The answer is π^2/6: What’s the question?(blog.republicofmath.com)
blog.republicofmath.com
The answer is π^2/6: What’s the question?
http://www.blog.republicofmath.com/archives/4801
11 comments
That's not really the geometric interpretation of the Riemann zeta function, but rather the series \sum_{k=1}^\infty 1/k^n. Riemann zeta function makes sense not only in positive integers greater than 1, but also on the almost whole complex plane, and identifying complex zeros of zeta function is of main interest.
Okay, so it's an interpretation of Riemann zeta on positive real integers. It's still neat. ;)
From the article: "Here’s a simple yet revealing question to ask people at all levels of mathematical attainment: 'The answer is 10. What is the question?'"
I know I'd love to hear HN's answers.
I know I'd love to hear HN's answers.
Here's one:
How do you express the number of digits that exist in any given base, in that base?
If you're using Arabic numerals, that is.
[deleted]
A related comic which gives away the answer:
http://cowbirdsinlove.com/43
http://cowbirdsinlove.com/43
what is the binary representation of 2?
Theorem without proof is unsatisfying.
This includes a brief sketch of Euler's argument as well as a several formal methods of proof:
http://en.wikipedia.org/wiki/Basel_problem
http://en.wikipedia.org/wiki/Basel_problem
You can find the proof by looking for documentation about the zeta function.
A little harder: the answer is pi^2/(12 log 2). What's the question?
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Standard way to do this is with Fourier series on the sawtooth wave.
http://math.mit.edu/classes/18.03/spr10/lec_22.txt
http://math.mit.edu/classes/18.03/spr10/lec_22.txt
[deleted]
I like this a lot, it seems like a classic example of divergent thinking, in a subject (math) that doesn't traditionally involve a lot of it.
http://en.wikipedia.org/wiki/Divergent_thinking
http://en.wikipedia.org/wiki/Divergent_thinking
> divergent thinking, in a subject (math) that doesn't traditionally involve a lot of it.
I strongly disagree with that.
Divergent thinking as well as creativity is very important in mathematics and has always been. Solving problems - this is what mathematics is all about. So creativity is vital, and more important in math than in many other professions that call themselves "creative".
It is a common misconception that math is about following calculation steps and simple algorithms. This is "math" as taught in school, and it is okay because those are the very basics. However, as soon as you join mathmatical competitions, or even start studying math, things are different: It is no longer about using math but about understanding and maybe even extending math.
That means: defintions, theorems and proofs. And, of course, always keeping some examples, visualizations, simplified views, and heuristics in mind. Those help you to stay on top of things, so you don't get lost in this forest.
To find proofs, or even just good definitions (which is sometimes as important), you have to be very creative. Learning other proofs means getting lots of ideas about what could work, but as soon as all well-known tricks are exhausted, there are still lots of open questions whose definite answer (i.e. proof or counter-proof) will need yet another genius. Or just another student whose thinking diverges into directions that others haven expoited yet. Or maybe they also thought about that, but didn't take this seriously or didn't try hard enough.
Dedication to an concrete problem at hand is in no way a contradiction to divergent thinking. It is more about the combination of convergent and divergent thinking. Sticking to only one of those, and you either won't get far enough, or be lost into totally secluded corners without even noticing.
Also don't be fooled by math competitions (which are very different from plain calculation competitions!). Students may get excercises that are fairly well-known to mathematicians. However, in the view of the students those are new, unknown problems they almost certainly haven't heard of up to that point. They might be well-prepared with lots of standard tricks and maybe know about similar-looking issues. But in the end of the day, they have to be creative - very creative - to find their proof.
It is not uncommon for the jury to receive a solution from a student the original creators haven't thought of. Usually, such an exceptionally creative solution leads to a special award on the competition.
Also, usually the students get multiple tasks, so when they get stuck with one excersice, they can skip to the next one. When they later come back to the first one, it usually makes a lot more sense and have some more ideas to explore.
I strongly disagree with that.
Divergent thinking as well as creativity is very important in mathematics and has always been. Solving problems - this is what mathematics is all about. So creativity is vital, and more important in math than in many other professions that call themselves "creative".
It is a common misconception that math is about following calculation steps and simple algorithms. This is "math" as taught in school, and it is okay because those are the very basics. However, as soon as you join mathmatical competitions, or even start studying math, things are different: It is no longer about using math but about understanding and maybe even extending math.
That means: defintions, theorems and proofs. And, of course, always keeping some examples, visualizations, simplified views, and heuristics in mind. Those help you to stay on top of things, so you don't get lost in this forest.
To find proofs, or even just good definitions (which is sometimes as important), you have to be very creative. Learning other proofs means getting lots of ideas about what could work, but as soon as all well-known tricks are exhausted, there are still lots of open questions whose definite answer (i.e. proof or counter-proof) will need yet another genius. Or just another student whose thinking diverges into directions that others haven expoited yet. Or maybe they also thought about that, but didn't take this seriously or didn't try hard enough.
Dedication to an concrete problem at hand is in no way a contradiction to divergent thinking. It is more about the combination of convergent and divergent thinking. Sticking to only one of those, and you either won't get far enough, or be lost into totally secluded corners without even noticing.
Also don't be fooled by math competitions (which are very different from plain calculation competitions!). Students may get excercises that are fairly well-known to mathematicians. However, in the view of the students those are new, unknown problems they almost certainly haven't heard of up to that point. They might be well-prepared with lots of standard tricks and maybe know about similar-looking issues. But in the end of the day, they have to be creative - very creative - to find their proof.
It is not uncommon for the jury to receive a solution from a student the original creators haven't thought of. Usually, such an exceptionally creative solution leads to a special award on the competition.
Also, usually the students get multiple tasks, so when they get stuck with one excersice, they can skip to the next one. When they later come back to the first one, it usually makes a lot more sense and have some more ideas to explore.
I think the author of the post was talking about middle and high school students, who don't really see proofs. (I don't think triangle proofs in Geometry class count) My high school math felt mostly mechanical, and involved nothing like the kind of activity the author wrote about.
I agree that math in high-school should emphasize more on the creative part. It is harder, but for those students interested in it, showing to them what math is really all about seems to be the only way to keep them motivated. And math competitions help a lot in that regard, at least in my country (Germany).
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"If a lattice point is selected at random in two dimensions, the probability that it is visible from the origin is 6/π^2. This is also the probability that two integers picked at random are relatively prime. If a lattice point is picked at random in n dimensions, the probability that it is visible from the origin is 1/ζ(n), where ζ(n) is the Riemann zeta function."
[1] http://mathworld.wolfram.com/VisiblePoint.html