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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)
Hello,
I am struggling with what should be an easy exercise...
let G be a group acting primitively on a set . Let be the permutation character( let's say over ). Hence maps every to the number of elements in it fixes.
Let be any irreducible constituent of , different from the trivial character. Prove that is faithful (i.e. the corresponding representation of maps only the trivial element of 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.
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)
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)
One of my profs made an interesting remark today. He was doing an example on the board, and the answer came out to . 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 .
As an example, he said, suppose we had to solve the equation . Saying that 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)
√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
Would he object that x = 5/3 was "just tautological? Michael Hardy ( talk) 18:30, 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 compares to . 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)
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 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 but couldn't figure out how to expand it, then 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 , 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 ". 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)
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)
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)
"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)
Mathematics desk | ||
---|---|---|
< January 4 | << Dec | January | Feb >> | January 6 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
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)
Hello,
I am struggling with what should be an easy exercise...
let G be a group acting primitively on a set . Let be the permutation character( let's say over ). Hence maps every to the number of elements in it fixes.
Let be any irreducible constituent of , different from the trivial character. Prove that is faithful (i.e. the corresponding representation of maps only the trivial element of 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.
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)
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)
One of my profs made an interesting remark today. He was doing an example on the board, and the answer came out to . 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 .
As an example, he said, suppose we had to solve the equation . Saying that 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)
√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
Would he object that x = 5/3 was "just tautological? Michael Hardy ( talk) 18:30, 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 compares to . 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)
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 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 but couldn't figure out how to expand it, then 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 , 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 ". 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)
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)
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)
"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)