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Wollstlings erschien The Ghost Games, abgerundet mit dem ganzen Krper. Wer hat 30 Tage lang das Glck leben beide Seiten findet ihr in Lebensgefahr.

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Thank you for your comments, its something that many math teachers have expressed to me including lecturers at university which is what prompted us to write the piece; Its clear to me where I would draw the moral line and it isn't in the same place they have.

Well, I think the Numberphile guys made an entertaining video and I wish that my school maths teachers had been even half as good with their explanations as these guys are.

For me it was the teachers, with their leaps of faith, blasting through the set books and omitting whole chapters homework : study chapter so and so and do the exercises who instilled the fear of maths and a sense of futility into all but the three or four kids, out of a class of 25, who could figure out what was going on.

Maths teachers take a look in the mirror! Isn't it more important to recognise that maths is about adventure, discovery, fun?

This is simply an example of an important aspect of in mathematics - you take some concept, abstract it and extend it, and see where it leads.

Shouldn't more maths teaching and learning emphasise this? A nice example that I use in my classroom, far simpler than analytic continuation and one that survives the journey more intact, concerns the index laws.

You take the well understood concept that a times a times a These are not meaningful under the initial view that "power means repeated multiplication", but the extension and subsequent exploration lead to important and very meaningful results.

Learning the index laws can be an exercise in rote memorisation, or it can be a wonderful journey of discovery, where seeming "nonsense" becomes clarified and empowering!

More important than truth, precision, clarity, correctness, understanding, etc.? Many Numberphile videos are fun and all that in addition to being essentially correct.

However, they really blew it with the nonconvergent-series videos. You might consider steering students to Martin Gardner's books based on his Scientific American column "Mathematical Games"; even though he wasn't a mathematician, he had lots of contacts in the mathematical community, and his writing was clear, entertaining, and essentially correct.

I have to disagree, and am quite bemused by the moral tone taken up by some of the detractors of this work. In context, they are a valid exploration of mathematical ideas, and thus important and valuable.

In the history of maths, this happened with irrational numbers, negative numbers, imaginary numbers. Formal manipulations that included "nonsense" ended up enriching mathematics.

And in modern maths, look at p-adics where, for example. There's the Umbral Calculus where the formal basis is still only being constructed.

Also, the extended complex plane where infinity is just a point like any other. Now you can say they sneakily departed from real numbers, but the whole question naturally goes beyond the reals because it involves infinity.

Infinity is not a real number, but more relevant, it is not just a "really big number". That it is qualitatively different is important to learn, and apparent from ideas like divergent and non-absolutely convergent series.

And I use similar things to both communicate my love of maths, and to encourage others to look at it differently and find their own sources of wonder.

And it is not so far removed from school maths. We teach that infinity - infinity is undefined arithmetically, but in terms of sets we often show that it can have many values easiest example is remove all even numbers from 1,2,3,4, Exposure to such conflicts is a great and fun way to learn about the limits and context of maths.

And a major source of error is not learning the limits, applying things unthinkingly out of context, because too much exposure is only to "nice" examples.

There is loads of education research on this. These paradoxical results force students to confront the limits, and thus can be used to enhance their mathematical thinking.

The authors of the article recognise the broader context although were clearly not entirely happy with the presentation.

The commenter above, Matt E, a maths teacher, seems to miss this broader context. Of course, approaching and crossing boundaries may well mean things need to be redefined, concepts generalised, but that's mathematics.

Use it to generate interest, provide historical and real-world context, and thus enrich teaching. Thinking about Grandi's series is like a first step on a journey.

Enjoy it, and let students enjoy it too. Relate it to Thomson's Lamp. Bring mathematics to life! As the leading professor at the university of Cambridge with 10 PhDs within the world of maths, I have to quite frankly state your opinion is invalid.

You forgot to differentiate the function zx : zy accordingly before putting it into partial differentials. I really just wish they had put some sort of disclaimer in their video about their math trickery.

Now, a while after seeing the video I decided to do my own research and came upon this article which was very helpful in understanding what was actually going on.

I suppose I have a different perspective on math than the average person, but I was much more satisfied with this, correct explanation, than their hand-wavy one.

Think about it. Being wrong, is simply wrong. Even if this is true, you are assuming that the sum ends with I agree. I was staring at their addition of the two infinite series for a while trying to make sense of it.

It doesn't make sense. Not sure how to answer you. Non-convergent series are unpleasant and subtle things most 1st year mathematics text books will tell you the convergence properties of this sum.

Instead let me just quote from Abel in , "Divergent series are the invention of the devil and it is shamful to base on them any demonstration whatsoever".

Hardy's excellent book on the subject explains the rights and wrongs; the good the bad and the ugly. The terms in round brackets sum to Z, as do the terms in square brackets.

No, your argument is invalid, because you must realize that infinite summation is not commutative or associative, so putting brackets around certain numbers, and then re-ordering those numbers to change the order of the sum is not a valid operation.

This is what I've seen: there are ways that rearranging infinite series sums doesn't work. The expression for S -3 two lines below the two plates draw is wrong.

The Numberphile folks have a video that goes through basically the same area, talks about the sleight-of-hand in the shorter video, discusses analytic continuation and all that good stuff, and it's linked at the end of the short version.

Well worth a watch: www. The sum is nonconvergent. However you can very simply analytically continue the function that the sum represents beyond the radius of convergence but that is a continuation.

The sum itself is not defined, it is not convergent. It doesn't equal anything The one thing mathematicians do when making this tricky problem work, is change an equal sign to mean "associated with.

While I understand the point you are trying to make, your assertion is flawed in a simple way: you assume that "converges to" and "equals" are the same thing, when discussing its value.

Of course, the video and thus Ramanujan has the same flaw in their reasoning - they're reordering the infinite sums in a way that is inconsistent Anonymous proves their own point by using nonsense themselves?

I'm sorry, but why did you substitute Z with zero? Where the hell did you get that? I think you forgot that the equation is not a function, and you can't substitute it with a value of zero, since we don't know the value of Z yet.

You can't contradict a statement and substitute it with a value since the person is still trying to find the value. The pattern of Z alternates addition and subtraction, while the addition of the 1 - doubles up on subtraction.

If we assume the pattern continues and you stop at a given point, the two results do not match perhaps someone smarter than I could prove.

I'm not sure what you mean by "doubles up on subtraction. The equation. You got all of the previous finite examples right, but they are irrelevant.

Ok so I have read the article and watched the shorthand proof on numberphile. However don't the natural numbers have a closure property, meaning if you add any of the natural numbers together you must get a natural number?

I think my idea of adding all the natural numbers doesn't match with what is going on in the article, so if anyone could fill in the gaps that would be great.

Very good, the naturals are closed under addition. Only the so called regularised sum where "an infinity" has been subtracted- that "infinity" is obviously not a natural.

Ok thanks for the response, sorta wanna get my head around this incase I tell people about it. Where this all loses me is where the extended function gets its values from when you plug in values less than 1 somewhere in the article it says less than -1, is this a mistake, what about So exactly where does it get its value from?

This is the technical bit about analytic continuation. There is a way, beyond the scope of this article to extend functions of complex numbers once you know the value of the function elsewhere.

I understand why the numberphile people didn't want to bring that up because it is hard to explain nontechnically and indeed to give them credit where it is due, they have a discussion about it in an adjoining video and a written bit about it now too.

So basically there is a bit of maths using function of complex numbers that allow you to assign a value to a function once you know its values elsewhere.

Only a function that is related to that sum has that value. Where by related I mean that function has the same value as that sum when that sum is convergent.

The closure of the natural numbers under addition means that the sum of any two natural numbers is a natural numbers.

We can apply this principle again and again finitely many times to see that the sum of any finite number of natural numbers is a natural number.

This however says nothing of the sum of an infinite number of naturals, which we have in the case of an infinite series; in fact we see that this need not exist at all.

The limit of a series of rationals can be irrational. It element of the epsilon neighborhood of the partial sums Sn for a lage n.

Sn is still part of Q for all n. Thanks very much for this post - I feel like this is the follow up that like Numberphile is missing!

I'm not quite sure what Numberphile were thinking I'm all for promoting discussion and argument, but even I have "limits" -: Seriously though, I hope that Numberphile come back with a follow up video at some stage explaining themselves!

This is a misconception as these are infinite series that have no end. Christian Whittaker email me at cpwhittaker hotmail.

I am far from a mathematical genius. However, this discards the aforementioned issue and issue raised in your comment that it must conveniently end with Same goes for the second 'Z.

Correct me if I am wrong or have mistakenly supported your argument I think you misunderstood. If you place a 1 at the end it would mean 1 of 2 things.

Since the series is infinite So im not completely sure what you meant, but the math doesn't work out for a series. Also, the article got that point wrong, as already mentioned.

Therefor, I am not inclined to agree with this article. Thank you for the interesting explanations. That series, while not convergent, is still "Cesaro summable" [which means, that the average value of its first n partial sums converges in the limit as n is taken to infinity].

The Cesaro sum of series has many nice properties, e. The fallacies enter elsewhere, in the various manipulations of divergent and non-Cesaro summable series.

Such manipulations can be used to arrive at any value at all for the sum of all the natural numbers. Perhaps others have already made this point, as I've not viewed all of the linked and related sites.

Like others, I too would have wished to see some indication in the Numberphile video of the "tongue in cheek" nature of the "proof," without losing the huge entertainment value.

Here, the video is understandable to many middle grades students as well as high school students; this is great! But their teachers then need to know it is "sleight of hand," and encourage their students to question the "authority" behind it.

Teaching mathematics by authority rather than through understanding is a continuing, unfortunate trend in our schools, at least in the USA, and I cannot help but feel an opportunity here might be being missed.

Skip to main content. You can easily convince yourself of this by tapping into your calculator the partial sums and so on. Or, to put it more loosely, that the sum is equal to infinity.

Srinivasa Ramanujan. Extending the Euler zeta function As it stands the Euler zeta function S x is defined for real numbers x that are greater than 1.

From 4. Re: Upcoming 1. The combined length of "prefix" and "message" must be less than or equal to bytes.

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