Wikipedia:Reference desk/Archives/Mathematics/2011 January 5

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January 5

Special magic squares

I'm reading a book that has a chapter on magic squares, and it gives the following special magic square:

 5 22 18
28 15  2
12  8 25

If you write the names for the numbers out in English and then count the number of letters in each name, you get another magic square:

 4  9  8
11  7  3
 6  5 10

It then says that, for totals of less than 200, English has seven of these squares, while French only has one. What are they? --75.60.13.19 (talk) 03:08, 5 January 2011 (UTC)

non-trivial irreducible character of primitive permutation character is faithful?

Hello,

I am struggling with what should be an easy exercise...

let G be a group acting primitively on a set $\Omega$. Let $\chi$ be the permutation character( let's say over $\mathbb {C}$). Hence $\chi$ maps every$g\in G$ to the number of elements in $\Omega$ it fixes.

Let$\chi _{1}$ be any irreducible constituent of $\chi$, different from the trivial character. Prove that $\chi _{1}$ is faithful (i.e. the corresponding representation of $G$ maps only the trivial element of $G$ to the trivial matrix).

I think it should be easy, but it appears I am missing a crucial observation. I already know why primitivity is important, because otherwise the non-faithful permutation character on the blocks of imprimitivity would be contained in the character.

Groups of order 24

Do all the 15 groups of order 24 have a subgroup of order 12? How can I prove or disprove this.-Shahab (talk) 09:41, 5 January 2011 (UTC)

Take a look at the Sylow theorems article. As the article says, they are "a collection of theorems... that give detailed information about the number of subgroups of fixed order that a given finite group contains.". Since 24 = 2³⋅3, Burnside's theorem tells us that G is solvable. There's lots more information to be had in the Solvable group article. — Fly by Night (talk) 15:48, 5 January 2011 (UTC)

I think that the semidirect product $G=C_{3}\ltimes Q_{8}$ has no subgroup of order 12, where Q₈ is the quaternion group, and C₃ acts on Q₈ by cyclic permutation of i, j, k. Any subgroup of G of index 2 would be normal, hence it would induce a nontrivial homomorphism f: G → C₂. However, f vanishes on C₃ (as it has order 3) and on at least one of i, j, k (as ij = k), and therefore on all of i, j, k (as they are conjugated in G, and C₂ is abelian). Thus f is trivial, a contradiction.—Emil J. 16:00, 5 January 2011 (UTC)

This is a central extension of A₄ by C₂. A₄ has no subgroups of order 6 and this implies that the extension has no subgroups of order 12. This is also the only group of order 24 that has no subgroup of order 12. I got this by checking generators and relations for all 15 groups, these are well known, but I suppose it wouldn't be too hard to prove it from scratch using Sylow etc.--RDBury (talk) 16:54, 5 January 2011 (UTC)

1+1=2

What were the definitions and axioms used in Principia Mathematica that required such a long proof of 1+1=2 and other basic arithmetic facts? A link to a page or website describing the foundations is sufficient; I just don't know where to find such a reference. --24.27.16.22 (talk) 15:54, 5 January 2011 (UTC)

At the very end of that page about Principia Mathematica you can find proposition 54.43 which is that 1+1=2 validated using a modern theorem checker Metamath. Dmcq (talk) 16:06, 5 January 2011 (UTC)

Qualitatively speaking, the issue is that the axioms are very low-level. For instance, 1+1=2 does not require proof if you are using the Peano axioms. But if you don't want to do that, you could start with Zermelo_Fraenkel_set_theory and derive the Peano system from there. This would be a much longer proof than that quoted on the PM page. Both Peano and ZFC schemes pre-date PM, and Whitehead even used much of Peano's notation. I think that PM axioms are lower-level / weaker than ZFC, but hopefully someone else can elaborate on that. I can't easily find an short itemized list of axioms in the PM, but you can download the whole book through google books here [1]. SemanticMantis (talk) 17:03, 5 January 2011 (UTC)

If we start with ZFC, and identify natural numbers with finite cardinals and define addition as a disjoint union of sets (using some silly trick like A+B = (Ax∅)∪(Bx{∅})), would a complete formal proof of 1+1=2 require a similar length? --24.27.16.22 (talk) 17:49, 5 January 2011 (UTC)

No, but that's an awful lot of "if"s. Russell and Whitehead were working at a time when there was no settled notions of things such as a logical theory, and so they had to build up everything from the very ground. Many tricks and tools that would be used in a modern work to simplify and streamline the presentation simply hadn't been invented yet (or had been used once or twice but their general applicability not appreciated). Furthermore, the theory they designed is/was markedly more cumbersome to work with and reason about than ZFC, and as a result didn't really catch on.

Also, nobody's saying that all of the 360 pages that preceded *54.43 were necessary prerequisites to that particular proposition. As a comparison with a newer text, Mendelson Introduction to Mathematical Logic (4th ed., 1997) reaches page 258 before it defines "$X{+_{c}}Y\mathrm {~for~} (X\times \{\emptyset \})\cup (Y\times \{1\})$". However, that includes 70 pages that develop formal number theory from the Peano axioms and are not used for the axiomatic set theory, so "number of pages before addition is defined" is not really a meaningful metric. –Henning Makholm (talk) 22:08, 5 January 2011 (UTC)

Does anyone know any source where real everyday mathematics is developed formally in ZFC? By real math I mean calculus, algebra, analysis etc. Money is tight (talk) 23:22, 5 January 2011 (UTC)

http://us.metamath.org/mpegif/mmset.html 67.122.209.190 (talk) 08:50, 6 January 2011 (UTC)

What constitutes a valid solution?

One of my profs made an interesting remark today. He was doing an example on the board, and the answer came out to $x={\sqrt {2}}$. But he stopped, and said that he didn't feel this was a complete solution, and that the fullest answer possible would be to say that $x=1.4142...$.

As an example, he said, suppose we had to solve the equation $x^{3}=4$. Saying that $x={\sqrt[{3}]{4}}$ is just tautological. The only meaningful solution would be to write out cube root of 4 as a decimal expansion.

At first I agreed, but thinking about it now, I'm not so sure. A decimal expansion of a number is, by definition, the expression of the number as an finite or infinite series of fractions with a denominator of a power of ten. How is this any more valid of an answer? —Preceding unsigned comment added by 74.15.138.87 (talk) 16:56, 5 January 2011 (UTC)

What constitutes a "complete solution" in mathematics is somewhat arbitrary, and most professors would be perfectly happy to accept $x={\sqrt {2}}$ in my experience, since that is the most precise answer one can provide. It's really not clear exactly what your professor is looking for in a complete solution, except possibly that he expects final answers to be written out in decimal notation? --COVIZAPIBETEFOKY (talk) 17:14, 5 January 2011 (UTC)

√2 is exact, whereas the decimal form is approximate. Of course all solutions of equations are in a sense tautological. Suppose the equation had been

$3x-5=0.\,$

Would he object that x = 5/3 was "just tautological? Michael Hardy (talk) 18:30, 5 January 2011 (UTC)

Right on Michael. Much of math can be seen as a search for tautologies. In my experience, "just a tautology" is used as a dismissal much more commonly in fields outside of math. SemanticMantis (talk) 19:02, 5 January 2011 (UTC)

Incidentally: I took two classes taught by Alphonse Vasquez, and remember he used to say that everything that's been proven in math is tautological, and if one doesn't see something as such then he hasn't sufficiently wrapped his head around it.—msh210℠ 19:16, 5 January 2011 (UTC)

When I teach math, I ask students for exact answers. Partially this is to make it easier on the grader, and partially because I think it is important to grasp that root two cannot be expressed exactly in decimal notation. But this is just personal preference. Also, asking for decimal approximations implicitly encourages students to use calculators when they are not required, and I believe this is counterproductive to really learning math. --But on to your prof's comments. I disagree completely that the decimal approximation is the "fullest possible answer", but perhaps this is not a direct quote from the instructor. Rather than talk of 'meaningful' or 'valid', we may consider whether an answer is *informative*. Consider an application where two quantities are to be compared. If solved exactly, it is not easy to see how ${\sqrt[{5}]{2}}$ compares to ${\sqrt[{10}]{5}}$. However, it's quite easy to see that 1.1487... < 1.1746... So really, the best form for an answer depends on why you're quantifying something in the first place. SemanticMantis (talk) 18:55, 5 January 2011 (UTC)

But before a student could reach for a calculator, she should notice that ${\sqrt[{5}]{2}}={\sqrt[{10}]{4}}$. -- 119.31.126.69 (talk) 16:38, 8 January 2011 (UTC)

There's an important point that my prof made that I forgot to mention. In the final solution, he doesn't want decimals; he wants radicals, because decimals aren't exact. His point was that the only reason $x={\sqrt {2}}$ is an acceptable answer is because someone could look up the decimal expansion to the desired precision. Presumably, he also means to say that if mathematicians were dumber, and had made the notation $x={\sqrt {2}}$ but couldn't figure out how to expand it, then $x={\sqrt {2}}$ would be meaningless. But I don't see why decimal representation should have validity than other representations. At the same time, if we accept, as Michael Hardy said (and which I agree with) that all solutions are tautological, then that would mean that when we say $x={\sqrt {2}}$, we are really saying "I have found that the solution to such-and-such equation also happens to be the positive solution to the equation $x^{2}-2=0$". I guess this makes sense, but it makes the whole business of solving equations kinda...arbitrary, no?74.15.138.87 (talk) 20:53, 5 January 2011 (UTC)

It depends why you're solving something. Do you need to know how many people it will take to change your lightbulb? Then you need a decimal representation (actually, the ceiling of one, usually). Do you need a number you can substitute into another expression? Then a form like${\sqrt {2}}$ is usually best. Do you need it as an element of ℂ? Then you want (perhaps) ${\sqrt {2}}+0i$. Etc.—msh210℠ 21:00, 5 January 2011 (UTC)

That's very dismissive of tautologies. I try and make everything I say logical and as nearly tautological as possible. Like for instance 'If I don't get some sleep I'll never wake up in the morning' ;-) Dmcq (talk) 21:30, 5 January 2011 (UTC)

I grappled with the same problem when initially introduced to square roots and logarithms at school (and arcsin...in fact, ANY inverse function). It seemed retarded and meaningless to say that the solution of $10^{x}=2$ is $x=\log(2)$ (decimal log obviously). What's the point of just inventing notation and calling these answers "solutions"? They are no more useful than the original equations. In a sense, ALL inverse functions are just tautologies, useless for ACTUALLY solving anything. :That was until I reached university and learned about Taylor series, which allows you to actually compute and evaluate those expressions in a meaningful way. That was when it all made sense. Zunaid 12:20, 6 January 2011 (UTC)

Math and prejudice

How many cases should you consider until you come up to the conclusion that an ethnic group has this or that feature? For example, if you take nations with 300 millions or 100 millions, is my personal experience of 200 interactions each year enough? Quest09 (talk) 17:03, 5 January 2011 (UTC)

There's a question above about elections that is related. What if the 200 people you meet all had a property that no-one else in the population had? I guess you want to answer the following question: If p% of a sample has a certain property, then what's the probability that (p±d)% of the whole population has that property. You need to decide what percentage of the sample/population needs to posses a property before you call it a characteristic. Take a look at my question, and Meni's answer, here. — Fly by Night (talk) 17:19, 5 January 2011 (UTC)

This is a question of statistics. A simple random sample of 1000 or so people is generally enough to establish with reasonable certainty the rough proportion of people in a given population who possess a feature/hold a particular opinion/etc, independently of the size of the population (assuming only that it is significantly larger than 1000). Your personal experiences most likely do not constitute a simple random sample, and therefore cannot be used for this purpose. --COVIZAPIBETEFOKY (talk) 17:24, 5 January 2011 (UTC)

What COVIZAPIBETEFOKY said. We all have social interactions strongly biased by our income, locality, profession, etc. Our own experience is never a good source from which to extrapolate to the general population. See Sampling bias#Historical examples for some famous cases where samples of millions that were not truly random and unbiased, were insufficient. Ray^Talk 18:35, 5 January 2011 (UTC)

The Real Answer

Let me put it this way. Let's say your IQ is 300, and you've just made a handful of major breakthroughs in your chosen field, however you are just an undergraduate at a huge state school. Let's say that you are able to prove yourself to be an extremely valuable researcher to a professor at your University, if he will speak to you for 10 minutes. Then he would be convinced by your ideas, be instantly swayed, and want to publish with you. As a result of this, you would be able to get admitted to the graduate program of your choice, even if you did nothing more than flesh out the ideas you just published as an undergrad, you would be set to get tenure based on that, if not at Harvard, then at least in some respectable state school such as the one you're attending. There's just one little problem: your state school is in Misouri, has low standards of admission, you are of a minority race, and the professor has had a LOT of experience with semi-literate members of that minority!! He might refuse the 10-minute interview on that grounds alone, just from remembering your face among the 300 faces he teaches at any one time!! So, let me ask you the question this way: how many members of your race that he had experiences with, who were semi-literate and had an IQ of more like 60 (one fifth of yours), would make you say: You know what, he shouldn't waste 10 minutes on an interview with me, it's just not a reasonable request. 100 such people? 1000? A million? How about if you're Indian, and there are one BILLION people who are all different from you, and you're the only Indian who can do Italian opera in all the world? Would you agree that the director of La Scala should refuse to even listen to you on that basis? 87.91.6.33 (talk) 19:05, 5 January 2011 (UTC)

I would argue that such an interview should never be refused. However, there are cases where the "judge" is in such demand that he can't spend the time, so then underlings should be enlisted to do a "pretest", to determine if you are anything worth bothering the "judge" about. StuRat (talk) 21:56, 5 January 2011 (UTC)

Exacrtly. the OP should come to the same conclusion, and, therefore, realize that he should not become racist even after a billion examples confirming his suspicions about a group. 87.91.6.33 (talk) 22:02, 5 January 2011 (UTC)

(ec) Please, there is no need for emotional language; just state your assertion clearly and with justification (and without fallacies like appeal to consequences).

The question Quest09 asked ("is my personal experience of 200 interactions each year enough [... to] come up [with] the conclusion that an ethnic group has this or that feature") is a simple question of statistics for which COVIZAPIBETEFOKY gave a direct answer ("Your personal experiences most likely do not constitute a simple random sample, and therefore cannot be used for this purpose."). Eric. 82.139.80.114 (talk) 22:05, 5 January 2011 (UTC)

Yes, this was a question about mathematics in the math RD. I was not asking about moral implications and actually not even thinking about discriminating people. Quest09 (talk) 23:04, 5 January 2011 (UTC)

Of course there also is an implicit assumption about why the interview was rejected. Maybe the professor can only grant so many interviews and cuts it off at an arbitrary number. Or maybe he grants no non-class-related interviews to undergrads at all. Or maybe he has already seen common misconceptions during the initial request for the interview. All of these are arguably beyond the realm of mathematics, of course.Unless you consider "There are only 16 working hours per day, if I grant one interview to every of my 600 students I'll never finish that research" as maths ;-) --Stephan Schulz (talk) 10:41, 6 January 2011 (UTC)

600 student won't ask you for an interview. And even if they do, if you spend 15 minutes with each, that would only make 30'/day each year. Quest09 (talk) 12:10, 6 January 2011 (UTC)

With 600 what I'd do is I wouldn't even bother reading the first line of the application for most. I'd do a random draw of a selection to check through and then winnow down to an even smaller number to interview. Actually I think interviews are very overrated so its mainly to eliminate the unsuitable rather than to pick the best. Dmcq (talk) 13:15, 6 January 2011 (UTC)

let me spell it out for you guys

"the conclusion that an ethnic group has this or that feature" is the definition of racism. Sorry. That thought is the definition of racism. In other words, this is a question about what level of statistical confidence justifies racism. The answer is: none. Even if you have a hundred billion examples of a member of an ethnic group with a certain feature, you still can't come to "the conlcusion that an ethnic group has this or that feature". Is that clear enough for you? How about this way: I grew up in a very poor part of Boston. I met literally thousands of black kids who were way below the required level in their grade. How many should I have met before I concluded that a black kid has features that make them, say, not qualified for a Presidential-track education? The answer is, there is no such number. (Math: Not a Number, NaN). Because even if you have 300,000,000 black kids who can't be president, because they're too stupid and all the ritalin in the world would not make them smart enough: it only takes one. You can never induce the rule. The rule is the definition of racism. Clear enough for you? 87.91.6.33 (talk) 20:38, 6 January 2011 (UTC)

in yet different terms, an ethnic group can't have any features (besides the tautological ones*). Except in the mind of a racist. A racist can list dozens of features for any given ethnic group. Go try a racist sometime, I am not making this up. 87.91.6.33 (talk) 20:43, 6 January 2011 (UTC)

* tautological = obviously if you group people in races based on criteria, the "race" will have member meeting that criteria... but this says nothing about them, and only about you and your grouping choices -- it's "begging the question".

It's not racist to believe or state a fact about demographics if there's truth behind it. Chances are, however, that the true statement will not be a universal; it will merely be that a certain percentage of the population shares a characteristic, and that this percentage happens to be unusually high compared to other demographics. Also, statement of a fact does not necessitate any particular response, positive or negative, to the fact.

For instance, it is definitely true that a higher percentage of black people in the United States are below poverty line than the US population as a whole. This does not necessarily mean that black people are somehow inherently poor, and that we should discriminate against them for being incompetent or, at the other end of the spectrum, that we should send more money to the cause of improving the welfare of black people; it is merely a statement of fact.

We may then speculate as to whether black people are in some way rendered incapable of finding and keeping a self-sustaining job as a consequence of genetic predispositions not directly/obviously related to the color of their skin, or if there are social and environmental pressures that cause them to fall below the poverty line, and they could have done better if they had been brought up differently. I think most people agree that the latter explanation is the more significant factor, in this case. We would also want to speculate what the proper response would be to encourage improvement in the welfare of black people, a problem which remains unsolved.

It is also important to be able to make observations about various characteristics of a population in order to better understand a variety of phenomena. For instance, again in the US, non-whites as a direct percentage are more likely to vote democrat than whites are. But if you correct for socioeconomic status, which is also seen to affect a person's vote, you actually find that non-whites are more likely to vote republican than whites are. Direct percentages are misleading, but you can't see this until you are willing to take in 'racist' data. --COVIZAPIBETEFOKY (talk) 02:40, 7 January 2011 (UTC)

It is important to separate what is true from what we believe. For example, I know it is true that black Americans are dumber -- have a lower IQ -- than white Americans. See the book called "The Bell Curve". However, even though I am aware of this fact, I do not believe it. I do not believe that black Americans are dumber than white Americans. Why? Because I'm not a racist. It's that simple guys. When it comes to groups of people, you simply can't internalize statistically "true" statements about that group, unless you're a racist. Only a racist would agree that black Americans are dumber than white Americans. Your very first sentence "It's not racist to believe or state a fact about demographics if there's truth behind it" is, simply false. Put another way, when you hear a racist statement, you need to realize that you are being asked to participate in racism -- and not react by questioning whether the statement is true. You have a store. "Well, we've just gotten through the first round of interviews, who should we call back for a second interview?" If someone says "let's not call back the black guys, because blacks are far poorer and more likely to steal from us, this will increase our chances of finding someone reliable." It doesn't matter what city in the world you're living in. It doesn't matter if 20%, 50%, 75%, or 98% of blacks will, in fact, steal from your store. (short the till). You can't believe that statement, because you are not a racist. (You can turn this around. If you're the first black guy in your family to get a college degree, and yours is in art history, what percent of black people in your city would be likely to be unreliable manpower in the art gallery, and short the till -- steal from their employers -- before you agreed that they shouldn't call you in for a second interview because you're black? 50%? 70%? 99%? No: the answer is, even if every single black man in the whole city except you would make a terrible employee in that shop, you still do not want the shop to make the conclusion that black people will steal from them). Frankly this whole discussion is extremely dirty, and I can't believe we're having it in English in 2011. Philosophically, everything I've written above could be interesting in 1947. It's 2011. All of this is way in the past. We have a black President. I don't know why we're even talking about why you can't generalize about a group of people no matter how "true" such a statement would be. 87.91.6.33 (talk) 11:32, 7 January 2011 (UTC)

This is the kind of discussion I expect in Germany. I recently lived in Munich for a year, and that city was the most racist, and thus awful, place of any location I've ever lived. Of course, if someone is from Munich, I will not generalize and say they a racist. It's simply my experience of that awful city. It's no wonder that Hitler, who actually wasn't German but Austrian, had to travel all the way to Munich, Germany before he could find an audience for his spiels. 87.91.6.33 (talk) 11:49, 7 January 2011 (UTC)

See this article. The guy is the next Hitler, right down to the Mustache. And, according to you, as long as his book doesn't make factual mistakes in its statistics, it is "right". Do you even have any idea what the consequences of what you're promoting are. 87.91.6.33 (talk) 11:54, 7 January 2011 (UTC)

This is a maths reference desk. Please take unrelated stuff elsewhere. It is not a soapbox see WP:SOAP. Dmcq (talk) 13:39, 7 January 2011 (UTC)

Actually, the question was entitled "Math and prejudice", and consists in whole oftwo short questions. The first reads "How many cases should you consider until you come up to the conclusion that an ethnic group has this or that feature?". The answer to this question is "Not a Number" - there is no number that justifies the conclusion. The second simple question is: "For example, if you take nations with 300 millions or 100 millions, is my personal experience of 200 interactions each year enough?". The answer is "no". In fact, for a nation of 300 million, not even personal interaction with 299,999,999 of them is enough to come to a conclusion about the one you didn't meet. The reason Obama got a shot at the presidency despite being elected by people who have met plenty of African-Americans who were not qualified to be president (very far from it) is that they understood this fact. Sorry, there is no mathematical question here, the original poster wants to know what sample size justifies racism or prejudice, and the answer is there is no such sample size. Even if you meet EVERY single member of a race, you still can't make any statements about that race. What? How??? Because someone new can be born ino that race who is the first counterexample. Sorry, your "prejudice" (OP's word fro mtitle) has no justification in mathematics or statistics at any sample size. 87.91.6.33 (talk) 14:39, 7 January 2011 (UTC)

Dmcq is right. If you must continue this discussion, I've responded to your silly post on my talk page. --COVIZAPIBETEFOKY (talk) 15:44, 7 January 2011 (UTC)

This is ridiculous. Believing things without sufficient evidence is the reason that racism exists, not the solution for it. The OP should not come to racist conclusions because A) Observed statistical differences between the races are due to sample bias and external conditions, not inherent genetic differences and B) Averages are averages. Even if a group did have an inherently lower average IQ (which is not the case) there would still be many intelligent members of that group and there are many ways that someone can contribute positively to society for which IQ play no or a very small role. We cannot solve any problems with society by believing things unjustified by evidence, only by looking at the world and coming up with informed solutions. Racism has no justification in fact, and we should point to the facts rather than making tolerance something entirely separate from critical thinking. 76.67.79.61 (talk) 01:23, 10 January 2011 (UTC)

You've missed the point. I'd restate it, but I don't think I was unclear to begin with. --COVIZAPIBETEFOKY (talk) 03:15, 10 January 2011 (UTC)

I was not addressing you; I was addressing 87.91.6.33, who said "I know it is true that black Americans are dumber -- have a lower IQ -- than white Americans. See the book called "The Bell Curve". However, even though I am aware of this fact, I do not believe it." I'm sorry if this was unclear. 74.14.110.15 (talk) 07:07, 10 January 2011 (UTC)

The fact that black americans have a lower IQ than white americans is frequently pointed out as an indication of a flaw in the standardized tests for IQ. In short we can't really draw any meaningfull conclusion from it because we can't know how well our tests actually messure intelligence. However there are several meaningful facts we can draw about a population. For example we know that few black americans take higher education than white americans. And I belive this holds even when you adjust for parents income. We can't draw any conclusions about genetics from this, and when we deal with individuals there will be better indicators. But we can still state that black americans tend to be less educated than white americans. This is a fact that we should be careful not to ignore because it doesn't just say something about the ethnic group, it also says something about American society as a whole. (Actually I can safely ignore it, but that's because I come from Norway). Note that OP did not say how much would you need to sample a group in order to draw conclusions about individuals of the group. He asked how much must you sample before you can say something about the group. And that has nothing to do with racism, it's a pure question of statistics. (Although as the example of IQ shows we should be careful about what we are actually testing for). Taemyr (talk) 20:10, 11 January 2011 (UTC)