by Matthew Cobb
According to Douglas Adams, Earth was created as a kind of super-computer to find the question to which the answer was 42, the secret of life, the universe and everything. The Earth was, of course, destroyed by the Vogons just before the programme was due to come up with an answer (or rather, a question).
As a result, ’42’ has taken on a humorous symbolic value – try asking Siri what the secret of life is. But now there’s a new kid on the block, a new number that will probably lead to all sorts of nerdy jokes. And it is a very weird number indeed: – 1/12.
This number is what comes out of attempts to calculate an *infinite* series: 1 + 2 + 3 + 4 + 5 … In other words, instead of the sum of infinite numbers being… infinity, it is in fact a negative fraction.
This is explained in this video from the excellent Numberphile team at Nottingham University. It’s already had > 1.5 million hits on YouTube, and has been covered in the New York Times.
This isn’t a joke, and although there’s some mathematical jiggery-pokery going on to help us poor mortals understand it, more formal and elaborate proofs are around, and have been known literally for centuries – this is the Reimann-Zeta function, first worked out by Euler in the early 18th century, and then developed by Reimann in 1859. It is used a lot in string theory, a highly speculative part of physics.
Before you all start chipping in below pointing out basic mistakes in the maths (I don’t think there are any), read this response by Tony Padilla (I got lost when the Greek letters began) or, for the truly mathematically endowed, watch this extra and more elaborate proof which takes you through Euler’s calculation and read this mind-bogglingly difficult piece.
The key point to remember before you get angry or think you’ve been tricked (you haven’t) is that while if you add up any set number of natural numbers (say 2, 5, 178 or a googleplex of numbers), you get a large positive number, whereas if you were to try and do this with an *infinite* number of natural numbers, you get – 1/12.
Roll over 42—your time is past.
108 thoughts on “One number to rule them all: -1/12”
I remember hearing Lawrence Krauss bring this up in a lecture.
I think it was the one that gave birth to his book The Universe from Nothing…
Of course, if you accept that sum(0,1,0,1,0…)=undefined, then the whole sum is undefined, and that matches all our intuitions. Alternatively, if you force this series to converge to a particular value, by invoking Cesaro summation, you can get the whole thing to converge.
… and Cesaro summation isn’t normal addition. It’s a way to assign some infinite sequences of real numbers to single real numbers, which acts like addition at times.
Brady Haran’s YouTube channels are great. Sixty Symbols & Computerphile are also on my regular watching list.
The University of Nottingham have done a fantastic thing by getting involved in this project.
Seconded. A great way to spend a few minutes when you got ’em.
The trick here is that somebody confuses Cesaro sum of a series with the series itself.
Yikes. I hope none of those maths trolls come over here! The comments (and the OP) would make Professor Ceiling Cat take executive action. They ought to lighten up instead of calling each other names!
Your content is wrong, and now you play the “tone” card. Shame.
Yeah, so basically this -1/12 thing is BS… Vive le quarante-deux ! 🙂
From your link (emphasis mine):
This matches my intuition; but I’m not a maths major either … (I got as far as 3-D integration (lovely, elegant), then differential equations and matrix math, both of which I hated.)
There is a raging philosophical debate as to whether infinity is a real (world) thing or only a mathematical construct with no real world correlate.
Much of that debate involves defining what infinity actually is.
The proof of this equation (the actual one, not the ‘for Dummies’ version in the video), may go a long way to answering that question.
It depends on whetehr you require the bits to be represented by bits of matter. If so, then no, there are a finite number of bits of matter (however small you go), a number which in theory could be known; but in practice can’t be.
If you don’t require it to be represented by bits of matter, then any finite object contains an infinite number of dimensional subdivisions. I contain multitudes!
It’s worth noting that physics has given us incredibly powerful reasons to think that there are limits to divisibility — specifically, the Planck units, especially Planck Length and Planck Time.
There’re other reasons to be confident that it should be so. For example, were reality infinitely divisible, then one could likely construct a perpetual motion machine.
Physics is not the be all of mathematical truth. We already had atomic theory to put limits of divisibility of actual physical objects, that doesn’t make infinitesimals any less useful for mathematics (and physics too in other contexts).
Physics can speak to the mathematical utility of certain ideas, which is how this video should have gone. Using the Cesaro summation of a divergent infinite series to get a useful value for quantum physics calculations is cool. Posturing that this is THE sum of the natural numbers is silly. Usually numberphile is much better than this.
Completely agreed. I personally would have much preferred for the video to have quickly stepped through the real proof with lots of handwaving and then spent most of its time on the practical applications of using -1/12 in place of the sum of natural numbers.
This was debunked recently after Phil Phait and PZ both mentioned it.
(a) There is no way you can add positive numbers together to get a negative sum
(b) The mistake in the ‘proof’ begins by claiming the sum of 0, 1, 0, 1… is 1/2. It isn’t.
It was “debunked” by a blogger who admittedly isn’t a mathematician and the errors are the same errors that other non mathematicians are making. They are using the standard high school algebra II definition of infinite sums. It’s the same mistake that people who would debunk the existence of negative square roots or factorials of irrational numbers because the domain of applicability of operators as they were taught are insufficient in certain cases. The result is not only non controversial in mathematics, but is useful in physics. The Casimir Effect involves a sum of infinite integer vibrational modes of vacuum fluctuation between two metal plates. The sum of those modes results in a negative value which equals the attractive (negative force) between the two plates.
I was going to bring the Casmir Effect up. It is this sort of addition of infinite sequences that gives physicists happy faces instead of angry frowns because the whole thing blows up to infinity. Renormalization for the win!
Thanks, I was thinking massive bullshit that the sum of an infinite number of positive numbers could come out negative.
0 + 1 + 0 + 1… is infinite but not quite as dense as 1 + 1 + 1… WTF (where the W is for why) would 0 +1 + 0 + 1 = 1/2??? On average maybe but it is tough to take an average over infinity. I’m no math genius but this sounds like a lot of BS.
To be fair, the infinite series they evaluate as being equal to 0.5 isn’t the one you describe, but rather:
1 – 1 + 1 – 1 + 1 – 1 …
They have an entire video on Grandi’s series here:
Watching those videos just makes me want to punch those guys in their big teeth. I’m not convinced that an infinite series of one’s and zeros equals one half. Just that the average of all the possible sums of ones and zeros is equal to one half. I don’t think that it makes sense to equate the two things.
A large part of the problem is in the way they’re describing the equality. It might help for them to use less-loaded terms, such as “can be substituted with in certain circumstances,” or “has a type of congruity with,” or the like.
Again, a concrete example might help. That video on Grandi’s Series ended with a comparison with Thompson’s Lamp, but got all hung up on Zeno and final steps and the like. But anybody familiar with the way that dimmable LED lights work knows that they’re just flickered on and off faster than the human persistence of vision. An hypothetical light switching on and off infinitely fast (not possible in the real physical universe) is, clearly, cycling faster than the human persistence of vision and, by definition, on and off for equal amounts of time (even if those time slices are infinitely short). In an alternate universe where you could construct such a light, an human observing it (assuming the rest of physics hasn’t significantly changed, which it would have) would see the light flicker before becoming steadily dimmed to half its linear brightness. Human vision itself isn’t linear, so it wouldn’t actually look half as bright, but that’s another story….
Anyway, that’s a not-bad demonstration of showing how the “1/2” answer can reasonably considered useful. I imagine real-world physics might have even better examples. What would be really exciting would be to pick that class of infinite series and show different real-world examples where different answers apply, and explore what leads to those differences.
I cannot watch the video here, but if that’s really what is going on, that’s the *textbook* example why divergent series are tricky.
This is wrong. First of all, the regular old “sum of numbers” works only on two operands at a time. You can extend it by repeating the operation any number of times needed. However you can’t do this with *infinite* number of natural numbers because it will NEVER end and the answer will NEVER be -1/12.
There are many ways to ***assign a value*** to the sum of infinite series. If the series is convergent then the limit is typically used as the assigned value. If the series doesn’t converge (which in this case doesn’t) there are other ways of ***assigning a value*** to the series. One of these methods assigns a value of -1/12 to the sum of natural numbers. That does not mean however that the ordinarily understood “sum of numbers” is mysterious in any or that it gives a result = -1/12 if you were to do the sum ad infinitum.
See Phil Plait’s post here and Mark Chu Carroll’s post here about the same topic.
The method used does not assign the value -1/12 to the sum of the natural numbers. The Riemann zeta function is undefined at 1. If however you plugged 1 into the formula for the function, it would give you the infinite sequence of natural numbers.
The analytic continuation of the zeta function has a value at 1 which is -1/12. This is not the same thing as saying that the sum of the natural numbers has value -1/12. It is however a useful and overall nice way to think about what is going on. We have to be careful about the way this is explained though.
“The analytic continuation of the zeta function has a value at 1 which is -1/12. This is not the same thing as saying that the sum of the natural numbers has value -1/12.”
This. (Sorry, Professor Ceiling Cat–I know you don’t like this kind of response. lol)
It’s the type of mathematical result that is best left to the discussion of mathematicians and physicists who can make use of its result, because as you can see in this post (and others), it just causes arguments over misunderstands of what is really meant by the finding.
The sum of the natural numbers doesn’t have a negative value–it’s a divergent series; the analytic continuation of the zeta function (which is itself undefined at 1) at 1 is different. And I won’t pretend to know its use (as james mckaskle above seems to), as my graduate math education didn’t get close enough to physics. 🙂
I assume you mean “-1” instead of 1 above.
Yes, you are right. In this case -1/12 is not a value assigned to the sum of natural numbers in the same way as we can assign the value 1 to the sum 1/2 + 1/4 + 1/8 + …. That makes the video worse. -1/12 is the value of Riemann zeta function at -1 and the sum of the infinite series of which Riemann zeta is the analytic continuation of, becomes the sum of natural numbers at -1. The video seemed to imply that physicists use these two interchangeably though by sticking an equal sign between these two results.
Yes you are right. I linguistically approximated and was wrong. I’m not a mathematician and it shows.
I started looking into this by searching a bit (I always loved being able to do those sums using complex analysis, for example) and soon ran into “assigned values” indeed. Interesting stuff. And there are some wiki pages out there that are informative, especially this one: http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7 (I quite like the derivation given by Ramanujan).
On a possibly related (to this site’s discussions on biblical historicity) note: Is there any evidence that Ramanujan really existed, outside the writings of Hardy?
His hospitalization records?
His notebooks in his own handwriting, his surviving family and friends (including a widow who lived until 1994, his appearance in photographs like this one
the fact that he’s been the subject of several biographies, and the utter lack of any plausible motive for Hardy to invent such a person.
Mark Chu Carroll’s post is worth reading.
Bottom line: assigning values to non-convergent series.
I feel compelled to quote 90s Barbie, “Math is tough!”
In the linked “more information” post, when the physicist started talking about sums of harmonic series…I wondered if this might explain why the chromatic musical scale has twelve tones….
No, but it may explain the number of dimensions. From sgo’s link:
“Ultimately it is this fact, combined with the Goddard–Thorn theorem, which leads to bosonic string theory failing to be consistent in dimensions other than 26.”
Then superstring theory folds that number into a couple of theories with d=10, which can be shown to exist in d=11 dimensions.
The reason a chromatic scale has 12 tones is AFAIK that you can scale it “good enough” to a similar set of overtones 7 scales up. E.g simply because (2/3)^12 ~ (1/2)^7; see here after running google translate: http://www.nyteknik.se/popular_teknik/kaianders/article3597095.ece .
Yeah, I’m quite familiar with temperament and all that it implies. Non-keyboard musicians are especially always on the lookout to lower the third of major triads, for example. Brass musicians who’ve played inner parts with very good brass chamber ensembles can tell all sorts of stories about times that they’ve held what was ostensibly a single pitch (as a “pedal tone”) but had to constantly fine-tune it as harmonies changed around them — and not because the other musicians couldn’t play in tune, but quite the contrary.
What I’m wondering is a bit more fundamental. That “-1/12” fraction appears to be fundamentally linked to an harmonic series in some sense. The limits of the overtone series where we quantize our scales are somewhat arbitrary. Could the fact that we settled on dividing octaves in twelve parts be somehow related? After all, it’s very shortly after the appearance of whole tones in the overtone series (1/6 octave) that we get semitones (1/12)…and, by that time, quarter-tones (1/24) practically immediately follow, at which point no form of chromatic quantization makes (human psychological) sense. (At least, not to Western ears.)
“very good brass chamber ensembles can tell all sorts of stories about times that they’ve held what was ostensibly a single pitch (as a “pedal tone”) but had to constantly fine-tune it as harmonies changed around them”
THAT is cool.
Believe me, it’s even more fun (and amazing) actually being in the middle of that sort of thing than it is to read about it.
You can even hear it coming, actually, which helps you remember to be ready to kick that slide or switch to an alternate fingering or just lip the hell out of it or whatever.
String players and choir members also have to make these kinds of adjustments.
In fact, before equal temperament became the norm, many keyboard instruments (organs, clavichords and harpsichords) had “split upper keys”, ie, where you would see only Csharp on a piano today, you would’ve seen a second “hump” for Dflat. It looked like a 3-tiered staircase from the back of the keyboard to the front.
I did not know that about older keyboards. I’d love to see such an instrument in person and hear somebody do a lecture / recital demonstration.
They were by no means the norm. Most of the time keyboardists had to deal with the temperaments of the time by playing onlyusic that did not wander too far afield harmonically, ie, didn’t use many accidentals. This is why Bach famously campaigned for “well temperament” (nb: not equal temp.) with “The 48”.
You can see a keyboard with split upper keys by doing a Google image search for “örgryte organ keyboard”. This instrument is actually new, but it is a breathtakingly faithful recreation of a Schnitger-style instrument. The first image is a good one.
Thanks for that. I can imagine modern organists both getting excited about the chance to play such an instrument, and getting headaches trying to adapt….
This is evident in some Baroque flutes, as well — some, for instance, had two separate keys for D# and Eb — although almost all modern copies now have just a single key for those notes.
As a one-time professional brass player (tuba and trombone, mostly, with the odd euphonium or Wagnertuben thrown in), making the switch to instruments and musics which are played with just intonation (and sometimes exotic modes) has been at times challenging. I mostly play folk winds now — Irish flute and Uilleann pipes, Armenian duduk, various other flute-like and reeded instruments — and it’s been as much a challenge for the ear as for the fingers and lips. I’ve heard examples of Uilleann pipes where the chanter has been made to play well in equal temperament, and it’s painful to hear the instrument go “out of tune” with itself against its own constant drones.
I’m trying to imagine somebody who knows enough to be able to build a set of pipes but not enough to know that they should be tuned with perfect temperament for the one-and-only key they’ll ever play in…and I’m flabbergasted….
That’s an interesting thought (or series of thoughts) about our base-12 scale.
Indeed, why twelve instead of 11 or 13? Because “they” had to nudge the perfect 4th goalpost a bit to make the 12 fit.
Where is Musical Beef when we need him?
I guess what I’m wondering is whether the magical -1/12 helps explain why those other pieces fall into place better.
Most theorists do not think the overtone series serves as the source for any pitches other than those that make up the major triad.
The reasons are that beyond the fourth overtone, they match up with the pitches in the scale based on the fundamental only approximately, and, as you note, the perceptual currency these higher overtones have for anything other than establishing timbre is pretty much zero. But most importantly, once you’ve derived the major triad, all other pitches can be demonstrated to be derivations of counterpoint or other compositional interactions/processes.
For instance: the minor seventh is not derived from the seventh overtone, it is a passing tone that has been chromatically altered to strengthen the motion from tonic to subdominant. This is the reason for the Voice-leading 101 directive always to resolve minor sevenths down. Really good composers can, of course, find ways to simultaneously satisfy AND flout “rules” like this. In fact, that’s in large part what makes them good.
As far as the ultimate reason for 12 pitches? Perhaps the math has something to do with it at an ultimate level, but closer to the surface, I’ve always thought this explanation made a lot of sense:
The strongest relationship in tonal music is tonic/dominant. A note, to wax poetic, almost has an urge to be a tonic or a dominant. So, if you construct a major triad on a pitch, on the fifth below (making your original pitch a dominant), and on the fifth above (making your original pitch a tonic), you wind up with the seven diatonic scale-degrees. Applying a dominant to each of these scale-degrees supplies the remainder of the chromatic pitches.
I’m not sure I’m following you as far as that second sentence goes.
(What follows is painfully spelled out in the hopes that the non-musicians won’t get hopelessly lost. If it was just us musicians, two sentences would get the point across.)
The first works out. Using C Major, you’ve got C-E-G for the “home” triad, F-A-C for the fifth below, and G-B-D for the fifth above. That gives you all the white keys on the piano, no more and no less.
What I don’t see is how you get from there to the complete chromatic scale.
If you’re using “dominant” to mean a Perfect Fifth, you get:
C => G D => A E => B F => C G => D A => E B => F#
That only gets you a single chromatic note.
If, instead, riffing off your earlier discussion of seventh chords, we step by a minor seventh, that gets us:
C => Bb D => C E => D F => Eb G => F A => G B => A
…and that only gets us two chromatic notes.
You can, of course, walk a circle of fifths or fourths to get the whole chromatic scale…but that brings us right back to questions of temperament. Long before you get to your first black key note, a “proper” (beat-less) tuning is going to be so far off that nobody’s going to think the two are the same note.
(For the non-musicians, the sequence is: B#/C G D A E B F#/Gb C#/Db G#/Ab D#/Eb A#/Bb E#/F. Left-to-right is fifths; right-to-left is fourths.)
That is, the B that you get by walking fifths and tuning each is a very different pitch you’d get than if you tuned the B as the major third of a G that itself is tuned as a perfect fifth of C.
I’m sure you’re not just smoking crack on your theory, so I’d appreciate a bit of clarification so I can spot where I went off the rails.
A dominant for each of the newly-derived diatonic pitches:
V of C is already done
V of D gives us C#
V of E gives us D#
V of F already done
V of G gives us F#
V of A gives us G#
V of B gives us A#
Ah — the leading tone. Yes, that would do it. For each of the white notes, an half step below.
That could do it, especially considering that would be a natural note for ornamentation as well — thus leading directly to the “color” sense of “chromatic.”
…though, of course, temperament is a perpetual Diablos in musica…
Thank you gentlemen very much! We/us guitarists usually don’t learn too much theory! I have only enough to make public errors (most of which pass unnoticed 🙂 ).
Yes, this system presupposes equal treatment, but even in the era of mean-tone etc, they recognized at leafy some kind of equivalency between, say, D# and Eflat, which is why they made split upper keys rather than just two entirely separate keys.
Wtf. “Leafy” = “least”
And “treatment” = “temperament”.
I don’t know exactly what it is, but my typo rate has dramatically increased since iOS 7 and the iPhone 5.
I think the important thing is, and the thing that makes the above derivation attractive, is that they thought of pitches as functions, as being defined in relationship to other pitches, and not absolutely, based on brute frequency.
Yes, this seems almost certain.
Seems to me that scales (other than a few points such as “perfect 4ths”, “perfect 5ths”, and octaves) are defined for convenience. It seems certain that for the well-tempered scale (which has a slightly imperfect 4th), this is the case.
When I start to think too hard about this, people nearby say, “Do you smell bacon?”
This almost makes me long for the Good ‘Ole Days when the Catholic Church decided what was true and what wasn’t. They’d decide that the sum of the natural numbers is equal to infinity and we’d be done with this -1/12 nonsense.
Then anyone who disagreed would, ‘smell like bacon’…literally
At first I felt -1/12 as confused as I normally do, but some of the comments made me feel infinitely better.
I’m going to make an attempt at mathematical logic and conclude the sum of the negative natural numbers should be 1/12 since positive and negative together should be 0.
Sameer is right. What this “sum” shows is that there is a well defined map from the sequences of real numbers that assigns (1,2,3,4..) the number -1/12. That is not at all controversial.
What is irritating is to hear the term “sum” assigned to that.
For the interested who enjoyed their calculus courses, I can recommend the chapter “divergent series” from Knopp’s book Theory and Application of Infinite Series.
It talks about the various ways the usual “sequence of partial sums” definition can be extended.
One final remark: to those offended by the 1-1+1-1+1….. “=” 1/2 , if one uses the formula and substitutes x = -1, one gets the result. Of course, -1 is not in the interval of convergence of the series by the usual definition of convergence.
“For the interested who enjoyed their calculus courses, I can recommend the chapter “divergent series” from Knopp’s book Theory and Application of Infinite Series.”
I’ll check that out. I hated everything that came after calculus (derivative and integral, up through 3-dimensions). Maybe it was because the computer guys were teaching those later courses instead of the maths guys?
That is very much possible. I am one of the computer guys. And we make a horrible mess out of math.
Unless you are working the theoretical branch of computer science the attitude most of the time is: Let’s just try it. If it doesn’t work we can still ask questions about why it didn’t work.
I seem to recall that Leibniz and a few others had opinions on this matter, and someone did defend the 1/2 thing for a while, until the radius of convergence idea became clear.
One thought on these posts.
I think the reason why this solution is being dissed is that it isn’t intuitive.
Well, guess what? Most of reality isn’t intuitive for us.
‘Intuitive’ for humans are numbers less than five and distances shorter than to the horizon. Intuitive is sky above and ground below. Intuitive is what we were used to the last time we had s significant brian upgrade about 100k years ago.
Not understanding how it could be does not mean that it isn’t. This is the error of the creationists and the climate change deniers.
“I shall not commit the fashionable stupidity of regarding everything I cannot explain as a fraud.” — C.G. Jung
“I think the reason why this solution is being dissed is that it isn’t intuitive.”
NO, the reason it is being dissed is that it is wrong. It’s unfortunate that you have not yet come to that realisation.
Yes. Also, they seem to be trying to hijack the term ‘counterintuitive’ the way the D.I. have hijacked ‘complexity’.
I meant to use an exclamation mark!
>>“I shall not commit the fashionable stupidity of regarding everything I cannot explain as a fraud.” — C.G. Jung<<
I would argue that the also quite fashionable stupidity of regarding everything you can't explain (/understand) as brilliant isn't actually all that much better…
+ ∞/∞ 🙂
Long ago sf story. A group of hermit monks up in the mountains are devoted to recording the nine billion names of god. They believe this is the purpose of the universe and the universe will cease to exist once accomplished.
They hire a couple of computer guys to set up a computer to speed the task along. These guys think the monks will be mad at them when the task is accomplished and nothing happens. So they take off riding down the trail One remarks that the task should be finished about now. His colleague does not answer because he is staring up and watching the stars go out, one by one.
The Nine Billion Names of God by Arthur C. Clarke (for the three people on this site who did not already know that). I believe it can be read on-line, with the help of The Google.
I was one–who are the two others? Identify yourselves! 🙂
You still can’t deny that 6×9=42 in base thirteen, even if Douglas Adams himself thought it was silly.
I am glad I did biochemistry! The video (and the discussion) hurt …
The good thing about the video is that the mistake occurs right in the second equation. That equation is true if and only if the absolute value (!) of x is strictly less than one. Therefore as soon as he sets x = -1, I could stop watching … more time for reading the new Pynchon.
I mistakenly asked Siri what the “meaning” of life was and got an entirely different answer.
I don’t know this Siri; but I’d like to hear the answer anyways! 🙂
“It’s nothing Nietzsche couldn’t teachya”.
I just asked her again and she said,
“I can’t answer that right now, but give me some very long time to write a play in which nothing happens”.
Siri is a bit silly.
There’s nothing Nietzsche
Couldn’t teach a-
-bout the raising of the wrist.
Socrates himself was permanently pissed!
Did you ask specifically for the meaning of ‘life, the universe and everything’?
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
It seems that, in order to get rid of infinity, they’ve replaced it with either stupidity or dishonesty.
In the first place, if they don’t like infinity, then they can count the finite sums and get finite, large and positive answers.
Alternately, they use poorly defined arithmetical operations like “just shift all these along a bit” or “just put a bracket in here, even tho we didn’y put the bracket in before”, the validity of which, if it even exists, relies on the series being infinite.
And even then, they do it inconsistently; performing one operation on one series and another on the other as if the operations commute.
+ a bunch. That let’s just shift this over bit bugged me a lot.
Also, from the Pasilla link above:
“In my opinion, as a physicist, infinity has no place in physical observables, and therefore no place in Nature. David Hilbert, one of the founding fathers of quantum mechanics, described infinity as “a mathematical abstraction that does not have a physical content””
Well, thank goodness reality doesn’y care for our opinions, eh?
[rant]It’s just not good enough to condemn string theory as rubbish. Everyone who took tax-payers’ monet for string theory research needs to be made to give it back![/rant]
There was something stuck in my mind about this “proof” as well as some of the reactions to it: I had to think of the Sokal affair which showed that you could impress many people – even with heavy duty academic background – by waving around some mathematical (or in Sokal’s case: theoretical-physics-related) nonsense. No one will ever want to be the first one to admit he/she doesn’t understand a thing, so everyone claps their hands and congratulates themselves on belonging to a true elite of thinkers.
To be fair, this isn’t a case of the math itself being worng; just of the description (especially the way the word, “sum,” is used) being questionable and the informal techniques used to “prove” it being invalid.
If your take-away is that there’s a significant relationship in certain mathematical systems between the summation of the infinite series of natural numbers and the fraction -1/12, and that that relationship proves useful in physics, then you’re in good shape. If you instead think that adding all the integers larger than zero will give you -1/12 and that you can prove it so by adding shifting offsets of various other infinite series, you’re in trouble.
Sokal, on the other hand, was (intentionally) spewing bullshit through and through, without any regards to reality whatsoever. These guys are well-intentioned and mostly correct, but their pedagogy is off the mark.
That video instantly reminded me of this SMBC: http://goo.gl/tQ2rXC
As many people have pointed out above, the infinite sum of all natural numbers is most definitely not equal to -1/12. I won’t reiterate what was already said, but I will add something for those who are not familiar enough with the mathematics, and the distinction between *assigning* a value to some quantity, and whether or not that quantity is algebraically or arithmetically *equal* to some value.
Again consider the offending object: S = 1-1+1-1-…. Now suppose I rewrite the sequence of integers as follows:
S = 1-1-1+1-1-1-1+1-1-1-1-1+1-….
The two descriptions of the object S contain all the same numbers, but the “average” value of these two descriptions is quite different. For the second description of S, the “average” value approaches negative infinity, since the nth occurrence of “+1” is followed by (n+1) occurrences of “-1”. In fact, we can rearrange the terms of this “sum” to arrive at any “average” value that we like.
At its foundation, all this speaks to the importance of the concept of *convergence* of an infinite sequence of numbers. Convergence does not have a unique definition though. There are many, many different types of convergence, some useful in some situations, others not. This is something important to remember.
Indeed. The fundamental mistake seems to be in confusing, maybe deliberately so, sequences and series of numbers.
In sequences, aka arithmetic progressions, each number is the result of performing an operation on the previous number. So ordering is important and shifting them all along a bit is just as bad as switching the order. In series, the order is arbitrary as they are just sets.
As I undersand it, the euler-maclaurin formula, and probly also the zeta function relies on there being a sequence and not a series.
You’re getting at the same thing I’m trying to get at I think, but a few things to keep in mind about these terms. Not all sequences need be arithmetic progressions (think of a geometric sequence, like 1,2,4,8,16,…).
Also, a series is typically understood as a particular kind of sequence. That is, we usually interpret a series (i.e. a sum of infinitely many terms) as a limit of its partial sums (i.e. the limit as n approaches infinity of the *well-defined* sum of the first n terms). With this understanding, we basically generate two kinds of possible convergence of a series: absolute and conditional. Absolute convergence can be characterized by the property that it doesn’t matter what order we add the terms up in, we always get the same number. Conditional convergence is when the ordering of the terms matters, but they will still sum to a finite number. Riemann showed that sums that are only conditionally convergent can have their terms rearranged to converge to any finite value you like. This is similar to the example the video looks at and the one I discussed above, except those series are not even conditionally convergent.
This inspires a need (or maybe just a desire) to understand convergence in another way. It certainly seems like the series given by 1-1+1-1+1-1+… should converge in *some* way. At least it doesn’t seem to blow up. So we look for other ways to understand a series (or a sequence) as a “limit of something.” Someone mentioned Cesaro summation above, which is another very common way of “summing” up infinitely many numbers. Cesaro summation interprets a series as a limit of the *average value* of its partial sums. This is what the posted video is really doing.
For absolutely convergent series (defined fully as series that when you take the absolute value of every term individually, then add all these new terms up, what you get is an honest, finite number), Cesaro and “classic” summability are equivalent. That is, they produce the same result. This does not hold for non-absolutely convergent series in general though. None of the series the video is playing around with are absolutely convergent.
Without absolute (or conditional) convergence, it makes no arithmetic sense to say that an infinite sum is “equal” to a number. The Riemann-Zeta function is something more, inspired by the study of familiar looking sequences of powers of reciprocals of integers, but you have to consider it in a complex analytic setting to really make sense of the equalities.
The thing about math is that empiricism often applies just like in science. You can often verify results, especially verifying that a series is likely to converge or not, using a computer. It is pretty obvious that any sum of positive numbers greater than one will diverge. You don’t need to run that on a computer to see that it is true. I’m by no means a math genius but that is an obvious result. Anything that says otherwise seems to me to be mathematical slight of hand.
Historically, 42 as of religious and mythical significance arose first in ancient Egypt, 3000 BC or so. Maybe Adams was aware of that?
I like 42 because it’s an even number and 4 is a nice number in its even-ness and 2×2=4. To me, it’s almost as good as 44.
I’m pretty sure Adams is on record as saying that there’s no special significance to why he picked the number, and he’s not otherwise know as being an Egyptologist.
What, no word pedants just mathematicians on this thread so far? OK, I’ll play the part since I have nothing to say about the math.
It is “googolplex”, born in 1938, at least a couple of years before everyone’s favorite search engine was a gleam in it’s daddy’s eye.
Skitt’s Law strikes again.
Phil Plait has a couple of posts on this video at Bad Astronomy. Apparently there is more to this than meets the eye….
Reblogged this on Mark Solock Blog.