Oh for the love of - does this really need to be said?
Really?
I mean, of course it's false! It's an instructive example of how apparently reasonable things go wrong, and why you sometimes really, really need to pay attention to the details.
Do we really need to be told that the sum of all the positive integers is a negative fraction? Of course we don't.
I despair sometimes, I really do. I need to go away and spend some time in my happy place.
Well... I don't understand the mindset of "of course it's false!" The first question that comes to my mind is "can I use normal-seeming algebra to make 1 + 2 + 3 + ... sum to a number other than -1/12"? If it's trivial to make it come out to anything at all, we can chalk it up to apparently reasonable things going wrong; if it seems to be mysteriously easier to add it all up to -1/12, it's worth investigating why that might be.
I mean, we can apply the good old formula for the geometric series,
1 + x + x^2 + x^3 + ... = 1/(1-x)
to x=2, and we learn that
1 + 2 + 4 + 8 + 16 + ... = -1
And before rejecting that as obviously absurd, maybe we should take a minute to reflect on how the computers we're using right now represent the quantity -1?
> It won't work for decimal 10s complement numbers:
I don't get why you say this -- you go on to demonstrate the opposite?
This is just the other-side-of-the-decimal-point inverse of the well known 0.9999999... = 1 equality.
One of my favorite proofs for 0.9999... being equal to 1 goes like so: imagine subtracting it from 1. You'll get a number which has a 0 at every decimal place: 0.0000000.... Obviously, the number with a 0 in every decimal place is 0 itself.
Similarly, if you have your integer represented decimally as a bunch of coefficients (0 <= c < 10) of powers of 10, and all of the coefficients are 9, it's fairly straightforward to see that adding 1 will get you a new number for which all of the coefficients are 0. Since adding 1 to the original number gave us 0, we can treat the original number as -1.
I think it needs to be said because the people promulgating the falsehood are legitimate mathematicians. And most of the time their blog has quality stuff. But they dropped the ball on this one.
Forgive me, but I really don't understand your point. Do you think they believe it?
This is proper geek fun with serious points underneath and a good smattering of why it's actually valid to consider these things. This isn't established people with proper reputations setting out to con the unsuspecting public. There is solid math going on here. See Terry Tao's recent blog post[0] about why we can and should play with these things.
And now it's late here and I'm going off-line for a while. As I said elsewhere, I need to detach for a bit.
Added in edit: Just checking before signing off for the night and I see that the submission of Terry Tao's blog entry on this topic has been killed. By mods, flags, whatever, I don't know. here's the link[1] if you're interested, and this is me, signing off.
Depends on what you mean by "they" and "it". The numberphile folks undoubtedly know that they are playing fast and loose with the rules. The readers of Slate almost certainly don't know it.
> There is solid math going on here.
No, there isn't. There is solid math going on in complex analysis where you have concepts like analytic extensions that produce unintuitive but useful and (more importantly) consistent results. But that is NOT what this video is about. This video is about using high school math to "prove" a result that is simply not true under the rules of high school math. The only thing separating this from outright crackpottery is that the result they derive happens to look like one that can be legitimately derived under the rules of complex analysis and analytic extensions. But that's a mighty thin reed. It doesn't change the fact that they present the result as if it were true under the rules of high school math, and under those rules it isn't true.
EDIT: Terry Tao's article is excellent, and I am appalled (but, sadly, not surprised) that your submission was killed.
Hm, my own submission seems to have fallen off the front page awfully quickly.
EDIT2: I have been corresponding with an HN mod who informed me that my submission triggered the voting ring detector. (It was a false positive, which ought to worry someone at YC.) Also, the original Terry Tao submission has now been unkilled. I encourage you to upvote it.
That’s what I secretly wished. If it where infinite then everything is explained.
But the Wikipedia article “said” that it is 0 (because 1^0+2^0+3^0+...=1+1+1, and 0 is even). But the [dead?!] article submitted by ColinWright from Tao’s blog says it’s -1/2. (I prefer not to disagree with Tao, just fixed Wikipedia.)
The problem is more deep. I should read the complete version of Tao’s article.
Yes, but you cannot assume that the values are finite in the "proof" that they are finite. You have to prove that they are finite before you are allowed to manipulate them as finite quantities.
G. Hardy had a hard time understanding it, and Ramanujan apparently knew it[1]. I'm not a mathematician, but if Hardy had a hard time understanding this at first glance, I don't think anyone should be ashamed for not knowing that the same of all numbers is -(1/12)
Really?
I mean, of course it's false! It's an instructive example of how apparently reasonable things go wrong, and why you sometimes really, really need to pay attention to the details.
Do we really need to be told that the sum of all the positive integers is a negative fraction? Of course we don't.
I despair sometimes, I really do. I need to go away and spend some time in my happy place.