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Hi. I haven't done much work on limits and so need a bit of help with this problem.
Given that where m is a positive integer and n is a non-negative integer, show that .
Do you just make the substitution ? Thanks 92.7.54.6 ( talk) 16:04, 23 June 2009 (UTC)
Hi, I am working on qualifying exam problems, so I may start asking a lot of analysis type questions, mostly real analysis. For now, I need to construct a bijection from the natural numbers onto the rationals... in other words, I need to construct a sequence of all rationals without repeating. That's easy. But, I need more. Say g(k) is this function. I need an enumeration such that
diverges. And, I also need to know if there exists any such enumeration such that it converges.
My first thought was to just try the obvious enumeration and see if I get anywhere: 0, 1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5... . But, then I could not show that series converged or diverged. I tried root test but you get a limsup of 1 which is inconclusive. Ratio test, limit must exist... or you can use the one in Baby Rudin but it does not work either. I thought about switching it up a bit to make it larger... just for each denominator, write the fractions in decreasing order instead of increasing. But, same thing. Any ideas? Thanks. StatisticsMan ( talk) 19:50, 23 June 2009 (UTC)
How would you go about showing that ≤1? I don't know where to start. Thanks Scanning time ( talk) 20:24, 23 June 2009 (UTC)
I'm reading a programming book in which the author gave an example method which takes an array of doubles and returns their mean. If the array is empty, the function returns 0, which the author explains with the comment "average of 0 is 0". This annoyed me quite a bit, since the author is obviously confusing the mean of no terms with the mean of a single zero term. IMO the function should return NaN (or throw an exception), since an empty mean is 0/0.
Does anyone have any further input on the matter? More importantly, is anyone aware of an online resource discussing this (my search in Wikipedia and Google didn't come up with anything)? Thanks. -- Meni Rosenfeld ( talk) 22:59, 23 June 2009 (UTC)
The mean is not just a linear combination of the values; it's an affine combination, i.e. a linear combination in which the sum of the coefficients is 1. As
John Baez likes to say, an affine space is a space that has forgotten its origin. It has no "zero". The point is that if you decide, e.g., that the zero point is located HERE, and measure the locations of the points you're averaging relative to that, and I decide the zero point is located somewhere else, and likewise compute the average, then we get the same result! That's not true of sums, and it's not true of linear combinations in which the sum of the coefficients is anything other than 1. If the "empty mean" were "0", then there would have to be some special point called "0".
Michael Hardy (
talk) 23:30, 23 June 2009 (UTC)
...Just for concreteness: Say we're measuring heights above sea level, and you get 0, 1, and 5 (after measuring the heights at high tide). Your average is
(0 + 1 + 5)/3 = 2 feet above sea level. I measure the heights at low tide when the water is 4 feet lower, so I get 4, 5, and 9. My average is (4 + 5 + 9)/3 = 6 feet above sea level. You got 2 feet. I got 6 feet, when the water is 4 feet lower. So we BOTH got the same average height. But if it were correct to say the average of 0 numbers is 0, then should that 0 be your sea level, or mine? There's no non-arbitrary answer. So it doesn't make sense to say the "empty average" is 0.
Michael Hardy (
talk) 23:36, 23 June 2009 (UTC)
Ok, thanks for the answers. Looks like we all agree, with different ways to look at it. Perhaps someone will be interested in adding a note about this to one of our articles (personally I've stopped editing articles)? -- Meni Rosenfeld ( talk) 08:55, 24 June 2009 (UTC)
Taking the mean of an empty set could be an arbitrary convention, and it is possible that there are circumstances where this would be useful. On the other hand I think it is more likely to arise as a result of a fundamental difficulty which many (perhaps most) people have in connection many ideas related to zero and empty sets. I have had considerable difficulty in trying to convey to non-mathematical people that a value not existing is not the same as the value existing and having the value zero (for example, how many daughters has the king of Germany?). Likewise conveying the distinction between an empty set and no set. A final example is people who are convinced that 12÷0 must be 0. It seems that to many people it is very difficult to conceive of anything involving 0 which produces anything other than 0. On the other hand it once took me a considerable time to persuade someone (of fairly average intelligence in most respects) that 6×0 was 0. He said "if you start with 6 and then do nothing you must still have 6". I eventually persuaded him that it was possible to meaningfully have an operation involving 0 which did not mean "do nothing", but it was hard work. Another example is people who feel 0!=1 is completely unnatural. Zero is a surprisingly confusing and difficult concept for many people. JamesBWatson ( talk) 11:47, 24 June 2009 (UTC)
A difficulty leaning mathematics is that the meaning of the word "many" changes from signifying three or more to signifying zero or more. Asked "how many?" a mathematician may answer "zero" while a nonmathematician answers "no, not many, not even one". To the nonmathematician "empty set" and "number zero" is nonsense because empty is not a set and zero is not a number, as you don't count to zero. Redefining common words must be carefully explained. Bo Jacoby ( talk) 06:48, 25 June 2009 (UTC).
Well sir, you wrote that you "have never met anyone", not that you did meet zero. So you yourself prefer to express zero by "never anyone". If I said that I knew many more examples, wouldn't you understand "many more" to mean more than zero more? Bo Jacoby ( talk) 23:13, 25 June 2009 (UTC).
I have a better answer than all of the above. People expect the mean to be a kind of average. Like, "what weight girl (average) is at that Linux users group meeting"... Well the word average in that question is ambiguous, but whether the person would find the mean, the median, or the mode the information of greatest interest, for any of the three THEY WOULD IMMEDIATELY understand the meaning of the answer "empty set", and this is the correct result returned. I'm therefore quite sure that this is the function's best response, as anyone asking the question would understand this answer to mean there are't any girls. By contrast 0 implies zero-G conditions (without any implication of nobody being there) while NaN would seem to say "the numbers on the scale don't go up that far"... :) 193.253.141.64 ( talk) 21:41, 25 June 2009 (UTC)
Let me address the zero-G comment first, then we could go on to the other parts of my argument if there is no dispute. The zero-G comment is to show that returning zero is absolutely wrong and should be out of the question in all cases - it should not even be considered for a second. I realize this isn't the physics ref desk, so let me illustrate with an even more forceful example. Go ahead and answer the following question:
“ | In reflecting over all the times you have shot somebody in the face, what is the average number of warning shots you fired first? | ” |
What is your answer? I don't think any of us here would answer 0 to that question, and that is because 0 is not an appropriate answer. So as mathematicians, how would YOU answer it? I know when *I* reflect over all the times I have shot somebody in the face, I come up with the empty set! The average distance in centimeters the tip of the barrel was from their face? Empty set. Average number of beers I had prior to shooting them? Empty set. Average number of times I shot each person? Empty set. Why... What are YOUR answers to these questions?
Can we agree on this much: zero is totally inappropriate for any of the above questions, and more generally, for the average of an empty set... -- 82.234.207.120 ( talk) 13:52, 26 June 2009 (UTC)
You can rationalize it all you want, but the fact remains that the concept of an average EXISTS to be a "typical" item that is middling in the data. Further the concept of the empty set exists to represent the absence of any items. I suppose you think asking for even primes above two should throw some obscure error too? A middling reduction of no items would be "no item" not a result like 0/0... 193.253.141.80 ( talk) 04:26, 27 June 2009 (UTC)
I realize what the problem is: you're mathematician, while I am smarter than mathematics. Here is an example: the fact that the definition of median is WRONG - for an even number of items the median should be the middle two items - NOT their average. Mathematics is wrong. But you, being a mathematician, and not smarter than math (unlike me), would say "a definition can't be 'wrong' - it just is.". Well you would be wrong in that assertion. Mathematics simply is wrong on the definition of the median, and as for the average of the empty set, I don't know what mathematics (and hence mathematicians like you) have to say on the subject, but the correct answer, the TRUTH is that the average is the empty set. 193.253.141.80 ( talk)|
Of course it is well-defined: I said I was smarter than mathematics, not that I was a crank! Anyway I don't need mathematics, mathematics needs me. Your argument about the data being a sample is ridiculously inapplicable to the question, as the empty set is certainly not a sample of anything.... It would mean you didn't collect data at all. The average data item you collected was likewise nothing at all. The only motive you might have for hiding this fact about the average data item you collected is if you wanted to bill for collecting the sample, by making it seem like there was work done collecting the samples, but you ended up excavating samples that weren't numbers. You want to bill for an average sample of NaN whereas the truth is that you did no work at all, which would if you were honest be reflected in the fact that your "average" sample is the null set. More fully, there is not a single sample in your set of samples. Have the balls to report it rather than trying to compromise mathematics with a disinformation campaign to serve your own (or your employer's) narrow interests. 193.253.141.64 ( talk) 23:07, 27 June 2009 (UTC)
Mathematics desk | ||
---|---|---|
< June 22 | << May | June | Jul >> | June 24 > |
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. |
Hi. I haven't done much work on limits and so need a bit of help with this problem.
Given that where m is a positive integer and n is a non-negative integer, show that .
Do you just make the substitution ? Thanks 92.7.54.6 ( talk) 16:04, 23 June 2009 (UTC)
Hi, I am working on qualifying exam problems, so I may start asking a lot of analysis type questions, mostly real analysis. For now, I need to construct a bijection from the natural numbers onto the rationals... in other words, I need to construct a sequence of all rationals without repeating. That's easy. But, I need more. Say g(k) is this function. I need an enumeration such that
diverges. And, I also need to know if there exists any such enumeration such that it converges.
My first thought was to just try the obvious enumeration and see if I get anywhere: 0, 1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5... . But, then I could not show that series converged or diverged. I tried root test but you get a limsup of 1 which is inconclusive. Ratio test, limit must exist... or you can use the one in Baby Rudin but it does not work either. I thought about switching it up a bit to make it larger... just for each denominator, write the fractions in decreasing order instead of increasing. But, same thing. Any ideas? Thanks. StatisticsMan ( talk) 19:50, 23 June 2009 (UTC)
How would you go about showing that ≤1? I don't know where to start. Thanks Scanning time ( talk) 20:24, 23 June 2009 (UTC)
I'm reading a programming book in which the author gave an example method which takes an array of doubles and returns their mean. If the array is empty, the function returns 0, which the author explains with the comment "average of 0 is 0". This annoyed me quite a bit, since the author is obviously confusing the mean of no terms with the mean of a single zero term. IMO the function should return NaN (or throw an exception), since an empty mean is 0/0.
Does anyone have any further input on the matter? More importantly, is anyone aware of an online resource discussing this (my search in Wikipedia and Google didn't come up with anything)? Thanks. -- Meni Rosenfeld ( talk) 22:59, 23 June 2009 (UTC)
The mean is not just a linear combination of the values; it's an affine combination, i.e. a linear combination in which the sum of the coefficients is 1. As
John Baez likes to say, an affine space is a space that has forgotten its origin. It has no "zero". The point is that if you decide, e.g., that the zero point is located HERE, and measure the locations of the points you're averaging relative to that, and I decide the zero point is located somewhere else, and likewise compute the average, then we get the same result! That's not true of sums, and it's not true of linear combinations in which the sum of the coefficients is anything other than 1. If the "empty mean" were "0", then there would have to be some special point called "0".
Michael Hardy (
talk) 23:30, 23 June 2009 (UTC)
...Just for concreteness: Say we're measuring heights above sea level, and you get 0, 1, and 5 (after measuring the heights at high tide). Your average is
(0 + 1 + 5)/3 = 2 feet above sea level. I measure the heights at low tide when the water is 4 feet lower, so I get 4, 5, and 9. My average is (4 + 5 + 9)/3 = 6 feet above sea level. You got 2 feet. I got 6 feet, when the water is 4 feet lower. So we BOTH got the same average height. But if it were correct to say the average of 0 numbers is 0, then should that 0 be your sea level, or mine? There's no non-arbitrary answer. So it doesn't make sense to say the "empty average" is 0.
Michael Hardy (
talk) 23:36, 23 June 2009 (UTC)
Ok, thanks for the answers. Looks like we all agree, with different ways to look at it. Perhaps someone will be interested in adding a note about this to one of our articles (personally I've stopped editing articles)? -- Meni Rosenfeld ( talk) 08:55, 24 June 2009 (UTC)
Taking the mean of an empty set could be an arbitrary convention, and it is possible that there are circumstances where this would be useful. On the other hand I think it is more likely to arise as a result of a fundamental difficulty which many (perhaps most) people have in connection many ideas related to zero and empty sets. I have had considerable difficulty in trying to convey to non-mathematical people that a value not existing is not the same as the value existing and having the value zero (for example, how many daughters has the king of Germany?). Likewise conveying the distinction between an empty set and no set. A final example is people who are convinced that 12÷0 must be 0. It seems that to many people it is very difficult to conceive of anything involving 0 which produces anything other than 0. On the other hand it once took me a considerable time to persuade someone (of fairly average intelligence in most respects) that 6×0 was 0. He said "if you start with 6 and then do nothing you must still have 6". I eventually persuaded him that it was possible to meaningfully have an operation involving 0 which did not mean "do nothing", but it was hard work. Another example is people who feel 0!=1 is completely unnatural. Zero is a surprisingly confusing and difficult concept for many people. JamesBWatson ( talk) 11:47, 24 June 2009 (UTC)
A difficulty leaning mathematics is that the meaning of the word "many" changes from signifying three or more to signifying zero or more. Asked "how many?" a mathematician may answer "zero" while a nonmathematician answers "no, not many, not even one". To the nonmathematician "empty set" and "number zero" is nonsense because empty is not a set and zero is not a number, as you don't count to zero. Redefining common words must be carefully explained. Bo Jacoby ( talk) 06:48, 25 June 2009 (UTC).
Well sir, you wrote that you "have never met anyone", not that you did meet zero. So you yourself prefer to express zero by "never anyone". If I said that I knew many more examples, wouldn't you understand "many more" to mean more than zero more? Bo Jacoby ( talk) 23:13, 25 June 2009 (UTC).
I have a better answer than all of the above. People expect the mean to be a kind of average. Like, "what weight girl (average) is at that Linux users group meeting"... Well the word average in that question is ambiguous, but whether the person would find the mean, the median, or the mode the information of greatest interest, for any of the three THEY WOULD IMMEDIATELY understand the meaning of the answer "empty set", and this is the correct result returned. I'm therefore quite sure that this is the function's best response, as anyone asking the question would understand this answer to mean there are't any girls. By contrast 0 implies zero-G conditions (without any implication of nobody being there) while NaN would seem to say "the numbers on the scale don't go up that far"... :) 193.253.141.64 ( talk) 21:41, 25 June 2009 (UTC)
Let me address the zero-G comment first, then we could go on to the other parts of my argument if there is no dispute. The zero-G comment is to show that returning zero is absolutely wrong and should be out of the question in all cases - it should not even be considered for a second. I realize this isn't the physics ref desk, so let me illustrate with an even more forceful example. Go ahead and answer the following question:
“ | In reflecting over all the times you have shot somebody in the face, what is the average number of warning shots you fired first? | ” |
What is your answer? I don't think any of us here would answer 0 to that question, and that is because 0 is not an appropriate answer. So as mathematicians, how would YOU answer it? I know when *I* reflect over all the times I have shot somebody in the face, I come up with the empty set! The average distance in centimeters the tip of the barrel was from their face? Empty set. Average number of beers I had prior to shooting them? Empty set. Average number of times I shot each person? Empty set. Why... What are YOUR answers to these questions?
Can we agree on this much: zero is totally inappropriate for any of the above questions, and more generally, for the average of an empty set... -- 82.234.207.120 ( talk) 13:52, 26 June 2009 (UTC)
You can rationalize it all you want, but the fact remains that the concept of an average EXISTS to be a "typical" item that is middling in the data. Further the concept of the empty set exists to represent the absence of any items. I suppose you think asking for even primes above two should throw some obscure error too? A middling reduction of no items would be "no item" not a result like 0/0... 193.253.141.80 ( talk) 04:26, 27 June 2009 (UTC)
I realize what the problem is: you're mathematician, while I am smarter than mathematics. Here is an example: the fact that the definition of median is WRONG - for an even number of items the median should be the middle two items - NOT their average. Mathematics is wrong. But you, being a mathematician, and not smarter than math (unlike me), would say "a definition can't be 'wrong' - it just is.". Well you would be wrong in that assertion. Mathematics simply is wrong on the definition of the median, and as for the average of the empty set, I don't know what mathematics (and hence mathematicians like you) have to say on the subject, but the correct answer, the TRUTH is that the average is the empty set. 193.253.141.80 ( talk)|
Of course it is well-defined: I said I was smarter than mathematics, not that I was a crank! Anyway I don't need mathematics, mathematics needs me. Your argument about the data being a sample is ridiculously inapplicable to the question, as the empty set is certainly not a sample of anything.... It would mean you didn't collect data at all. The average data item you collected was likewise nothing at all. The only motive you might have for hiding this fact about the average data item you collected is if you wanted to bill for collecting the sample, by making it seem like there was work done collecting the samples, but you ended up excavating samples that weren't numbers. You want to bill for an average sample of NaN whereas the truth is that you did no work at all, which would if you were honest be reflected in the fact that your "average" sample is the null set. More fully, there is not a single sample in your set of samples. Have the balls to report it rather than trying to compromise mathematics with a disinformation campaign to serve your own (or your employer's) narrow interests. 193.253.141.64 ( talk) 23:07, 27 June 2009 (UTC)