Tuesday, February 4, 2014

astounding mathematics

this numberphile video shows a fun phenomenon of mathematics that completely blows my mind every time:

1 + 2 + 3 + 4 + 5 + ... = -1/12


first of all, infinity is crazy.   we use infinity as a tool to understand physics, but we dont really get answers that are infinite.  well, i would describe to someone that the universe is infinitely large, but that doesnt mean i'm completely satisfied by it, it's just the best way to describe it right now.

this quote from dennis overbye in his new york times article on the video puts the issue slightly differently:

"Cosmologists do not know if the universe is physically infinite in either space or time, or what it means if it is or isn’t. Or if these are even sensible questions. They don’t know whether someday they will find that higher orders of infinity are unreasonably effective in understanding existence, whatever that is."

we dont understand existence; we dont understand consciousness; we dont understand infinity.   but we do understand a lot about physics - enough to gravitationally slingshot little robots around some planets in order to land on and explore other planets.   that's awesome.   

let's keep exploring and pushing the boundaries of existing knowledge!


Archie Kohmover said...

Well, don't believe everything you read, and don't automatically believe people with impressive sounding titles.

This physicist, Tony Padilla, seems to have a few serious gaps in his understanding of mathematics, and clearly, Dennis Overbye did not understand enough of anything anyone told him to write anything but a ridiculously uninformed article. He should remember not to write on topics he doesn't understand.

First, it is NOT mathematically valid to shift the position of terms of a sequence or series in just any case and declare that the series is the same as the one you started with. Therefore, Padilla's 2*S2 does NOT equal his S1 ( 1-1 + 1 ...) because he has actually added 1 + 2 + 3 + ... to 0 + 1 + 2 + 3 + ... If he understood real analysis, he would know better. The actual series 2*S2 is 2 - 4 + 6 - ..., but that doesn't fit his purpose. This is a serious fallacy, and it's not the only one.

Not wishing to write a whole article on his silly "proof," I'll just finish by noting that he starts out by talking about the natural numbers and claiming he is summing them to equal -1/12. Well, if he knew his modern algebra, he would see that that is impossible right off the bat. The natural numbers are a subset of the ring of the integers, and so every number that he is summing is an integer. Now, ring theory tells us that addition of two or more (up to infinity) integers cannot yield any number outside the ring of the integers. This is known as being closed under addition. It is a violation of ring theory to suggest that an infinite series of integers could yield a number that is not an integer. Furthermore, the ring of the integers is a subset of the field of the real numbers. In the field of the real numbers, the sum 1 + 2 + 3 + ... does not add up to a real number; it "diverges," which means, for most series, it goes to infinity, meaning it has no bound. If it equaled -1/12, we would say that the sum "converges" to -1/12, which means that as you keep summing the terms of the series, the sum comes arbitrarily close to -1/12. This is not true of this series of real numbers.

Now, at the beginning, Padilla mentions Rieman zeta functions, which he doesn't want to get into. Instead, he talks about summing "natural numbers." Well, maybe he doesn't want to talk about the Rieman zeta function because he doesn't know enough about it. The sum of the Riemann zeta function with terms that look very much like the terms he is summing would be a different matter. These sums converge to -1/12 BECAUSE THEY ARE COMPLEX NUMBERS--NOT REAL NUMBERS. Complex numbers behave in ways that can be astounding when you are unfamiliar with them. BUT PADILLA WASN'T TALKING ABOUT THE FIELD OF THE COMPLEX NUMBERS! He specifically sais he was summing natural numbers. And the numbers he presented were, in fact, a series of positive integers (or natural numbers). His "proof," his statements, and his whole presentation were ridiculously fallacious and false. Moreover, waving a book in front of the camera that appeared to show the same thing he was saying just allowed him to gloss over the differences in theory (which, one hopes, would have been explained adquately in that book) from what he was claiming, which was simply false.

By the way, his S1 series also diverges, but it has two bounds, an upper bound and a lower bound. We don't average them; the series simply diverges.

Finally, it's pathetic that a New York Times journalist actually coughed up such a hairball of an article on this topic, which he clearly failed to understand.

StuntTrader said...

-1/12 = 0.083333333333

StuntTrader said...

I'm not buying it.
There is no way that the sum of a series of positive numbers can result in a number smaller than any of the series constituents.

StuntTrader said...

After much googling I've found that there is a difference between the sum of a finite series of natural numbers & the sum of an infinite series of same.
So maybe 1+2+3+4+5... = -1/12 as the 3 dots after the 5 denote an infinite series and when you're dealing with infinity you're in Alice in Wonderland territory!
Perhaps 1+2+3+4+5... can just as easily equal a jam sandwich!
But, the sum of a finite series is logical, say you want to know the sum of all the numbers up to 100, then you can use
[(100 *100)+100]/2 which = 5050
You can do that in your head faster that you can do it on a calculator.
Live & learn...