x2-1000x-1000 and explain why they have those values. Use the result to estimate the roots of x2-(10100)x-10100, and approximate the error this estimate gives."
Use the little button up top to make your powers look nicer. It's so much more confusing when there's carats all over the place. As for the actual math problem...is that going to the the official problem of the week for this week?
It's a reasonable problem. Since it is indeed relatively easy, I'll give people some time as many should be able to do it.
Incidentally, if we're having "problems of the week", how do we handle them? One problem is, if we use equations as images we can't even use white text to hide them.
How about a spoiler-free problem thread and separate solution thread? Then again, that stops you from looking up solutions to one problem without seeing those for another. That leaves separate solution threads for each problem, which is rather annoying,
In short: no. The long answer is, that the sum of all positive integers is a really cool thing to get people interested in math, but, as has been pointed out, the result's derivation is not a good introduction into math.
I can, however, give a short sketch of how it is done, and some of the concepts involved. A caveat to all those of you with degrees in math: I am a theoretical physicist, what I do to math is condemned by several international treaties as well as the Geneva convention.
Lets start by thinking about the series 1+1/2+1/4+1/8..., it is very easy to see that this series continued infinitely will still have a finite value, which is 2 (draw a square, add half a square, add half of that, then again, continue until bored). Now think about the series 1+1/2+1/3+1/4+..., clearly there is more stuff there than in the previous series. It turns out that the sum of that series is actually infinite.
Now, if you raise a number that is greater than 1 to a power n, that is also greater than 1, the number grows (21.01=2.0139, 22=4, etc.), conversely the reciprocal of that number gets smaller. So you can ask yourself the following question: If I sum 1+1/2n+1/3n+1/4n+..., and for n=1 this sum is infinite, and I know that all those numbers get smaller for n > 1, then how big an "n" do I need before the sum is finite again?
The answer is, that for any value of n larger than 1, the above series is finite. The series is, in fact, one of the most common forms of the Riemann Zeta Function. This is, in my opinion, one of the most important functions in mathematics, with deep relations to cryptography, physics and number theory.
Now, how does this help us with our 1+2+3+4+... series? It is not difficult to see, that if you take n=-1 then 1+1/2-1+1/3-1+1/4-1+... = 1+2+3+4+..., however, the way I have described the Zeta function it is only valid for values of n > 1. At n=0 it is infinite, and raising a number that is greater than 1 to a power that is less than 1 makes that number smaller 20.99=1.986) -> the reciprocal is larger -> for values of n<1 the series is even more infinite! Or at least it looks that way.
The mathematical "trick" which helps us is something called analytic continuation. For this you need to know about complex numbers. If you don't, here is what you need to know: Think about all real numbers as a straight line. Think about a function, like x<sup>2 for example. This function "lives" on the line, meaning, you feed it a number from the line and it'll tell you a different number which will still be on the line. E.g., the number 5 -> 52 = 25.
Now think about a flat surface. The straight line of real numbers goes through it, but now instead of just choosing from points on the line, you can "feed" your function (lets say x2 again) any point from the surface, even ones that are away from the line. Not surprisingly, the function will give you back a new number which also lies on the surface, but not necessarily on the line. Points on the surface are complex numbers and the surface is called the complex plane.
Analytic continuation states that, if you know exactly how a function behaves somewhere on the "line", then you can often infer how the function behaves anywhere on the plane, including bits of the line that you previously didn't know about or had trouble with.
Finally, we can come back to our series 1+1/2n+1/3n+1/4n+..., which is a function of n, meaning you plug in an n, and as long as it is larger than 1 and lies on the real line, you get back another number that is also on the real line. There are, of course, fiddly bits, and you have to write the series in a much more complicated way than the above one to make any sense of what happens when you plug in a complex number for n, but essentially, what analytic continuation lets you do is "step out" onto the complex plane and circle around the infinity at n=1. Lo and behold, on the other side of n=1 you can step back onto the line and find out what things look like at n=-1 or other points of interest. This is how you get 1+2+3+4+...=-1/12. Also, just for kicks, n=0 gives 1+1+1+1+...=-1/2.
The link I provided earlier has some plots of what the Zeta function looks like when you give n complex values. It is slightly hard to visualize, since instead of going from a point on a line to another point on a line, you go from a point on a plane to another point on a plane, but the singularity (i.e. infinite part) of the function sitting at n=1 can be seen nicely.
That's amazing. What is this other one you speak of?
Someone made a version where the bear is replaced by an orifice of internet infamy. It is to date, one of the most disturbing things I have ever seen. Sadly, I can't find it on the internets right now, so you'll have to take my word for it.
Did anyone here take any of the High School Math competitions that University of Waterloo does every year? They have the biggest math department in the world. UWaterloo
Sorry guys. The LaTeX extension to Vanilla doesn't work. If it ever updates, I'll go back and take another look at it. For now you'll have to do it the hard way. that means use LaTeX somewhere else to generate an image. Then host that image online somewhere, like imageshack or whatnot. Then link to that image in the forum.
I could code something to make this easier, but I have too many other things on the plate right now.
Comments
As for the actual math problem...is that going to the the official problem of the week for this week?
Incidentally, if we're having "problems of the week", how do we handle them?
One problem is, if we use equations as images we can't even use white text to hide them.
How about a spoiler-free problem thread and separate solution thread?
Then again, that stops you from looking up solutions to one problem without seeing those for another.
That leaves separate solution threads for each problem, which is rather annoying,
I can, however, give a short sketch of how it is done, and some of the concepts involved. A caveat to all those of you with degrees in math: I am a theoretical physicist, what I do to math is condemned by several international treaties as well as the Geneva convention.
Lets start by thinking about the series 1+1/2+1/4+1/8..., it is very easy to see that this series continued infinitely will still have a finite value, which is 2 (draw a square, add half a square, add half of that, then again, continue until bored). Now think about the series 1+1/2+1/3+1/4+..., clearly there is more stuff there than in the previous series. It turns out that the sum of that series is actually infinite.
Now, if you raise a number that is greater than 1 to a power n, that is also greater than 1, the number grows (21.01=2.0139, 22=4, etc.), conversely the reciprocal of that number gets smaller. So you can ask yourself the following question: If I sum 1+1/2n+1/3n+1/4n+..., and for n=1 this sum is infinite, and I know that all those numbers get smaller for n > 1, then how big an "n" do I need before the sum is finite again?
The answer is, that for any value of n larger than 1, the above series is finite. The series is, in fact, one of the most common forms of the Riemann Zeta Function. This is, in my opinion, one of the most important functions in mathematics, with deep relations to cryptography, physics and number theory.
Now, how does this help us with our 1+2+3+4+... series? It is not difficult to see, that if you take n=-1 then 1+1/2-1+1/3-1+1/4-1+... = 1+2+3+4+..., however, the way I have described the Zeta function it is only valid for values of n > 1. At n=0 it is infinite, and raising a number that is greater than 1 to a power that is less than 1 makes that number smaller 20.99=1.986) -> the reciprocal is larger -> for values of n<1 the series is even more infinite! Or at least it looks that way.
The mathematical "trick" which helps us is something called analytic continuation. For this you need to know about complex numbers. If you don't, here is what you need to know: Think about all real numbers as a straight line. Think about a function, like x<sup>2 for example. This function "lives" on the line, meaning, you feed it a number from the line and it'll tell you a different number which will still be on the line. E.g., the number 5 -> 52 = 25.
Now think about a flat surface. The straight line of real numbers goes through it, but now instead of just choosing from points on the line, you can "feed" your function (lets say x2 again) any point from the surface, even ones that are away from the line. Not surprisingly, the function will give you back a new number which also lies on the surface, but not necessarily on the line. Points on the surface are complex numbers and the surface is called the complex plane.
Analytic continuation states that, if you know exactly how a function behaves somewhere on the "line", then you can often infer how the function behaves anywhere on the plane, including bits of the line that you previously didn't know about or had trouble with.
Finally, we can come back to our series 1+1/2n+1/3n+1/4n+..., which is a function of n, meaning you plug in an n, and as long as it is larger than 1 and lies on the real line, you get back another number that is also on the real line. There are, of course, fiddly bits, and you have to write the series in a much more complicated way than the above one to make any sense of what happens when you plug in a complex number for n, but essentially, what analytic continuation lets you do is "step out" onto the complex plane and circle around the infinity at n=1. Lo and behold, on the other side of n=1 you can step back onto the line and find out what things look like at n=-1 or other points of interest. This is how you get 1+2+3+4+...=-1/12. Also, just for kicks, n=0 gives 1+1+1+1+...=-1/2.
The link I provided earlier has some plots of what the Zeta function looks like when you give n complex values. It is slightly hard to visualize, since instead of going from a point on a line to another point on a line, you go from a point on a plane to another point on a plane, but the singularity (i.e. infinite part) of the function sitting at n=1 can be seen nicely.
I could code something to make this easier, but I have too many other things on the plate right now.
It's MATHcore! HA HA! Get it? GET IT?