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  1. #1 Math Joke 
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    This came up in another forum. I thought it might be of interest here to people more attuned to pure mathematics. This story is true.

    'Scientists tend to overcompress, to make their arguments difficult to follow by leaving out too many steps. They do this because they have a hard time writing and they would like to get it over with as soon as possible.... Six weeks of work are subsumed into the word “obviously.” '—Sidney Coleman

    There is a related joke about a mathematics professor. The professor is giving a lecture and has made an assertion as part of his presentation. A student, not understanding the basis for the assertion asks why it is true. The professor responds that "It is obvious." Then the professor steps back, stares at the board and ponders for several minutes. Then he turns and walks out of the lecture hall. He is absent for a fairly long time and finally one of the students goes to look for him. He sees the professor in his office working on the blackboard which he has covered with mathematics. The student returns and reports to the class. Finally, just before the class is scheduled to end the professor reappears, and announces "Yes, it is obvious." (You're supposed to laugh here, since this is usually the end of the joke.)

    But it gets better. I once told this joke to a man I know who was at one time the head of the Aeronatucial Engineering Department at MIT. His response was, "That is not a joke. The professor was Norbert Weiner. I was in the class."


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    LOL! That was funny!


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    Good one, Doc.
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  5. #4 Re: Math Joke 
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    Quote Originally Posted by DrRocket
    This came up in another forum. I thought it might be of interest here to people more attuned to pure mathematics. This story is true.

    'Scientists tend to overcompress, to make their arguments difficult to follow by leaving out too many steps. They do this because they have a hard time writing and they would like to get it over with as soon as possible.... Six weeks of work are subsumed into the word “obviously.” '—Sidney Coleman

    There is a related joke about a mathematics professor. The professor is giving a lecture and has made an assertion as part of his presentation. A student, not understanding the basis for the assertion asks why it is true. The professor responds that "It is obvious." Then the professor steps back, stares at the board and ponders for several minutes. Then he turns and walks out of the lecture hall. He is absent for a fairly long time and finally one of the students goes to look for him. He sees the professor in his office working on the blackboard which he has covered with mathematics. The student returns and reports to the class. Finally, just before the class is scheduled to end the professor reappears, and announces "Yes, it is obvious." (You're supposed to laugh here, since this is usually the end of the joke.)

    But it gets better. I once told this joke to a man I know who was at one time the head of the Aeronatucial Engineering Department at MIT. His response was, "That is not a joke. The professor was Norbert Weiner. I was in the class."

    Ha ha I get the joke as I was looking at a proof of Calculus just the other night, I was quite pleased I had managed with some struggle to follow it untill it came to a particular stage when he seemed to make a quantum leap in his calculation, it basically looked like he had just gave up and writtne down the answer.

    Actually I have found it
    http://www.sosmath.com/calculus/inte...3/integ03.html

    It's going from the second line to the third

    I really don't see how he gets the 'faint' f(x) inside the brackets and indeed inside the integral on the right of the equals sign (=).

    Prehaps someone could explain!!

    I think I might be able to grasp it but I think I would need a diagram or two to help me.
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  6. #5 Re: Math Joke 
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    Quote Originally Posted by esbo

    I really don't see how he gets the 'faint' f(x) inside the brackets and indeed inside the integral on the right of the equals sign (=).

    Prehaps someone could explain!!
    Since you are integrating with respect to t, f(x) is a constant inside the integral and what he is saying is that integrating a constant over an interval of length h and then dividing by h yields the constant.
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  7. #6 Re: Math Joke 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo

    I really don't see how he gets the 'faint' f(x) inside the brackets and indeed inside the integral on the right of the equals sign (=).

    Prehaps someone could explain!!
    Since you are integrating with respect to t, f(x) is a constant inside the integral and what he is saying is that integrating a constant over an interval of length h and then dividing by h yields the constant.

    Hmmmm obviously....

    Which brings me to another little matter, where does the 't' come from in the first place? That is not explained - obviously!!

    So what is 't' and what is it's relationship to 'x'?
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  8. #7 Re: Math Joke 
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    Quote Originally Posted by esbo
    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo

    I really don't see how he gets the 'faint' f(x) inside the brackets and indeed inside the integral on the right of the equals sign (=).

    Prehaps someone could explain!!
    Since you are integrating with respect to t, f(x) is a constant inside the integral and what he is saying is that integrating a constant over an interval of length h and then dividing by h yields the constant.

    Hmmmm obviously....

    Which brings me to another little matter, where does the 't' come from in the first place? That is not explained - obviously!!

    So what is 't' and what is it's relationship to 'x'?
    t is a dummy variable -- the variable of integration if you will. You can use any other symbol that suits your fancy. In fact, in more advanced classes you dispense with the t entirely. It really doesn't mean anything. I thas no relation to x or to anything else.
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  9. #8 Re: Math Joke 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo
    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo

    I really don't see how he gets the 'faint' f(x) inside the brackets and indeed inside the integral on the right of the equals sign (=).

    Prehaps someone could explain!!
    Since you are integrating with respect to t, f(x) is a constant inside the integral and what he is saying is that integrating a constant over an interval of length h and then dividing by h yields the constant.

    Hmmmm obviously....

    Which brings me to another little matter, where does the 't' come from in the first place? That is not explained - obviously!!

    So what is 't' and what is it's relationship to 'x'?
    t is a dummy variable -- the variable of integration if you will. You can use any other symbol that suits your fancy. In fact, in more advanced classes you dispense with the t entirely. It really doesn't mean anything. I thas no relation to x or to anything else.
    Ah now we are getting somewhere, what is a 'dummy variable'?

    Seem to me s if it is being made up as it goes along!!

    x squared = x cubed + t!!

    I like these dummy variables, I can solve any equation with then...obviously!!

    I suppose you in really advanced classes you can dispense with all the variables, all the logic and indeed all the maths and just pluck the answer out of thin air...obviously!!
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  10. #9 Re: Math Joke 
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    Quote Originally Posted by esbo
    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo
    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo

    I really don't see how he gets the 'faint' f(x) inside the brackets and indeed inside the integral on the right of the equals sign (=).

    Prehaps someone could explain!!
    Since you are integrating with respect to t, f(x) is a constant inside the integral and what he is saying is that integrating a constant over an interval of length h and then dividing by h yields the constant.

    Hmmmm obviously....

    Which brings me to another little matter, where does the 't' come from in the first place? That is not explained - obviously!!

    So what is 't' and what is it's relationship to 'x'?
    t is a dummy variable -- the variable of integration if you will. You can use any other symbol that suits your fancy. In fact, in more advanced classes you dispense with the t entirely. It really doesn't mean anything. I thas no relation to x or to anything else.
    Ah now we are getting somewhere, what is a 'dummy variable'?

    Seem to me s if it is being made up as it goes along!!

    x squared = x cubed + t!!

    I like these dummy variables, I can solve any equation with then...obviously!!

    I suppose you in really advanced classes you can dispense with all the variables, all the logic and indeed all the maths and just pluck the answer out of thin air...obviously!!
    A dummy variable is just a symbol ued to help you keep track of that with which you are integrating. It has no meaning outside of the integral sign. As I used to tell students you can replace "x" inside the integral with a "headless duck". It doesn't really mean anything.
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  11. #10 Re: Math Joke 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo
    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo
    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo

    I really don't see how he gets the 'faint' f(x) inside the brackets and indeed inside the integral on the right of the equals sign (=).

    Prehaps someone could explain!!
    Since you are integrating with respect to t, f(x) is a constant inside the integral and what he is saying is that integrating a constant over an interval of length h and then dividing by h yields the constant.

    Hmmmm obviously....

    Which brings me to another little matter, where does the 't' come from in the first place? That is not explained - obviously!!

    So what is 't' and what is it's relationship to 'x'?
    t is a dummy variable -- the variable of integration if you will. You can use any other symbol that suits your fancy. In fact, in more advanced classes you dispense with the t entirely. It really doesn't mean anything. I thas no relation to x or to anything else.
    Ah now we are getting somewhere, what is a 'dummy variable'?

    Seem to me s if it is being made up as it goes along!!

    x squared = x cubed + t!!

    I like these dummy variables, I can solve any equation with then...obviously!!

    I suppose you in really advanced classes you can dispense with all the variables, all the logic and indeed all the maths and just pluck the answer out of thin air...obviously!!
    A dummy variable is just a symbol ued to help you keep track of that with which you are integrating. It has no meaning outside of the integral sign. As I used to tell students you can replace "x" inside the integral with a "headless duck". It doesn't really mean anything.
    There is only one answer to that...obviously!!

    I am afraid I do ot find your explaination convinciing the next line is even worses,
    it seme to me I could prove virtualy anything using that method, I could start off with any hypothisis and arrive at any result I wanted.

    Not convinicing at all.
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  12. #11 Re: Math Joke 
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    Quote Originally Posted by esbo

    There is only one answer to that...obviously!!

    I am afraid I do ot find your explaination convinciing the next line is even worses,
    it seme to me I could prove virtualy anything using that method, I could start off with any hypothisis and arrive at any result I wanted.

    Not convinicing at all.
    Sorry that you are not convinced. Too bad. The explanation is quite correct.

    You, in fact, cannot prove anything at all using that method. NOTHING. Go aheae and try.

    The situation is quite simple. You integrate a function over an interval. could be just as easily written as and is often in more advanced applications written as where is Lebesgue measure on the liine.

    The whole thing goes back to the definition of the integral in terms of Riemann sums. What you are doing is multiplying the value of a function taken at some points within a set of small intervals times the length of the intervals. There are other weighted sums that apply to more advanced integrals that use measures of the intervals as something other than their length (these come up in the theory of probability for instance and are related to probability density functions). The dx is only a reminder that we are using length of an interval as a measure of the interval. The "x" really doesn't mean anything other than that.

    So LOL all that you want. You may come to understand this better if you learn a lot more mathematics, for instance if you study the general theory of integration.

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

    http://www.math.usu.edu/~dhs/MeasTheory.pdf

    http://www.worldscibooks.com/mathematics/1040.html

    http://web.media.mit.edu/~lifton/sni...ure_theory.pdf

    http://www.math.uconn.edu/~bass/lecture.html

    http://mathworld.wolfram.com/ProbabilityMeasure.html

    You might even try actually reading a book.


    http://books.google.com/books?id=n5v...esult&resnum=3
    http://www.elsevier.com/wps/find/boo...on#description
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    Of course another problem with it is that it starts off with the 'answer' and does a bit of dubiouos maths and arrives back at where it started.

    I don't think need all your links or terms I have never heard of to be able to do that - obviously!!
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    Well... I found the original Joke, and subsequent failure to comprehend, quite amusing. My current instructer does something similar to that, though. Whenever he's solving one of the examples to theoroms that he's presenting, he'll only go to the point where the problem get's ugly, and then invoke the 'power of the dot dot dot' as he calls it. He writes an ... on the board and calmy states something like, 'I have faith that you all could do the problem from here, so I'm going to move on to something else.' I think it's quite amusing at times, only doing the first 4 steps of a rather lengthy problem, and then just copping out on the 'I have faith in you guys' response.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  15. #14  
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    Quote Originally Posted by esbo
    Of course another problem with it is that it starts off with the 'answer' and does a bit of dubiouos maths and arrives back at where it started.

    I don't think need all your links or terms I have never heard of to be able to do that - obviously!!
    Have you noticed that your statements are factually incorrect and that you don't know what you are talking about ?

    Nothing started off with the "answer". There is no dubious mathematics, and in fact the mathematics is quite rigorous and rather standard. It does most certainly not arrive "back where it started".

    Apparently you just like to see your own typing.

    Try reading something that is both useful and correct. The novelty might amuse you.
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo
    Of course another problem with it is that it starts off with the 'answer' and does a bit of dubiouos maths and arrives back at where it started.

    I don't think need all your links or terms I have never heard of to be able to do that - obviously!!
    Have you noticed that your statements are factually incorrect and that you don't know what you are talking about ?

    Nothing started off with the "answer". There is no dubious mathematics, and in fact the mathematics is quite rigorous and rather standard. It does most certainly not arrive "back where it started".

    Apparently you just like to see your own typing.

    Try reading something that is both useful and correct. The novelty might amuse you.
    Well you failed to explain it and now you ar becomming rather rude and abusive.
    I have seen that before.
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    Quote Originally Posted by esbo

    Well you failed to explain it and now you ar becomming rather rude and abusive.
    I have seen that before.
    Pay attention. This is important to your future education.

    My ability to explain or not, and the ability of your professors to explain or not, is not the issue.

    What matters is that you have failed to comprehend.

    No one "teaches" you anything. The best that anyone can do is to help you to learn for yourself. You can't learn with your mouth open or your fingers moving on the keyboard. You have to stop and actually think for yourself, and think until you "get it".
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo

    Well you failed to explain it and now you ar becomming rather rude and abusive.
    I have seen that before.
    Pay attention. This is important to your future education.

    My ability to explain or not, and the ability of your professors to explain or not, is not the issue.

    What matters is that you have failed to comprehend.

    No one "teaches" you anything. The best that anyone can do is to help you to learn for yourself. You can't learn with your mouth open or your fingers moving on the keyboard. You have to stop and actually think for yourself, and think until you "get it".
    No I believe your explaination was inadaquate. trying to blame me for that is a cheap shot. what I believe I have comprehened is that your eplaination was rubbish.
    I learnt that with my fingers on the keyboard, it was obvious. The most convincing bit was your resort to being abusive.
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    wow... I'm on the Dr.'s side here esbo, you are just failing to grasp his explanation. I'm got it (mostly) and that should say that he is capable of explaining himself adequately enough for the purpose of his explanation. You, sir, are just incapable of understanding it, and resort to making silly jokes of it and insults to discredit him, to which he replies in kind.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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    Quote Originally Posted by Arcane_Mathamatition
    wow... I'm on the Dr.'s side here esbo, you are just failing to grasp his explanation. I'm got it (mostly) and that should say that he is capable of explaining himself adequately enough for the purpose of his explanation. You, sir, are just incapable of understanding it, and resort to making silly jokes of it and insults to discredit him, to which he replies in kind.
    Not really the joke is about teachers not explaing things so people can understand them, thats as much the teachers fault as the pupils I think.
    Or more to the pint teachers pretending they understand things, being able to explain something is a test of how well you understand something.
    And basicaly he is saying I should be able to understand it by myself which is kind of an insult to teachers.

    Maybe the problem is I think it is expplaining something more complicated when it is explainng the obvious.

    Prove the integral of x cubed is x to the forth divided by 4 for example.
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    Quote Originally Posted by esbo
    Or more to the pint teachers pretending they understand things, being able to explain something is a test of how well you understand something.
    And basicaly he is saying I should be able to understand it by myself which is kind of an insult to teachers.

    .
    One way in which mathematics is taught is called the "Moore method" or the "Texas method" after R.L. Moore who was a topologist at the University of Texas.

    In Moore method classes there are no lectures, and there is no text book. Students are provided with a set of definitions, theorems and examples. The theorems are provided without proof and the examples are not worked out. It is the job of the students to prove all of the theorems and work out all of the examples. They are to come to class prepared to present the proofs and the examples on the blackboard. They are not permitted to use any references or to talk to one another (or to anyone else). It is an excellent way to learn mathematics.

    That is how I learned toplogy and real analysis.

    Yes, you should be able to understand things by yourself.
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    the most you should ever need is a gental push, not a massive explaination as if you were incapable of thinking.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  23. #22 Re: Math Joke 
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    Quote Originally Posted by esbo
    Actually I have found it
    http://www.sosmath.com/calculus/inte...3/integ03.html

    It's going from the second line to the third

    I really don't see how he gets the 'faint' f(x) inside the brackets and indeed inside the integral on the right of the equals sign (=).

    Prehaps someone could explain!!

    I think I might be able to grasp it but I think I would need a diagram or two to help me.
    I don't blame you for being confused there. It strikes me as an absolutely awful "proof" of the fundamental theorem of calculus. It's not exactly wrong, but it's leaving out the essential step.

    The right way to proceed is as follows:



    Therefore



    This is less than or equal to



    Now since f is continuous at x, for sufficiently small h we can guarantee that the integrand in the interval [x,x+h]. It follows that the whole expression must be bounded by for h sufficiently small. This completes the proof.
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    Quote Originally Posted by esbo
    Prove the integral of x cubed is x to the forth divided by 4 for example.
    using the limit of riemann sums would give you the proof quite nicely.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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    I think the issue is that esbo was having trouble understanding the proof of the FT of calculus that was provided in the link he referenced. In addition to asking about that proof, he also asked a side question about dummy variables. It looks to me like people tried to address the second question without giving any thought to the first, to the point where everyone is now talking at cross purposes with each other.

    Now even I found the proof in the link rather incomplete and unsatisfying. So it's not just laziness on the part of the poster.

    Btw, having students come up with all the proofs and examples for themselves sounds really nice on paper. But anyone who would seriously suggest that hasn't spent much time in a real-life teaching environment. I'm sure it works great for students who are 3 standard deviations above the mean. But trust me when I say that most students, even smart and hard-working ones--will crash and burn. Take this from someone who has spent a lot of time grading calculus midterms and finals at a rather good university. And anyway what's the point of having a teacher if they're not going to teach?
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    Quote Originally Posted by salsaonline

    Btw, having students come up with all the proofs and examples for themselves sounds really nice on paper. But anyone who would seriously suggest that hasn't spent much time in a real-life teaching environment. I'm sure it works great for students who are 3 standard deviations above the mean. But trust me when I say that most students, even smart and hard-working ones--will crash and burn. Take this from someone who has spent a lot of time grading calculus midterms and finals at a rather good university. And anyway what's the point of having a teacher if they're not going to teach?
    I have also spent a lot of time in the past teaching calculus and other classes. In addition I have seen the Moore method in action, up close and personal.

    The Moore method is probably not a good way to teach introductory calculus. Even at good universities most students are not up to it. Introductory calculus classes are commonly long on computation and short on proofs in any case. Since the purpose of the Moore method is to teach the ability to understand theoretical mathematics and to do proofs, calculus is a bad application.

    IMO the purpose of a professor is not to teach, but to aid students in learning. There is quite a difference. That statement also applies to universities, which are centers of learniing and not of teaching.

    Wile the Moore methodis not appropriate for introductory calculus, it works quite well in some other classes. It is very effective in point-set topology, even at a fairly high level. It works well in real analysis, measure and integration in particular. It works very well in introductory abstract algebra.

    In those classes when a student is "stuck" the professor does not provide the correct proof except in very unusual circumstances. The student is simply given more time to find a correct proof. Thinking for oneself is one of the skills being developed.

    A modification, in which there is a text for purposes of reading but in which the lectures are presented by the students is also effective. In some applications the students deliver the lectures, with no prior notice of which student is to do the talking and they receive and answer questions from the other students and sometimes from the professor. In other situations the student knows ahead of time who will be speaking. Been there, done that. Probably the least effective classes in my experience are traditional lectures. Been there, done that too.

    The main point is that students learn mathematics by doing it, not by listening to it.

    Your example is a case in point. I agree that the proof was not all that it should be. But the final step, which you explained quite clearly, is all that was really lacking. To recognize that that final step was the only sticking point and to fill it in strikes me as a reasonable expectation for a good student; as you demonstrated it is a very simple exercise in "epsilonics" (once one realizes what a dummy variable is).

    For pedagogical purposes, and pedagogical purposes only, it is actually sometimes useful to have a text that contains such mistakes, or even some more serious mistakes. Then the students have the experience of finding and correcting errors in proofs, while still having the hint that the theorem being stated, because it is standard, is true. Lang's Algebra book was favored for this reason by some professors (and besides it is a pretty good book).
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    With point set topology, I can see how that program might work well (although, as Munkres notes, probably not for the proof of Urysohn's Lemma).

    I guess I differ from you in the following sense: I think that students should be left to their own devices to do homework problems. However, I think that the professor should prove enough things in class so as to lead by example. My sense is that most students don't really know what it means to "prove" something the first time they take an upper div math class. I don't think you do any real harm providing lots of examples of, first, how to think about math, and second, how to write a complete proof. Students will learn to think for themselves in the course of solving homework problems.
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    Quote Originally Posted by salsaonline
    With point set topology, I can see how that program might work well (although, as Munkres notes, probably not for the proof of Urysohn's Lemma).
    If I recall correctly Urysohn's Lemma was on the list of things we had to prove.

    I guess I differ from you in the following sense: I think that students should be left to their own devices to do homework problems. However, I think that the professor should prove enough things in class so as to lead by example. My sense is that most students don't really know what it means to "prove" something the first time they take an upper div math class. I don't think you do any real harm providing lots of examples of, first, how to think about math, and second, how to write a complete proof. Students will learn to think for themselves in the course of solving homework problems.
    You can do it that way, and many do. It works. I did not say that it is wrong. I did find courses like that less interesting than courses in which the burden was on the students. I disliked pure lecture classes.

    At the far end of the spectrum are really advanced classes where the professor lectures and the students take notes. No tests. No homework. The prof tends to learn a lot. Really interested students who put in a lot of time their own also do OK, but on the whole you wind up with not much more than "exposure". I have never seen a class like this except at the advanced graduate level -- it is pretty irrelevant to the undergraduate curriculum.

    But the Moore method also works. And in those classes the professor proves nothing and gives no lectures. They also work. The whole class is a homework problem. There are no other problems and no formal tests. Every day is a test.

    If you are iinterested here is an actual book designed for Moore method topology (It has no proofs and no worked out examples. It does have hints for some of the proofs. Some of the hints are bad advice. It does contain Urysohn's lemma as an exercise, with a hint regarding the usual construction using the dyadic rationals.) It is now pretty cheap on the used book market. http://www.alibris.com/booksearch?qw...*listing*title

    You might be surprised at how little preparation is required, and how little previous classes help. My introduction to topology was in a Moore method class (we actually used the book above as the list of theorems to be proved). I had essentially no undergraduate mathematics preparation -- calculus, matrix algebra, and one semester of a junion-senior real analysis class out of "Brown Bartle" (Elements of Real Analysis by Bartle). (I switched into mathematics rather late, after studying electrical engineering.) The lack of "prerequisites" was a non-issue. All that was needed was some hard work.

    Hybrids, as mentioned earlier, where the professors and the students both give lectures also work.

    I don't think there is a right or a wrong way. But I do hold a strong opinion that the only way students learn mathematics is by doing mathematics. You don't understand a proof until you have tried to do it or at least explain it yourself.
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    Well, we had something similar to this methodology in grad school. It was called qualifying exams.
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    Quote Originally Posted by Arcane_Mathamatition
    Quote Originally Posted by esbo
    Prove the integral of x cubed is x to the forth divided by 4 for example.
    using the limit of riemann sums would give you the proof quite nicely.
    OK use tthat if you must, please post you solution asap, preferebaly this year or whenever you have completed it
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    Quote Originally Posted by salsaonline
    Well, we had something similar to this methodology in grad school. It was called qualifying exams.
    My qualifiying and general exams were oral. There was no specific time limit. The subject was limited to anything that any of the members of the examing committee knew and thought that you should know or should be able to figure out.

    That method has been dropped in favor of more conventional written exams I am told.
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    We had written exams, with a wide range of possible topics that we could be questioned on for each exam. The only good way to prepare was to spend an entire summer doing nothing but solving problems from old exams and from textbooks, day in, day out.

    That is how I learned manifold theory and algebraic topology. Not from taking the courses, but from knowing that I was expected to be able to answer virtually any question that could be asked on the subject at the beginning graduate level.

    Still, I don't think this works with undergraduates, except for exceptional students. I don't doubt that you can put a Moore-style undergraduate syllabus together. The real question is how effective does the course end up being? Maybe the brightest students learn more than they would otherwise. But what about everyone else--are they left in the dust?

    To be convinced that this method is in any way preferable, I would need more than anecdotal evidence from someone would was probably in the top 99th percentile while in college.
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    Quote Originally Posted by salsaonline

    Still, I don't think this works with undergraduates, except for exceptional students. I don't doubt that you can put a Moore-style undergraduate syllabus together. The real question is how effective does the course end up being? Maybe the brightest students learn more than they would otherwise. But what about everyone else--are they left in the dust?
    It is usually done for beginning graduate students. I don't think students learn more in a Moore method class, and in fact the classes usually cover less than what can be covered in a lecture course. But they learn it better and learn how to think and construct proofs for themselves. Retention tends to be high so that you keep what you learned.

    Also it is usually done with relatively small classes -- a dozen students would be a large class. That essentially eliminates most undergraduate classes.

    While I like the method, there are people with equally valid opinions who dislike it too. I have heard one opinion to the effect that "It takes a mediocre student and makes him a little better." But the method turned out some pretty good topologists for Moore -- Dick Anderson, Mary Ellen Rudin, R.H. Bing for instance.

    My point in bringing up the method is to emphasize that the onus is on the student to learn and not on the professor to teach. Learning is not just absorbing lectures. It is an active endeavor and does not even require a teacher. The most that a teacher can do is help, and the responsibility (and the credit) for the outcome lie with the student. That observaton is consistent with your remarks regarding preparation for your qualifying exams.
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    With regard to grad level classes, we are pretty much in agreement. Although, in that case, I think the ideal set up is for the professor to at least convey how one is supposed to think about the subject, providing outlines of proofs, but allowing students to fill in the details.
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    Quote Originally Posted by esbo
    Quote Originally Posted by Arcane_Mathamatition
    Quote Originally Posted by esbo
    Prove the integral of x cubed is x to the forth divided by 4 for example.
    using the limit of riemann sums would give you the proof quite nicely.
    OK use tthat if you must, please post you solution asap, preferebaly this year or whenever you have completed it
    alright, no problem. Assuming you will agree that the best way to approximate the area under a curve is to 'carve' it up into a bunch of rectangles, then the smaller the width of each rectangle, the more accurate the approximation is, correct? going on that, we have a sum, and this is going to equal the added area of all of our rectangles if we start at some point, 'a' and take the area up until some other point 'b'. so, taking right end-points on our graph, we will get area's such that such that the area of any given rectangle, is the change in x, multiplied by the function evaluated at it's right end point. So, yes? Now, the only problem I currently have is that this only applies to a rectangle, but it clearly shows how, when the difference between the points goes to zero, the total area will be a function of one order higher, which is also obvious if you consider the are to be the 'height' (y) times the 'width' (x). That is to say, when , it should be obvious that the area function will be some function of The problem being finding the constant that will give you the proper area. If anyone would like to give me a push in the right direction here, by all means
    Wise men speak because they have something to say; Fools, because they have to say something.
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    Quote Originally Posted by Arcane_Mathamatition
    Quote Originally Posted by esbo
    Quote Originally Posted by Arcane_Mathamatition
    Quote Originally Posted by esbo
    Prove the integral of x cubed is x to the forth divided by 4 for example.
    using the limit of riemann sums would give you the proof quite nicely.
    OK use tthat if you must, please post you solution asap, preferebaly this year or whenever you have completed it
    alright, no problem. Assuming you will agree that the best way to approximate the area under a curve is to 'carve' it up into a bunch of rectangles, then the smaller the width of each rectangle, the more accurate the approximation is, correct? going on that, we have a sum, and this is going to equal the added area of all of our rectangles if we start at some point, 'a' and take the area up until some other point 'b'. so, taking right end-points on our graph, we will get area's such that such that the area of any given rectangle, is the change in x, multiplied by the function evaluated at it's right end point. So, yes? Now, the only problem I currently have is that this only applies to a rectangle, but it clearly shows how, when the difference between the points goes to zero, the total area will be a function of one order higher, which is also obvious if you consider the are to be the 'height' (y) times the 'width' (x). That is to say, when , it should be obvious that the area function will be some function of The problem being finding the constant that will give you the proper area. If anyone would like to give me a push in the right direction here, by all means
    I would dnot have approached the problem in this way, but you have gotten me to thinking and you can actually do the proof along these lines. What you need to do is take the integral over an arbitrary interval [0,b] and you need the fact (easily proved by induction) that



    You also need to know that if f is continuous then the function F defined by is differentiable and

    So now you proceed as you suggested to integrate x^3 as a limit of Riemann sums evaluating each value of on an interval of length at the right-hand endpoint


    Q.E.D.
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    That was good.
    -Haku Midori Shadowsong
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    Quote Originally Posted by DrRocket


    .
    Does this work?
    For n=2

    =

    Annd if you are adding the first term for n=1 it becomes 10 whereas as I understand it,

    So I would have to say thats 'not so good' unless I have misunderstood or miscalculated
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    It works. As DrR noted, it is easy to prove this formula by induction. I also checked the cases 1 through 5.
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    Quote Originally Posted by salsaonline
    It works. As DrR noted, it is easy to prove this formula by induction. I also checked the cases 1 through 5.
    Well I have just proved it does not work!!

    CAn you do a bit better than saying it works.

    That would hardly get you and marks in an exam!!

    Are you saying 2x 2 x 2=10 if so prove it!!
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    The difference here is I'm not taking an exam.
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    The similarity here is that you have failed to get any marks.
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    I can live with that.
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    you haven't proved crap Esbo. You did the summation wrong. is another way of writing 1^3+2^3+3^3+...+k^3. At k=2, we have

    It works.
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    Quote Originally Posted by esbo
    Quote Originally Posted by salsaonline
    It works. As DrR noted, it is easy to prove this formula by induction. I also checked the cases 1 through 5.
    Well I have just proved it does not work!!

    CAn you do a bit better than saying it works.

    That would hardly get you and marks in an exam!!

    Are you saying 2x 2 x 2=10 if so prove it!!
    No you did not prove that the formula does not work. You proved that you cannot do mathematics, or else that you cannot read. You have managed to both misunderstand and miscalculate. For somebody who has declared that he will write a professional quality explanation of Wiles's proof of Fermat's Last theorem which involves as proof of the Taniyama-Shimura conjecture, you are not instilling a lot of confidence in your ability to complete the task. It is a bit more difficult than the proof of this little formula.



    is correct and can be proved by induction just as I said.

    Your counter-example is incorrect



    As far as marks on exams go, I would observe that I have had students like you in several classes in the dim past. I think salsaonline is probably in a similar situation, but with more recent experience. We are both beyond the need to get marks on exams, and have been or are in the position of giving them. The students of mine whom you resemble flunked convincingly.
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    I understand that induction is an important and powerful way to prove things, but I find it a hard concept to put my head around. You prove that, if the case is true for 'n', that it is also true for 'n+1'. The significance of this is lost on me.
    Wise men speak because they have something to say; Fools, because they have to say something.
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    n^3= k^2(k+1)^2/4

    (n+1)^3= (k+1)^2(k+2)^2/4 <---ignoer this line


    OK putting in n+1 instead of n gives

    =(n+1)(n+1)(n+1)

    = (n+1)(n^2+2n+4)=n^3+2n^2+4n +n^2 +2n +4

    =n^3+ 3n^2+4n+4.


    So.....as you have proved it for n, you now have written n+1, in terms of n
    so..it must be correct!!

    Is that a proof?
    I think it might be!! But I am not 100% sure


    There is a possibilty the above maybe garbage!!
    Say yea or nay!!
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    the problem is, it's a sum. not a direct equation.
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    Quote Originally Posted by esbo
    n^3= k^2(k+1)^2/4

    (n+1)^3= (k+1)^2(k+2)^2/4 <---ignoer this line


    OK putting in n+1 instead of n gives

    =(n+1)(n+1)(n+1)

    = (n+1)(n^2+2n+4)=n^3+2n^2+4n +n^2 +2n +4

    =n^3+ 3n^2+4n+4.


    So.....as you have proved it for n, you now have written n+1, in terms of n
    so..it must be correct!!

    Is that a proof?
    I think it might be!! But I am not 100% sure


    There is a possibilty the above maybe garbage!!
    Say yea or nay!!
    It is clearly garbage. Not even very high quality garbage.

    Do you have any idea what a summation is ? You know, that sigma thing ?
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    Quote Originally Posted by Arcane_Mathamatition
    I understand that induction is an important and powerful way to prove things, but I find it a hard concept to put my head around. You prove that, if the case is true for 'n', that it is also true for 'n+1'. The significance of this is lost on me.
    Think of it this way. I have a proposition, call it A, that I claim is true for every natural number n.

    First I show it is true for n=1. So now it is true for some non-empty subset of the natural numbers.

    Next I show that if it is true for n then it is also true for n+1.

    I then claim that it is true for all natural numbers.

    Why ?

    Well suppose that it is not true for all natural numbers. Then there is a smallest natural number, call it K, for which it is not true.

    K>1 since we showed that A was true for 1. So then K-1 is a natural number since and A is true for K-1 since A is true for all natural numbers less than.

    But we also showed that if A was true for n then it was also true for n+1. Since now A is true for K-1 it is also true for (K-1) +1 = K which is a contradiction. Hence A is true for all natural numbers.

    Induction turns out to be a very powerful technique for proving theorems.
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    Quote Originally Posted by DrRocket

    As far as marks on exams go, I would observe that I have had students like you in several classes in the dim past. I think salsaonline is probably in a similar situation, but with more recent experience. We are both beyond the need to get marks on exams, and have been or are in the position of giving them. The students of mine whom you resemble flunked convincingly.
    As it happens I have never flunked a maths exam, I did maths up to A level, which is and exam taken at age 18, I got a grade B, nobody in my school got a higher grade
    (typically 180 in a year but not every one stayed on to do the higher level) about 15% maybe. I actually calculated I had achieved enouogh points for a grade A so I did a question not covered by my course on the exam paper for 'fun'. obviously not a very clever idea I suppose but there you go!!
    Maybe I would I have got an A if I did something we covered but I will never know!!
    Anyway I studied electronics after that(or rather didn't study it if you know what I mean!!) so it is a very long time since I have done any maths.
    So I have never flunked any maths exams, I still managed to get a degree in electronics basically without turning up or having any notes to revise from howeer I still got a degree ableit not a very good one.
    You probably have students who flunk their exams because you are a poor teacher. So you are wrong on your conjecture
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    Please pay better attention to how you enclose quotes, since I never wrote the lines you are attributing to me.
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    Quote Originally Posted by salsaonline
    Please pay better attention to how you enclose quotes, since I never wrote the lines you are attributing to me.
    Fixed!
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    Quote Originally Posted by esbo
    Quote Originally Posted by DrRocket

    As far as marks on exams go, I would observe that I have had students like you in several classes in the dim past. I think salsaonline is probably in a similar situation, but with more recent experience. We are both beyond the need to get marks on exams, and have been or are in the position of giving them. The students of mine whom you resemble flunked convincingly.
    As it happens I have never flunked a maths exam, I did maths up to A level, which is and exam taken at age 18, I got a grade B, nobody in my school got a higher grade
    (typically 180 in a year but not every one stayed on to do the higher level) about 15% maybe. I actually calculated I had achieved enouogh points for a grade A so I did a question not covered by my course on the exam paper for 'fun'. obviously not a very clever idea I suppose but there you go!!
    Maybe I would I have got an A if I did something we covered but I will never know!!
    Anyway I studied electronics after that(or rather didn't study it if you know what I mean!!) so it is a very long time since I have done any maths.
    So I have never flunked any maths exams, I still managed to get a degree in electronics basically without turning up or having any notes to revise from howeer I still got a degree ableit not a very good one.
    You probably have students who flunk their exams because you are a poor teacher. So you are wrong on your conjecture
    1) It was not a conjecture it was an assertion.

    2) You have just supported my assertion quite convincingly.

    3) If you received high marks on an exam for the sort of erroneous junk that you have been posting in this thread then that exam was not very well graded.

    4) If this thread were a math examand if I or any mathematician of my acquaintance were grading it, you most certainly would have flunked.

    5) If you want to be treated with deference you might stop your tactic of feigning superiority, exhibiting ignorance by making blatantly false assertions, and criticizing those who know better.

    6) You still have not learned that the responsibility for learning is born by the student. The teacher is only there to help. Until you learn that simple lesson you will forever blame your shortcomings on someone else. Teachers don't flunk courses, students do. Even if the teacher is bad, it is up to the student to learn. After all, who benefits or looses from failure to learn -- the student.

    7) I may have been a poor teacher. Or maybe not. It doesn't matter since I am now retired, except for some consulting work. In any case you really cannot effectively insult me since you have no basis for an opinion and since I don't care what you think anyway.

    You might take a lesson from Arcanemathematician. He is trying to learn something.

    You are fighting WAY out of your weight class.
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    Quote Originally Posted by DrRocket



    OK I will take you up on your offer.
    Easily prove it!!

    (I think I have got him here )

    and try to avoid using terms such as 'obviously' and 'it can easily be shown that' etc...
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    Actually I think I may have done it my self but I need to check

    (that might take a while!)
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    please let me do this one Dr.
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    Quote Originally Posted by esbo
    Quote Originally Posted by DrRocket



    OK I will take you up on your offer.
    Easily prove it!!

    (I think I have got him here )

    and try to avoid using terms such as 'obviously' and 'it can easily be shown that' etc...

    Lemma.

    PROOF

    First consider the case k=1.



    So the lemma is true for k=1. Now assume inductively that it is true for all values up to k and we will show that is must then be true for k+1 to complete the proof by induction.

    Under that assumption



    and




    Which completes the proof of the lemma by induction.


    So, as expected, no you don't "have hiim here". Moreover, the fact that you were unable to do this for youself speaks volumes. This is something that I would expect a good junior in a U.S. high school to be able to do.

    From now on please work these little details out for yourself, and don't waste the time of the rest of us. I don't intend to do trivial exercises for you again.
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    OK I wil have a bash at writing it out but its difficult.

    Firstly I will expand (a)

    OK to make it easy to write I will drop the /4 here.(1)

    So that equals (2)

    Now we need to add to that for the next term
    That expands to
    I am also going to times that by 4 as I did that in step(1)

    So it becomes (3)

    Now adding (2) + (3) we get ...drum roll....



    Now k=n I believe so we get

    (4)


    Right.

    Now if I subsitute k= k+1 in the orignal equation (a) we get

    (note I have dropped the /4 here to, it is easier to do it we take 4 times the sum.

    so....that expands to...



    Which expands to....drumroll.......


    And simplifying that we get....

    which is the same as (4)

    Which I think you can easily and obviously see is a proof by induction.

    Note I use easilly and obviously in the true sense of the words, as opposed to the sense where they mean completely the opposite!!

    DA!! DA!!
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    Obviusly I can do it because I never could type that in in 2 minutes and I have shown all the working which you did not
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    Incidently proof by induction is not done at age you suggest, 4th grade which I think means 14-15? It is done at advance level whihc is 16-18 I believe.

    Incidently I would not say your proof is easy to follow because it misses out some of the detail which makes it harder to follow especailly when your students may be confused about other things, so no surprise they flunked their exams....
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    I could follow his proof with no problems at all, which leads me to wonder how they are hard for you to follow?
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    Quote Originally Posted by esbo
    Obviusly I can do it because I never could type that in in 2 minutes and I have shown all the working which you did not
    But what you did DID NOT prove the lemma.

    In fact what you did was quite pointless. You expanded a bunch of polynomials with no particular purpose and proved absolutely NOTHING. What a bunch of crap.

    I don't know or care how long it took you. Whatever time you spent was completely wasted.

    The objective was to prove the stated formula for the sum of the first k cubes not to expand a bunch of polynomials with no objective.

    You quite clearly don't understand what the lemma says or what proof by induction is.

    Look, let's just be perfectly clear here. You have inserted yourself into a mathematics forum, and brayed like a jackass while proclaiming yourself a near expert. You are about as far from an expert and anyone that I have ever seen, and I have quite few complete incompetents.

    You need to go back and learn basic grammar school and high school mathematics and quit playing like you understand mathematics at all until the time comes, if ever, when you actually comprehend at least the rudiments of the subject.

    This just ridiculous. You don't know what you are talking about, and you don't even understand that you don't understand.

    Your nonsense is hurting people like Arcanemathematician who is actually trying to learn something.

    Since you are contributing nothing and are quite possibly doing some harm you would do better to remain silent unless and until you can say something intelligent. That certainly has not occurred in any of your posts thus far. If you are not embarrassed, you ought to be. If you keep this up you will be, assuming that you have the perceptive capability to actually be embarrassed.

    This is supposed to be a serious mathematics forum in which people can exhange meaningful ideas and learn some real mathematics. You are detracting from by posting nonsense and gibberish.

    Expect to be ignored in the future, except when your idiocy requires correction for the benefit of innocent lurkers, or when it suits someone's fancy to address you, for whatever purpose or mild amusement.
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    Rocket you are talking nonsense and you know it.
    I did prove it and that is a fact.

    I expect Arcanemathematician will find my proof more convincing than yours although he probably won't admit it, (and indeed he hasn't, or at least won't in public)) however he does not need to as I know it already.

    If you don't believe me I will send it so some maths insitute to have it verified for you, agreed?

    Anyway, it's no surprise all your students failed when you tell them they are wrong when they are right!!

    And I did not just expand things, I added the next term to the sum and proved
    it was the same as the sum of (n+1) when put into the original equation.

    A cast iron solution.

    It might take you a while to see that if you are more acustomed to coping your solutions from other web pages.
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  65. #64  
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    No. I side with Dr. Rocket. You are obviously full of crap and understand much less than I do about proofs and just about all of mathematics. Please, if nothing else, don't speak for me, because you don't know me. DrR's Proof was concise, easy to follow, and complete. All of the proofs I've seen him post are concise, easy to follow where applicable for me (sometimes they are beyond me because the subject is beyond me), and always complete. just as he said, all you did was expand polynomials, not even really in any way whatsoever proving what you claimed to prove. I understood this induction proof, because it was one of the proofs we had in high school geometry. plus, the only thing in it that was even a LITTLE confusing was the notation, simply because I'm not familiar with it. you contribute next to nothing and only show your own ignorance with every post, both in math and in your ability to understand those around you.
    Wise men speak because they have something to say; Fools, because they have to say something.
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    All he did was expand polynomials, infact he could not even be arsed to expand them in the first place, I showed every stage of the working, nothing left to the imagination,
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    There's two parts to proving a theorem: (1) Figuring out what the proof is, and (2) writing out the proof in a way that's convincing to other mathematicians. It doesn't matter how good you are at (1) if you can't do (2).
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    noting the most powerful tools to be induction and contradiction, I have noticed.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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    Quote Originally Posted by salsaonline
    There's two parts to proving a theorem: (1) Figuring out what the proof is, and (2) writing out the proof in a way that's convincing to other mathematicians. It doesn't matter how good you are at (1) if you can't do (2).
    If you don't understand my proof you don't understand maths.
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  70. #69  
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    Quote Originally Posted by esbo
    Quote Originally Posted by salsaonline
    There's two parts to proving a theorem: (1) Figuring out what the proof is, and (2) writing out the proof in a way that's convincing to other mathematicians. It doesn't matter how good you are at (1) if you can't do (2).
    If you don't understand my proof you don't understand maths.
    A bold statement, I'll give it that.
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  71. #70  
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    I understand exactly what you did and I understand that it shows nothing.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  72. #71  
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    duplicate post due to glitch deleted.
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  73. #72  
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    Quote Originally Posted by esbo
    Quote Originally Posted by salsaonline
    There's two parts to proving a theorem: (1) Figuring out what the proof is, and (2) writing out the proof in a way that's convincing to other mathematicians. It doesn't matter how good you are at (1) if you can't do (2).
    If you don't understand my proof you don't understand maths.
    Let me put the record quite straight here. Salsaonline understands maths (to use the British term) quite well. He has a Ph.D. in the subject. He is not the only one here who does.

    The problem is probably that salsaonline DOES understand what you posted as your proof, while you do not. You accomplished absolutely nothing except for wasting everyone's time with meaningless manipulation of symbols. And you don't understand that simple fact. You don't understand your own "proof" and why it proves nothing.

    I have had enough contact with salsaonline to be able to assert with complete confidence that he could do more and deeper matheematics at lunch, on napkin, with a crayon, suffering from a hangover, than you could do in a decade under the best of circumstances.

    Attacking salsaonline, or anyone else. with such drivel is not the way to remain ignored, which the best for which you can hope.


    Quote Originally Posted by esbo
    If you don't believe me I will send it so some maths insitute to have it verified for you, agreed?
    Send it, or put it, anywhere that you like. If you need a suggestion as to where to put it, just ask for it. I would be surprised if any reputable institution even botherd to reply. It has already been read and evaluated professionally, and found badly wanting, in any case.[/quote]
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    Hha ha Dr Rocket calls himself a professionial however he either has the inability to count up to two or his memory is so bad that he can't remember what he has just posted


    Any how my proof is correct and unless he made a mistake in his proof, it is the same as mine, which I can show fairly easilly so De Rocket is "bidding against himself" on this matter
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  75. #74  
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    Quote Originally Posted by salsaonline
    Quote Originally Posted by esbo
    Quote Originally Posted by salsaonline
    There's two parts to proving a theorem: (1) Figuring out what the proof is, and (2) writing out the proof in a way that's convincing to other mathematicians. It doesn't matter how good you are at (1) if you can't do (2).
    If you don't understand my proof you don't understand maths.
    A bold statement, I'll give it that.
    Bold in the same sense that playing in traffic is bold, with about the same general impression of awareness of the environment.
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  76. #75  
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    Quote Originally Posted by esbo
    Hha ha Dr Rocket calls himself a professionial however he either has the inability to count up to two or his memory is so bad that he can't remember what he has just posted


    Any how my proof is correct and unless he made a mistake in his proof, it is the same as mine, which I can show fairly easilly so De Rocket is "bidding against himself" on this matter
    Yep, you correctly noticed a problem that occurred when I made a post. Yep, it came it out twice.

    Congratulations, that is the very first correct observation that you have made in this forum.

    I corrected my mistake. Do you intend to undertake the Herculean task of correcting yours ?
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  77. #76  
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by esbo
    Hha ha Dr Rocket calls himself a professionial however he either has the inability to count up to two or his memory is so bad that he can't remember what he has just posted


    Any how my proof is correct and unless he made a mistake in his proof, it is the same as mine, which I can show fairly easilly so De Rocket is "bidding against himself" on this matter
    Yep, you correctly noticed a problem that occurred when I made a post. Yep, it came it out twice.

    Congratulations, that is the very first correct observation that you have made in this forum.

    I corrected my mistake. Do you intend to undertake the Herculean task of correcting yours ?
    Well as I have not made any mistakes it will be a Herculean task task indeed.
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  78. #77  
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    Mod note esbo: Your personal attacks are inappropriate in this sub-forum. I happen to think they are unfounded, but that is beside the point; if you want to discuss mathematics, this is the place for you, but I will delete any further insults from you to other members here, whether they are to those of good standing and proven expertise or any others.

    Is that clear?

    PS to all: I apologize for not being active here recently, and most specifically for failing to moderate theis sub-forum with sufficient rigour.

    My only excuse, if it is one, is that I have been really busy in my so-called real life
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    Quote Originally Posted by Guitarist
    Mod note esbo: Your personal attacks are inappropriate in this sub-forum. I happen to think they are unfounded, but that is beside the point; if you want to discuss mathematics, this is the place for you, but I will delete any further insults from you to other members here, whether they are to those of good standing and proven expertise or any others.

    Is that clear?

    PS to all: I apologize for not being active here recently, and most specifically for failing to moderate theis sub-forum with sufficient rigour.

    My only excuse, if it is one, is that I have been really busy in my so-called real life
    If you are to delete 'insults' made by me then I trust you will also be deleting insultss made against me.
    If you have a rule you must apply it to all.

    If you let people insult me then I have the right to respond, that's fair isn't it?

    Or wil you allow people to insult me and me not to respond, as is the norm for mods?

    Furthermore perhaps you would like to list my 'insults' because I am struggling to find them butI can find a fair few aimed at me.
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    OK you will be pleased to know I have over come my difficulty with the concept of integration, or the fundamental theorum of calculus or whatever it is called if it is indeed called that. I have always had a problem with it and that would be over 20 years now, although I could quite happily do and integration because I knew it was the reverse of differentiation (or whatever).
    Anyway after much time struggling with the and and and integrals and signs and what have you etc... I finally got to a solution I am happy with and it did not really involve much at all.

    What I did was to turn it into something real rather than some abstract dense fiddly maths. So what I did was replace 'x' with 'time', and y with distance so I would have a graph where y was the distance and it would be the shape of '' for example.
    Now I know when I differentiate this graph I get speed which would be so if I want to integrate the area under I can say well the area under 3x^2 is the distance because if you chop it up into small sections of time, the distance in each section is simply the time times the speed at that time.
    It won't be accurate because the speed changes constantly but if I make the time intival smaller it will be more accurate, and if I take the limit as it approaches zero time it will indeed be accurate.
    However I already know the distance from the initial graph so that the answer!!

    I wish my maths teacher had explained it to me like that at the time, it would have saved me and indeed you a lot of trouble and grief

    However he did not explain it like that at all, he did something probably similar to what Dr Rocket did and other text perhaps and left me with loads of mathematical symbols spinning around my head making me more confused

    I would say Dr Rocket was correct on one point though, that is if you want to learn maths you have to figure it out for yourself.

    Up until that point I used to figure out all the answers myself anyway, so whilst the teacher was 'explaning' how to do things I basically did not listen and just thought about what they were trying to explain and figured out the answer myself.

    I think as that maths got harder I stopped doing this, perhaps because they made it sound so complicated in the first place.

    But anyway I can understand it for myself now and I don't need fancy words and symbols to do it with

    I don't need leemas or indeed possums

    I suppose as a bonus I learned to do a particualar proof by induction
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  81. #80  
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    Quote Originally Posted by esbo

    What I did was to turn it into something real rather than some abstract dense fiddly maths. So what I did was replace 'x' with 'time', and y with distance so I would have a graph where y was the distance and it would be the shape of '
    That is actually a good way to think of derivatives and integrals, and it is a very common topic in introductory calculus classes, and also in introductions to differential equations.

    It is close to the motivation of Newton when he invented calculus. His purpose was, in large part, to develop a theory of mechanics, particularly celestial mechanics governed by gravity, but in any case he developed what is now called classical mechanics.

    Classical mechanics, boiled down to its bare essentials is simply studying the position, velocity and acceleration of a point mass as a function of time. So you start with position as a function of time, differentiate it to get velocity, and differentiate again to get acceleration. To go backwards from acceleration to velocity to position you integrate repeatedly.

    If that model helps to understand the integral and derivative in the abstract then it is a good model for you. Understanding of these things is a somewhat personal matter, and whatever works for an individual is good for that individual. Fiunding a perspective that works for you is part of learning for yourself.
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  82. #81 Loved the discussion 
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    Loved the discussion;
    It reminded me of my child hood days in the projects of South Boston. We were all as dumb as rocks there. We used to think that NASA used vaginal fluid for the lubricants on the Space ships because it was the best lubricant to be had.

    The last time I heard a conversation as good as this one was when my dog got into it with a rock. You guys were arguing with a blind man about what color the leaves on a tree are.

    Like I said, loved the conversation, thank Gods it's over though I got a ton of work to do.

    Thanks for the fun
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    If they had used knicker elastic to hold the tiles on they might not have lost so many space shuttles. What on earth were they thinking off? That's another story though.
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  84. #83  
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    HAHAHA

    that actually made me laugh aloud


    everyone else who laughed, we are ALL nerds!, I went and told my family, not a single one laughed


    I can't belive I'm a nerd :? :?
    It's not how many questions you ask, but the answers you get - Booms

    This is the Acadamy of Science! we don't need to 'prove' anything!
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    Your familly just have no sense of humour.
    Probably works best in text.
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