairly intelligent and knowledgeable person.

The IS Theory wiki is now hosted by BYU at http://istheory.byu.edu/wiki/Main_Page. There is a link on the original page to the new Wiki, but I figured a direct link from this thread would be handy.

I would have just added this to my previous post, but for whatever reason (maybe browser compatibility) I'm unable to edit the thing. I just came across this:

http://www.amazon.co...ch_tsr_rpsy_alt

which might be worth browsing. It's a collection of books recommended for learning various mathematics.

Infinity is not the multiplicative inverse of zero. Also, I think you're slightly misunderstanding how the expressions at the end of your post are used.

You're obviously intelligent, and it's good that you're thinking about these concepts rather than, for instance, obsessing about some reality TV show. But you have to tread carefully when dealing with infinity, and you seem to be making a few missteps.

John (edit: sorry for the broken tex quotes below, not sure why it does that!)

No worries. Also, as others have said, welcome to the forums.

As I said, let's assume that there exists a valid, numerical value for infinity (but not call it necessarily a real number, more like a special value in an extension of the RNS). What happens when we add 1 to infinity? Or 2? Or 3?

 

Well, of course this would make no sense in the, should I say, standard view of mathematics, but let's say it still remains infinite. Because infinity plus 1 should still be infinite. And we can apply this to all real numbers (again + and - infinity are excluded and treated as valid values exceptional properties).

 

 

I defined [math]n\pm\infty=\pm\infty, -\infty<n<\infty[/math] where the positive/negative are independently respective.

Why bother with all this?

For the finite real numbers, any number times 0 is zero, but infinity's different in this case, since remember that [math]0\times n=n-n[/math] making [math]0\times\infty=\infty-\infty[/math], which would be satisfied by any real number (again, excluding the special values of +/- infinity).

This strikes me as self-contradictory, as you've essentially said a) infinity is different from the real numbers, b) infinity follows the same pattern as a real number n, then c) but infinity won't follow this pattern.

Yes, sorry for that. I was referencing to the property I defined in the beginning... I meant [math]n\in\mathbb{R}[/math]. And I don't find this to in any way demote the credibility of my (quite unorthodox) proof; I hope you don't see it like that.

Not at all. I pointed it out in case you were intending some special meaning that I didn't see.

0 is a satisfying value, but as I showed above, it could also be 1, -1, googol, e, pi ... in an indeterminate form, like 0/0.

I see how you came to the conclusion you did based on your clarification of your intentions, but what do you mean here, i.e. what does something like "e in an indeterminate form" mean?

When I made that statement, I was referring to the idea that "infinity is merely a concept, and makes no sense in arithmetic, like 5+justice". I just meant that we should accept it (for the purpose of this proof) with a valid mathematical meaning.

 

I didn't necessarily mean to include it with the real numbers in the sense that it would have the same universal properties of real numbers. I do hope that it would be seen as a real number in this proof (so it would seem comparable to the finite reals and not abstract), but of course with different rules.

Are you saying you were basically going for a proof by contradiction to show that [math]0\times\infty\neq1[/math]?

I'm pretty sure it's not about "lack of interest". I think any mathematician would be exceedingly interested to expand infinity to areas of math where it is forbidden (in a figurative sense).

As mentioned previously, there are areas in which division by zero can be defined, but in these cases one has to be careful with exactly how the operation is defined, and in what cases it's valid.

And yes, it's the contradictions. [math]0\times1=0\times2[/math], by "simplifying out" the zero you would get 1 = 2. Which, since 0 times any real number is 0, would make every number in the real number system equal to each other, causing a collapse of mathematics so disastrous as to cause an annoying internet meme of nuke-like explosions, bottomless pits in the ground, and black holes.

 

Really, I don't think that argument is very convincing, since pretty much anything involving zero and an arbitrary value also involves indeterminate forms. It's a given that you will get different values equal to each other if you try reducing the equation. It's as valid as this...

 

[math]x^2=9[/math]

 

If I square root both sides, I'll get roots of 3 and -3. Does that mean I could say "Well, that would make 3=-3. Meaning every real number is equal to its negative. That doesn't make sense, so square rooting is undefined.

It can be shown that squaring either 3 or -3 will result in 9. It's obvious, however, that 1 does not equal 2. "Simplifying out the 0" in this case amounts to dividing both sides of the equation by 0, which is not valid as the operation is undefined.

In fact, this argument can be applied to any equation where two different values share a common function. That does not make them equal though, most especially in the case of indeterminate forms.

The argument can be applied, yes, but it's an invalid argument, which of course is the whole point.

[math]0\times n=n-n[/math] by the iterative definition of multiplication. True for all numbers.

 

Assume the existence of a valid, numerical infinity, as well as obvious properties like [math]n\pm\infty=\pm\infty[/math] positive/negative respectively. True for all real numbers.

If infinity is supposed to be some real number, then [math]n\pm\infty = \pm\infty \implies n = 0[/math].

Now we want to find the value of [math]0 \times \infty[/math]. Let's call it [math]x[/math]

If [math]\infty[/math] and [math]x[/math] are real numbers, then [math]x = 0 \times \infty \implies x = 0[/math].

[math]0\times \infty=\infty - \infty[/math]

 

[math]x=\infty-\infty[/math]

 

[math]x+\infty=\infty[/math]

 

[math]x\in\mathbb{R}[/math], since we inductively established the postulate [math]n+\infty=\infty, x\in\mathbb{R}[/math].

Mhm. Notationally, [math]n+\infty=\infty, x\in\mathbb{R}[/math] strikes me as a bit odd, since the fact that x is a real number doesn't have much bearing on equations not involving x.

In conclusion, [math]0\times\infty=\mathbb{R}[/math] indeterminate.

Given the assumptions you've made so far, again, [math]0 \times \infty = 0[/math].

Demonstratur ut putantur. Of course, this is only valid by establishing new grounds. Infinity itself isn't even a number in most cases, so why not go this far? Still, it's all meaningless without formally accepting infinity and solving the consequential contradictions we run into. And that would be pointless, since it wouldn't really help us in any way.

I've not studied very advanced mathematics, and so someone better educated may find problems with my reasoning above. In any case, as far as I know, systems in which division by zero is defined have slightly different definitions of the division operation. You, and maybe mississippichem based on his reply, might find http://en.wikipedia....l_number_system interesting.

Edit: This post was made assuming that by "assume the existence of a valid, numerical infinity," you mean "include infinity in the set of real numbers."

Here are a few titles that spring to mind, though most are admittedly ones I haven't read, instead being included based on recommendations from many others more experienced than I. I'll add more over time, probably.

General mathematics:

• Richard Courant and Herbert Robbins, What Is Mathematics?
• Keith Devlin, The Language of Mathematics: Making the Invisible Visible

Proof and logic:

• Daniel Velleman, How to Prove It: A Structured Approach

Calculus:

• Michael Spivak, Calculus
• Tom Apostol, Calculus vol. 1 and Calculus vol. 2
• Richard Courant and Fritz John, Introduction to Calculus and Analysis I and Introduction to Calculus and Analysis II (the latter is divided into two volumes, the second of which is here)
• J.E. Thompson, Calculus for the Practical Man (Richard Feynman apparently taught himself calculus from this book)

I also have and somewhat enjoy a few of Debra Anne Ross' Master Math series, though they're more reference books than textbooks.

The most groups of five that can be covered by a set of 10 is [math] \frac{10!}{5!5!} =252 [/math]. As the total number of groups of 5 numbers is [math] \frac{20!}{15!5!} =15504 [/math] you need at least 62 groups of 10 if your coverage is perfect - whether that is possible I do not know yet.

That sounds more like the right track. I guess for whatever reason I had it in my head that the ten-element sets would also be random but unique (though I'm still not sure my proposed strategy is correct in that case--haven't done a whole lot with combinatorics).

If I'm understanding the question correctly, then the first person chooses five numbers of the twenty. The second person then builds unique sets of ten numbers from the same twenty. The question is how many ten-element sets the second person would need to choose to ensure that at least one contains all five numbers from the first person's set.

It seems to me the solution is to figure out how many sets can be created without using all five of the first person's numbers, then add one.

Here's a more simply worded proof:

http://www.remote.org/frederik/projects/ziege/beweis.html

and a Java applet to simulate as many games as you like:

http://www.remote.org/frederik/projects/ziege/empirie.html

Xittenn, this isn't some competition. I, and everyone else, would be just as happy if the probability did turn out to be 1/2. However, you have been shown in many different ways (including intuitive explanations, mathematical proofs, and simulations) that the probability of winning is 2/3 if the contestant switches, and 1/3 if he sticks with his original door. You've always seemed fairly intelligent, so I'm sincerely not sure if you're just messing with us now or if you've fallen into the classic trap of being too attached to what you think the answer should be. If the former, then good job. If the latter, then perhaps we've all been posting in vain.

As for the whole "it's not a probability puzzle" bit, that doesn't really matter. If we're seeking to decide what the contestant should do (i.e. what the correct action is), then, assuming the contestant wishes to win a car, he should switch. This is based on the probabilities involved.

That's what I originally thought too, but there's something about the modulus that would make it 11, which would set it back to 1 since it exceeded 10 by 1. 10+1 mod 10 =1?

Perhaps I'm thinking of a specific setup, too specific,

You could do (1+1) mod 2, but that's just 0. 1+1 in binary is, in fact, 10.

And yes, (10+1) mod 10 = 1.