Knuth changes his mind on Bernoulli number B_1 | Hacker News Reader

Knuth changes his mind on Bernoulli number B_1

www-cs-faculty.stanford.edu

300 pointsColinWright112 comments

Related tweet from Russ Cox: https://twitter.com/_rsc/status/1483899960684863493

> Knuth is making backwards incompatible changes to fix bugs in something he wrote 54 years ago (TAOCP volume 1, 1968)!

And mine:

Apart from the care Knuth takes, what's remarkable is that he has basically put out a permanent invitation to a DDoS on his time and attention—everyone in the world is invited to contact him about every word he has ever written—and somehow still continues to produce new material.

From Wilf's toast/roast of Knuth (https://www2.math.upenn.edu/~wilf/website/dek.pdf): “[…]your letter will be placed on a stack that already has 5,379 letters that reached him before yours did,[…] while he completes his latest additions to 47 new manuscripts and 311 revisions of already existing books.”

I think the secret is he uses letters.

Try that invitation to a DDoS on his life with Twitter or email.

If you’re willing to post a physical letter, chances of the content being worth reading are a lot higher.

It's hashcash! But the hash is literal cash of about 50 cents

Moreso the effort. But yeah.

idk, there are APIs for this now. https://www.postgrid.com/pricing-print-mail/#perunit

Is it handwritten though?

With a further optimisation in the batch handling of snail mail:

"I have a wonderful secretary who looks at the incoming postal mail and separates out anything that she knows I've been looking forward to seeing urgently. Everything else goes into a buffer storage area, which I empty periodically."

from: https://www-cs-faculty.stanford.edu/~knuth/email.html

Just to clarify, you can email him to report errors in any of his books (and I have done so, several times). It's only for other general communication that he insists you use physical email.

That is a fantastic and highly amusing speech, thank you for linking it.

You may also like this other speech by the same author (Herb Wilf) that I love too (on an unrelated topic):

• Epsilon Sandwiches https://www2.math.upenn.edu/~wilf/website/MAASpeech

In fact everything by Wilf that I've read is lovely: the paper "Recounting the rationals" (https://www2.math.upenn.edu/~wilf/website/recounting.pdf with Neil Calkin), and the book generatingfunctionology https://www2.math.upenn.edu/~wilf/DownldGF.html

Epsilon sandwiches is a great speech, with advice that I wish I could follow—but it pre-supposes students for whom the referenced very easy proofs are indeed very easy conceptually, and the struggle is only to put those concepts into words. I can believe that this is true of students in a UPenn first analysis course, but it is not true at the less prestigious university where I teach students who are encountering proofs for the first time (well before analysis)—and I have long struggled with how to break down this two-step complication into separate manageable steps with students for whom, say, it is still a real challenge to understand (in the context of proving facts about sums and products of even numbers) why 2(x + y) = 2x + 2y is true, but 2(xy) = (2x)(2y) is not. If anyone knows how to adapt Wilf's advice to such students, then I—and they, in my fall class!—will be grateful to hear it!

I case it helps, this is how it was taught to me when introducing proofs.

For your example case, it's simply the matter of seeing multiplication as repeated addition. So, 2x is x+x, and from this follows:

2(x+y) = (x+y)+(x+y) = x+y+x+y = x+x+y+y = 2x+2y

However, for the other case, when we convert the multiplication by 2 to an addition and back:

2(xy) = (xy)+(xy) = xy+xy = 2xy

Now we can show that this works for n instead of two:

n(x+y) = (x+y)+..[n times]..+(x+y) = x+..[n times]..+x + y+..[n times]..+y = nx+ny

And for multiplication:

n(xy) = xy+..[n times]..+xy = nxy

Thanks for the suggestion, which I think is exactly the right way to address this particular issue.

Indeed, now that I've learned from a few times teaching the course that it always comes up, I will address it. But, if I spend my time building up basic-algebra proficiency at this level, then I'm never going to get into the meat of proving things. Perhaps the answer is to view the very act of writing your argument as a proof—and I like that perspective (because it genuinely is!); but, if I were to break that down to its full details, then I would have to write 2(x + y) = (x + y) + (x + y) = x + (y + (x + y)) = x + (y + (y + x)) = x + ((y + y) + x) = x + (x + (y + y)) = (x + x) + (y + y) = 2x + 2y, and writing down proofs of that sort risks alienating students who are eager for a conceptual picture, and guarantees giving the wrong idea of what sort of activity proving is. (And it also risks confusing why later I will say that, for a silly example, π(x + y) = πx + πy "by the distributive law"—I can't think of multiplication by pi as repeated addition, so I must cite the distributive law; and then why didn't I just do that before?)

This has confused me a bit about the US education system. Shouldn't this be taught in highschool? Where I live if someone wants to study maths in university, and they haven't done the B2 math track, the university would simply not accept them into enroll into that major. The student would have to get a highschool degree that includes the maths B2 track, if they're under 18 they could simply return to highschool, or if they're older then there's adult (night) schools.

> This has confused me a bit about the US education system. Shouldn't this be taught in highschool? Where I live if someone wants to study maths in university, and they haven't done the B2 math track, the university would simply not accept them into enroll into that major. The student would have to get a highschool degree that includes the maths B2 track, if they're under 18 they could simply return to highschool, or if they're older then there's adult (night) schools.

It's not that they haven't been taught this; it's that they haven't conceptualised it, so that 2(x + y) = 2x + 2y to them is just a meaningless rule, and there's no particular reason why it should be true but not 2(xy) = (2x)(2y), or 2^(x + y) = 2^x + 2^y, or whatever. Of course ideally one would only have students in the course who have conceptualised this, but that's not how it happens, and it's not fair for me to teach the course to the students I wish I had.

You can find proofs by making analogies in almost any direction, but it helps to know which direction leads towards solid intuition the student already has and can build on, rather than just leading to more abstract nonsense that's equally unfamiliar.

In the case of having no intuition about algebraic manipulation, you can suggest a geometric interpretation to connect to intuition that's more likely to be there. For 2xy != 2x2y, draw two xy rectangles and one 2x by 2y rectangle.

Now the students all see the problem. Now they just have to connect the geometric intuition back to the algebra. This helps motivate the algebraic rules and shows why they must be what they are. Just the idea that geometric intuition exists -- that you can solve problems by putting pictures together in your head -- this isn't something every incoming freshman already consciously knows they have as a technique always available to them.

(This is just a wordy re-telling of Polya's "Draw a figure", from How to Solve It; if you haven't read, drop everything and get a copy.)

> Apart from the care Knuth takes, what's remarkable is that he has basically put out a permanent invitation to a DDoS on his time and attention—everyone in the world is invited to contact him about every word he has ever written—and somehow still continues to produce new material.

I mean, this is just a best practice among academics -- every paper has at least one corresponding author to whom you should write for all inquiries about it. Of course, Knuth's immense popularity and the chance you may get a reward [0] contribute to the intensity of the DDoS.

[0]: https://en.wikipedia.org/wiki/Knuth_reward_check (of course, the "reward" is not really the money, but rather getting recognition from Don Knuth himself)

> everyone in the world is invited to contact him about every word he has ever written

But not by email.

Well, as others have pointed out, at his scale this is likely a necessary rate-limiting countermeasure.

Even relatively obscure academics easily end raking up hundreds of emails in their inboxes once they are senior enough. With someone as popular as Knuth, it would be madness.

Yes by email (search the posted page for "@" or look under "Rewards" at https://cs.stanford.edu/~knuth/taocp.html#:~:text=Rewards )

And when I said "invited" I meant "strongly incentivized": by offering cash rewards (initially) and those highly prized "Knuth checks", he surely gets orders of magnitude more scrutiny and error reports than a typical academic.

I feel like this man is as close to being a monk for computer science as one could ever be. I envy his daily routines and discipline that keeps him going well beyond typical retirement age.

I can only hope to have a sliver of his discipline to lead my daily life.

> I feel like this man is as close to being a monk for computer science as one could ever be.

He has a wife, two grown kids, and some grandkids too. He’s a believing, practising Lutheran [0] (some have even said a “devout” one) and published a book on the Bible [1]. He plays piano and organ - he sometimes plays the organ at his church, and he and his wife even have a pipe organ in their house [2], and he has composed an original work for it [3].

Calling him a “monk for computer science” makes him sound very one-dimensional, when he isn’t.

[0] https://www.livinglutheran.org/2018/12/im-a-lutheran-don-knu...

[1] https://www-cs-faculty.stanford.edu/~knuth/316.html

[2] https://www-cs-faculty.stanford.edu/~knuth/organ.html

[3] https://www-cs-faculty.stanford.edu/~knuth/fant.html

Real life monks aren't one dimensional either. Buddhism in its essence is about balance and right path not extreme focus on one thing

Real life Christian monks weren't / aren't one dimensional either.

Other than the "has a wife and kids" part, that sounds a lot like a monk?

I mean, it sounds more like a church organist than a monk.

Ah yes, the unmistakable sound of an organist.

You're confusing monks with aesthetics.

And you are confusing aesthetics for ascetics :)

I would graciously assume an autocorrect failure on a phone, rather than earnest confusion.

Or, my phone is at any rate.

You mean ascetics, not aesthetics. Wikipedia’s article on Christian monasticism [0] starts with (my emphasis):

> Christian monasticism is the devotional practice of Christians who live ascetic and typically cloistered lives that are dedicated to Christian worship

So, am I really confusing monasticism with asceticism, when the later is a key part of many definitions of the former?

Many people confuse monks with members of non-monastic Christian religious orders. Many of the famous Catholic religious orders - such as Dominicans, Franciscans and Jesuits - are not monks. In the traditional understanding, monks withdraw from the world to focus primarily on prayer and contemplation, and are not actively involved in activities such as preaching, education or missionary work; those religious orders whose members are actively involved in the world in those ways are not classified as “monks”, rather “religious”. Some Jesuits are even university professors in secular subjects - such as the Australian law professor Father Frank Brennan SJ (who was once awarded the accolade “Living National Treasure”) - something fundamentally incompatible with the contemplative life of a genuine monastic.

Is that true of non-Christian “monks”, such as Buddhist bhikkhus? I don’t think I understand Buddhism well enough to confidently venture an opinion. But, I do think that using a Christian-origin word to describe a concept in a radically different religious tradition is a questionable decision; using the indigenous term (bhikkhu) is preferable since it avoids that issue entirely.

[0] https://en.m.wikipedia.org/wiki/Christian_monasticism

Do you mean ascetics?

Sure did!

When I think of Lutherans, “lay-monks” is a phrase that comes to mind.

Email is a wonderful thing for people whose role in life is to be on top of things. But not for me; my role is to be on the bottom of things.

Source: "Knuth versus Email" https://www-cs-faculty.stanford.edu/~knuth/email.html

I think more developers should be like that.

Too much energy is wasted for being on top of new framework of the month.

> It was their well-intentioned but ultimately poor choice that had led to what I'd been taught in the 1950s. (...) Yet Luschny convinced me that we have all been wrong, and that it's high time to change back to the correct definition before the situation gets even worse.

I have no doubts that Knuth's ability to keep producing novel material is strongly related to this willingness to be proven wrong on ideas he's held for almost seventy years. I can only hope I get to his age and still have the flexibility to do that.

Another older fella in the game I know says: "strong opinions, loosely held".

This phrase bothers me.

It's sort of a contradiction in terms: what does it mean to hold a "strong" opinion "loosely"? In my experience, those tend to be highly correlated. The people who have strong opinions hold them firmly, and the people who have weak opinions (or more charitably, for whom having opinions---whether strong or weak---is not important to their identity) are much more able to accept reality as it exists rather than as they'd like it to be.

I think the phrase ends up getting used to justify arguing for things (opinions) passionately, but then (supposedly) being able to turn around when proven wrong. I say "supposedly" because again my experience is that this is more wishful thinking than reality, and that in practice the correlation I identified above holds the vast majority of the time.

Does anyone else feel this way?

Since the best we can hope for for an opinion is evidence based, experience based and from deductive reasoning, and new evidence and experience is always being uncovered (in science and elsewhere) let alone the amount of material a person has been exposed to, you can have a strong opinion but be willing to change your mind if you see new evidence.

I mean, you aren't wrong. But part of the point is that you should be accepting of feeling uncomfortable.

That is, the contradictions are part of the point. Logic on few variables fails to capture life.

Holding an opinion in the present should not beholden one to guard it even when presented with evidence that may not have been available or evaluated when forming the opinion.

That's what I take it to mean.

"""In simpler terms, ‘strong opinions, weakly held’ sometimes becomes a license to hold on to a bad opinion strongly, with downside protection, against the spirit and intent of Saffo’s original framework."""

https://commoncog.com/strong-opinions-weakly-held-is-bad/

I have something of a love-hate relationship with that saying. The idea is good, but it’s been tarnished by the sheer number of assholes who have used it to justify bullying people with their opinions.

I had not realized there’s now an estimated release date for 4B of this October. I think it may finally be time (after 4B) to buy the box set and leave it lying around in hopes that by 2030 I’ll get up the courage to start it.

Worth noting: there’s currently a discount available via a code displayed prominently on the Pearson site, so preordering it came to about $210 USD for me.

https://www.informit.com/store/art-of-computer-programming-v...

I bought it, got stuck on the maths in chapter 1 of volume 1, and it's now in my friends Attic in another country. Maybe one day! But not yet.

just skip ahead!

This is such great advice at least for reading mathematics, and I assume for reading most science. Skip the introductory part you don't understand, which is almost certainly explanatory throat-clearing, and get to the part that interests you. Now you won't understand it! But you'll see the reason you don't understand it, and you can go back to the introduction with the mission specifically of understanding that part. It is consistently amazing to me how the same sentence that was baffling when you had no context in which to put it becomes enlightening when you have a problem to which to apply it.

(That is also, if I may air a personal pet peeve here, why, though I am always delighted to see people understanding a concept, I get defensive as a teacher when later-life learners say "why didn't my teacher just say that?" Because the answer is often some combination of (1) even if they had said that exact thing, it wouldn't have reached you then, because you weren't then in the head space that you are now; and (2) they probably did say that, or something like it, and, because of (1), it seemed so far from relevant that you don't even remember it.)

I also am a teacher and I also find that exact thing is true.

I think it at least party accounts for second book syndrome, where someone asks the Internet, "Hey, I'm having trouble with Calc I, can someone recommend a good text?" and people in all seriousness suggest Spivak, or Rudin. "I also didn't understand it until I did Rudin" they'll say, having overlooked your point that headspace changes over time.

> people in all seriousness suggest Spivak, or Rudin

What's wrong with them?

> > people in all seriousness suggest Spivak, or Rudin

> What's wrong with them?

They are excellent books in their own right, and can be excellent books for mathematically mature students looking for a mathematically mature perspective, but their likelihood of serving well someone who is struggling with an introductory calculus course is, I would suspect, vanishingly small.

You don't have to read them linearly. Volume 2 (Seminumerical Algorithms) starts with an awesome discussion of random number sequences. I think all programmers who use random number generators could benefit from reading it. You can just pick and choose other topics throughout the series.

Also, used copies can be really cheap. Some of the volumes in my back up set were just $20 each, and in excellent condition. Apparently lots of people buy the books and never read them :)

I finaly had an opportunity to use my box this year for an interview step which was a home challenge. It was about debugging a flowed code to return all valid anagrams of a word according to /usr/share/dict and then propose optimizations. The fix was easy, and after implementing my own ideas, I searched for other approaches online as I might usually do. And only after that I had the idea to look at my TAOC box. And I did find some pertaining material. Not something I would usually do at various positions I had until now, where the shelf, if any, would only contains books on the frameworks, languages and tools used by the team. And when going back to home, there are plenty of other things to do.

Anyway, this helped me I think to go through this step of the recruitment process. My application went through the whole process, and didn't lead to a hire, but I really enjoyed it.

Now in the meantime I got a new job mostly on full remote, so maybe I might take the habit to reach my shelve for TAOCP more often.

I already tried a linear reading starting with volume one, but honestly it's not my favorite prose to go through like that alone. Knowing that you'll be given solutions in the out of date MIX, waiting for an MMIX covered edition come out, plus you might better first go through Concrete Math first, and so on, that is a lot of material to climb. The view up there might be wonderful, but finding the time and attention required can be a great challenge per itself.

They make great bathroom books; open at random and you're likely to find something interesting. Also confuses guests.

> These changes to Concrete Mathematics are too numerous to incorporate into the online errata. Therefore I've prepared replacement pages for anybody who wants to upgrade their copy of the second edition.

I have a beloved copy of CM displayed in a prominent place on my living room bookcase. Do I just stuff the replacement pages in between the old ones? Is there actually a proper procedure to replace pages and rebind a book like that?

Yes, but it is an art. I would recommend taking it to a reputable used book store. They will sometimes have traditional bookbinding skills, if they carry lots of old books.

A library might also have people with the right skills?

There are book binders that will do one offs, and the binding will probably be better than the original if you get it sowed & glued correctly!

The conventional way to add a page to a book is to glue it along the inside edge. https://en.wikipedia.org/wiki/Tipped-in_page

There are several versions of this article floating around the web showing the process if you happen to have bookbinding equipment laying around https://www.lineco.com/book-repair-3/

let's just get some perspective here

HE'S 84 YEARS OLD AND STILL DEFINING COMPUTER SCIENCE

I'd stopped checking for updates a few years back and just assumed that TAOCP was done because he'd run out of steam... boy was I wrong

Yup, this individual is epochally awesome, and he still lets us try and correct him.

So humble, so calm, so happy.

I don’t get it. We certainly wouldn’t let someone younger do so.

Puts into perspective the behaviour of companies developing reputations for firing older staff in favour of fresh graduates, doesn't it?

The talent is used for different parts of the process.

Which companies have that reputation?

IBM.

https://bigthink.com/the-present/ibm-ageism/

Also Google, Verizon, and Goldman: https://www.forbes.com/sites/jackkelly/2019/07/23/google-set...

Your link talks about age discrimination in hiring.

I was interested in a reputation for firing based on age.

(The link about IBM explicitly talked about firing based on age.)

2022-06-27 IBM settles age discrimination case that sought top execs' emails https://www.theregister.com/2022/06/27/ibm_age_discriminatio...

2021-04-30 Intel laid me off for being too old, engineer claims in lawsuit https://www.theregister.com/2021/04/30/intel_age_discriminat...

2021-04-16 Age discrimination class-action against HP and HPE gets green light to proceed: https://www.theregister.com/2021/04/16/age_discrimination_cl...

2019-11-25 Take a Big Blue cheque and go: IBM settles 281 UK age discrim cases https://www.theregister.com/2019/11/25/ibm_uk_settles_281_ag...

2019-09-17 Google age discrimination case: Supervisor called me 'grandpa', engineer claims https://www.theregister.com/2019/09/17/google_age_discrimina...

2019-07-22 Google settles a four-year age-discrimination battle with 227 engineers by dishing out... $11m https://www.theregister.com/2019/07/22/google_settles_discri...

2019-05-09 Oracle's legal woes deepen: Big Red sued (again) for age and medical 'discrimination' https://www.theregister.com/2019/05/09/oracle_sued_age_discr...

Thanks.

So this suggests that IBM (and maybe Oracle) have fired people for being too old.

For Google the complaints seems to be mostly about hiring, not firing.

(Just for disclosure: I expected in general to see more complaints about age discrimination in hiring than in firing.)

Knuth and Cliff Stoll are some of the people that I feel like are amazing treasures of our generation.

One of them hangs out on this site, too.

I wonder if we can get him on Team Tau: https://tauday.com/tau-manifesto

Very unlikely. I think that one you've become accustomed to using 2π everywhere there isn't a big technical reason to change. I love tau though and I use and promote it at work.

Wild link. That's a well reasoned argument and a good geometry mind bender. Thanks.

I’m a mathematician. Tau is bikeshedding. It’s far removed from being either interesting or useful. But it is a useful litmus test for whether a person’s options on math are worth listening to or not.

Disclaimer: I'm not a mathematician, and I only have a bachelor degree in science.

With respect: It's not bikeshedding.

Pi is wrong, and you know it, it's just that you (likely) feel that it is too late to change it, so why fight against history?

One of the best quotes I've ever heard is: "The standard you walk past is the standard you accept."

There are several problems with simply letting things remain as they are. I'm sure you've heard them already, but allow me to reiterate them for the benefit of the non-mathematicians here:

1. Pedagogical: I never "got" radians, or had a really firm grasp on trigonometry until I had heard of tau-vs-pi. Suddenly, everything just clicked. Sure, once you know it, it's a trivial mental substitution to replace every instance of pi with tau/2, but... only if you know to do this. I'm not the only one. Persisting on using pi is doing millions of students a disservice. A tau is one turn. A pi is half a turn! Wat? Do you buy bread half a loaf at a time and tell everyone that this is fine because you just need to by two half-loafs and it all works out?

2. Theoretical: Something that opened my eyes about physical theories is that it's easy to confuse oneself by cancelling small constants... such as when making the tau -> 2 pi substitution. The '2' gets cancelled all over the place, making other integer constants "wrong", and not always in an obvious way. They then get hand-waved away because they stop making sense, and so important things can be missed.

3. Boundary pushing: Most endeavours are limited by physics. Mathematics is unique in that it is limited only by human brain capacity. Ergo, the cutting edge of mathematics is bounded by how far a single human brain can go, but no further. Progress is achievable only through efficiency of thought, via shortcuts, elegance, abstraction, consistency, simplicity, and other similar means. Unnecessary complexity at the bottom is unnecessary and should be dropped to make room at the top.

I could go on and on, but better articles have been written on the topic.

The point is, if you see someone turning right three times to turn left, you immediately, indistinctively think less of them.

You're saying that no-no-no-no... the triple turners are just fine the way the are, and nobody should listen to those strange people in the minority who insist on turning directly towards the intended destination. Sure, it's faster, but we've never done it that way...

> Ergo, the cutting edge of mathematics is bounded by how far a single human brain can go, but no further. Progress is achievable only through efficiency of thought, via shortcuts, elegance, abstraction, consistency, simplicity, and other similar means.

I think this is perhaps the strongest pro-tau argument. How many intuitive insights may have been missed or delayed, because a 2 was factored out of 2pi?

Perhaps a softer stance on the way to tauism is to encourage folks to keep the 2 and the pi together as a single pseudo-symbol 2pi. So you'd never factor out the 2 (A = 1/2(2pi)r^2) nor would you distribute over it ((2pi)^2 would never become 4pi^2). I don't think that would work well in the educational space - now there is this extra weird rule about reductions - but it might help in other fields. At that point I guess you might as well use tau though.

> I think this is perhaps the strongest pro-tau argument. How many intuitive insights may have been missed or delayed, because a 2 was factored out of 2pi?

About as many as were missed or delayed because a 1/2 was lost from 1/2tau.

> Pi is wrong, and you know it, it's just that you (likely) feel that it is too late to change it, so why fight against history?

I'm also a mathematician, and pi is just fine.

> 1. Pedagogical: I never "got" radians, or had a really firm grasp on trigonometry until I had heard of tau-vs-pi.

The way I remember it, I learned trigonometry via triangles, whose internal angles sum to tau/2. This is absurd! Why don't they sum to a whole thing instead of a half thing?

> 3. Boundary pushing:

By the time one gets to the cutting-edge, of all the things that are holding one back a missing or extra factor of two is certainly not one of them.

The problem with pi vs tau is that it is pretty much impossible to convince everybody that one of them is the best and also that there are no other better choices.

So, in the absence of unanimity, the incumbent pi wins by default.

As an example of the diversity of opinions in this matter, while I agree that tau is better than pi, I believe that an even better choice than either of them is pi/2.

Any choice of the constant is equivalent to a choice of the measurement unit for the plane angle. Choosing tau is equivalent to choosing the cycle as the measurement unit for the plane angle ("cycle" was actually a standard name for this unit of plane angle, even if it is now out of fashion, e.g. in the old literature there are many references to cycles per second, cycles per meter etc.), while choosing (pi/2) as the constant, is equivalent to choosing the right angle as the measurement unit for the plane angle.

There are several arguments why the right angle is a more appropriate unit than the cycle, besides the fact that the right angle was already the unit of plane angle used by Euclid.

Even in a plane, but much more so in 3D space, a rotation of 1 cycle is ambiguous, it is not associated with a certain axis of rotation, like a rotation of 1 right angle. Given a rotation of 1 right angle, a rotation of an arbitrary angle is just the rotation corresponding to a multiple of that right angle, which can be non-ambiguously defined, like you expect for any physical quantity, to have a value that is a multiple of its unit.

Also in the trigonometric functions, the right angle is the most convenient unit for the arguments (making their reductions equivalent with taking the fractional part) and I am dismayed that the floating-point standard has chosen to recommend the functions sinPi, cosPi and the like, to be included in the standard libraries of the programming languages, instead of the much more convenient functions where the angles are measured in right angles, i.e. sin((pi/2)*x) and so on (such functions do not have the argument reduction problems that plague the functions with arguments in radians; the former are more convenient for large angles, while the latter are more convenient for very small angles).

The pi/2 argument is interesting.

Could you explain why a cycle is ambiguous in 3D? It seems like for any choice of coordinates, one cycle would always entail a return to an original point. But I'm probably misunderstanding.

There you go, arguing a trivial point at length instead of doing interesting math. This is why tau is not worth anyone’s time. There’s no one weird trick to understanding math.

I believe you just empirically verified this. :-)

> "A digraph is weakly connected if and only if cannot be divided into two nonempty parts"

Is there an "it" missing here?

(I wouldn't normally, but this is the closest I'll ever come to finding a Professor Knuth mistake!)

Email him. I got a check for noticing some macro barf in a preprint.

He doesn't have an email address. He accepts letters though:

https://www-cs-faculty.stanford.edu/~knuth/email.html

Well, the article that is being discussed [0], mentions an "address for bug reports (taocp@cs.stanford.edu)".

[0]: https://www-cs-faculty.stanford.edu/~knuth/news22.html

Not quite. From that very page:

> Email is a no-no EXCEPT for reporting errors in books.

I can't read apparently...

OK manchestercalling it seems you've caught a pretty major grammatical error unprotected in the wild. Now, do you replace the second if with it or add an it post second if?

"if and only if" is a standard and well understood term in our rather wooly language, so we will probably need your missing "it" to complete the sentence. Note that nothing was lost anyway - both you and I could see what was going on. English is pretty close to stripped to the waist and holding both fists up already. There isn't much left to remove for conciseness with clarity and unambiguity. I think we need to keep the pronouns but they can often be dropped with minimal loss of clarity. If we lost pronouns wholesale, we'd have to get pretty weird on suffixes or the like. Before you know it - you are speaking German or Dutch instead of our unholy brew of pretty much every language that has been seen on the continent and cuddled up to nearly every other language in the world.

(The last time I saw JB in the flesh was 1977 when I was seven and lived in Wythenshawe for a year in between parental postings to exotic lands like W Germany - it looked like science fiction to me then and it still looks rather impressive now)

as a side note, publishing errata and pages is such a lovely practice and i wish it was more widely applicable and used.

For anyone interested in a magical introduction to Bernoulli numbers, there is Mathologer's video [0].

[0]: https://youtu.be/fw1kRz83Fj0

Welp, time to throw my set of AOCP in the trash.

They say you can edit/modify an actual book with replacement pages!

Nice! This implies cutting the old page to a stub and pasting the new one onto it, doesn'it?

This reminds me somewhat of Nicod redefining Sheffer's Stroke as NAND instead of NOR.

(edit: Use the correct name)

Nicod

Good lord, I've had that wrong for ten years! Thanks for the correction. I think I was blending NAND into his name.

> It would be nice to believe that I actually got the details right in my first attempt. But that seems unlikely, because I had hundreds of chances to make mistakes. So I fear that the most probable hypothesis is that nobody has been sufficiently motivated to check these things out carefully as yet.

Even the pros have woes.

On a side note: Bernoulli was born in my home town Groningen (NL). Brilliant mathematician.

It wasn’t just one man either. I don’t know of a more famous family of mathematicians.

There is also a very nice link to CS since the first computer program was by Ada Lovelace and was written to compute Bernoulli numbers.

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