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A Wordsworth moment
Yes, April — Spring!
Taking Basil for his walk one pleasant, warm morning in mid-April, I had a Wordsworth moment. It wasn’t actually daffodils, it was buttercups: a lo-o-ong row of them on an otherwise nondescript roadside verge.
In fact the “I” there should really be “we”: Basil was entranced, too.
Now oft when on his couch he lies, in vacant or in pensive mood …
Check death
I have long since gotten used to writing “check” instead of “cheque.” For how much longer will this tiny morsel of Americanization avail me? Will checks soon disappear?
Whatever may be the case with checks, cheques seem to be dying a slow death. From across the pond:
Cheque usage peaked in 1990, when four billion were written, according to banking trade body UK Finance.
In 2010, more than a billion cheques were paid in. But this fell to 185 million during the pandemic in 2020, a 32 per cent decline on the previous year.
Barclays [bank] says the average number of cheques written by its personal customers is down by 44 per cent compared with before Covid-19 struck in early 2020.
Today, none of the High Street providers offers a chequebook to customers as standard. Those who want one must make a request. [Don’t write off our cheques! Millions rely on them, yet dozens of firms now refuse to take them or charge extra to use them … but you are fighting back by Amelia Murray; Daily Mail, April 12th 2022.]
If checks are going to fade away, I doubt credit cards will be far behind. We’ll then be where China is today: digital payment apps for all everyday purposes, supplemented by an occasional cash transaction (probably for something illicit).
China, said The New York Times in 2020, “skipped over a generation of finance and went straight to smartphone-based digital payments.” Indeed she did. Visiting my country-in-law at 18-year intervals, I’ve watched it happen.
When I lived there 1982-3 there was nothing but cash. It’s possible that checks were used in some corners of the Chinese economy — factory managers paying suppliers, perhaps — but none of my middle-class colleagues on the college staff had a checkbook. Credit cards were perfectly unknown.
When the college paid me off at the end of the year they did so in cash. I went to the bursar’s office and she carefully counted out the 100-RMB bills into my hand — an entire year’s salary. (And much more than a year’s working-class wages in China. A little knot of college workers stood watching the counting, slack-jawed and silent at the sight of so much money changing hands.)
Unfortunately the RMB couldn’t be exchanged for Western currency at the time — well, not legally — so I had to spend it all before I left. Among my purchases were the two stone lions that grace my front doorstep to this day (although badly in need of a steam-cleaning).
Returning in 2001, with China’s opening-up well under way, it was still the case that no-one of my acquaintance was writing checks. People knew about them, but only in a business context. China Daily reported in 2004 that
In Beijing alone, the clearing centre handles more than 140,000 cheques daily, and in Guangzhou, about 130,000 cheques per day. A bank such as Industrial and Commercial Bank of China in Shanghai handles around 40,000 cheques on a daily basis.
That was all business, though. Personal cheques — sorry: checks — were a rarity.
Credit cards likewise. They did take off: CNN Business reported in 2018 that “China had nearly 6.7 billion credit and debit cards in circulation” the previous year. That’s a surprising number — five cards per citizen — and I wonder if “billion” is a typo for “million.”
Whatever: there definitely are Chinese credit cards; I’ve seen them used. There is, however, no credit card culture, no expectation on checking out at a store that there will be a gadget to register your card with by swiping or inserting. Usually — well-nigh invariably outside the big cities — there won’t be.
So yes, there were checks and credit cards in 2019 China, and there still are. Before they could really settle in, however, digital payments came up and took over for most everyday purposes. The New York Times got it right: from the perspective of ordinary consumers, China went more or less directly from cash to digital.
Sometimes it’s an advantage to be behind everyone else. You have no installed base holding you back.
Amis Centenary
The British novelist Sir Kingsley Amis was born a hundred years ago this April 16th. I am an Amis fan from way back; I wrote an appreciation of him for National Review 24 years ago. From which:
Like all sensible people, Sir Kingsley regarded the Political Correctness movement with utter derision and cheerfully confessed to impure thoughts of the minor sort. He even wrote novels around such thoughts. The main character in Stanley and the Women (1985) wrestles with a question every man has pondered at some time or other: Are women all mad? Similarly, when asked in an interview whether he was antisemitic, Sir Kingsley replied: “Very, very mildly.” Urged to explain this, he added: “Well, when I’m watching the credits roll at the end of a TV program, I say to myself ‘Oh, there’s another one’.” Let him that is without sin cast the first stone.
Amis died in 1995. We should be glad, for his sake, that he was spared the horrors of wokeness. I’m not even sure that 1998 appreciation of mine would be publishable nowadays.
Several of Amis’s novels wouldn’t, for sure, as my reference to Stanley and the Women makes clear. That’s our loss. The raucous, bawdy, irreverent tradition in English-language fiction, the tradition of Chaucer and Shakespeare, is unacceptable to our enstupidated age, which prefers simplistic moral dramas.
Probably this will change. The pendulum will swing. Oliver Cromwell’s Puritans closed the theaters for twenty years; but the old Chaucer-Shakespeare outlook then re-asserted itself in Restoration Comedy and the novels of Fielding and Smollett.
It seems to me we’ve suffered twenty years of Puritan lunacy already, though. Isn’t it time for the pendulum to swing back? Perhaps it’s stuck on something.
Chitty chitty hama hama
In last month’s diary I included a link to what I called every Second Amendment enthusiast’s favorite page in the 1979 Xinhua Zidian Chinese-Chinese pocket dictionary.
That brought in an email from a Japanese reader:
Dear Mr. Derbyshire,
I saw my name in your “second amendment” page, of all places. Did not know it was pronounced [i.e. in Chinese] “bang.” In Japanese it’s either “Hama” or “Hin.”
I need to explain that. The dictionary consists of Chinese characters, arranged by their Chinese pronunciation. The page I linked to shows fourteen characters with pronunciations from bāng to bàng, the diacritical marks there indicting the tone (1st, 4th). The page is headed with “bāng-bàng“; that was my joke.
Chinese and Japanese are utterly different spoken languages, but for writing purposes Japan imported Chinese characters, giving them Japanese pronunciations. One of the characters on that “bāng-bàng” dictionary page is 浜, pronounced in Chinese as bāng, meaning a creek. In Japanese, however, it’s pronounced Hama and it’s a surname.
Got it? I replied to my reader that if he should ever decide to settle in Denmark, he will be pleased to find that “Bang” is quite a common Danish surname.
How did I come to know that? From reading Kingsley Amis’ 1963 novel One Fat Englishman in which the Danish surname Bang is used for comic effect.
The comic effect Amis was striving for was of course salacious. With that in mind I may as well reproduce one of the oldest and best-known British music-hall (i.e. vaudeville) routines.
In North Wales there is a pleasant town named Bangor. So the music-hall comedian tells his straight man:
“We’re going away for a few days. It’s our anniversary. I’m taking the wife to the little town in North Wales where we first met — to Bangor.”
There is of course a Bangor here in the U.S.A. Was that same joke current in vaudeville? Anyone here remember vaudeville?
Mongolians ♥ Russia
In my February 18th podcast I said the following thing:
Politically, there isn’t much worse a fate than to be ruled by Russians. Nations that have suffered that fate and then escaped from it, like the Baltic states, were always very glad to see the Russians go.
At the VDARE conference on the weekend of April 23rd I got into conversation with an attendee who took issue with that. He had recently been in Mongolia. That country was a Soviet satellite state from the early 1920s to the fall of the U.S.S.R., since when it has been independent and decently well-governed, with regular elections and a market economy.
Mongolians, my friend told me, speak well of the Russians. They are grateful to them for modernizing the country — building roads, raising literacy, and such.
Checking around on the internet I find a Mongol guide telling Time magazine in 1962 that, “Everything new here is Russian.”
That’s interesting, but also somewhat surprising. I know next to nothing about modern Mongolian history — come on, how much do you know? — but some quick scanning of internet sources tells me that Mongolia suffered horribly under Stalin’s puppet rulers. Things seem to have lightened up some after Stalin’s passing, as in the U.S.S.R. itself; but it’s hard to believe they left no resentment.
One factor may be China. Mongolia was de facto part of China until 1921, and de jure so for much longer in the imaginations of China’s leaders. When Khrushchev and other Soviet leaders visited Peking in 1954, Mao Tse-tung opened discussions by demanding the “return” of Mongolia. Chiang Kai-shek’s rump “Republic of China” in Taiwan only formally recognized Mongolian independence in 1961; I don’t know the current ChiCom position.
Why might this be a factor in Mongolians speaking fondly of the Russians? Because as awful as Russian government is, Chinese government is worse.
Glad to see the Russians go
In support of my original assertion, I offer the April issue of Literary Review.
That issue has a review by British historian Robert Service, who has himself written a shelf-full of books about modern Russia. The book he’s reviewing is Not One Inch: America, Russia, and the Making of the Post-Cold War Stalemate by Mary Sarotte, a Professor at Yale.
A passage in Service’s review tweaked my attention because I happened to read it shortly after watching some TV talking head telling us how, following the end of the Cold War, the U.S.A. had prodded, persuaded, and pushed former Soviet satellites in the Baltic and Eastern Europe into joining NATO. Here’s what Service wrote.
More than any previous historian, she [i.e. Sarotte] also emphasises the campaigns by most other countries in Europe’s eastern half to join NATO. This is a welcome antidote to a widely held assumption that the alliance’s expansion was exclusively the product of American initiatives. From Estonia to Albania, in fact, there stood a queue of national leaders demanding admittance. They pleaded on the basis of experience. The USSR had oppressed them in the years after the Second World War. They feared what might happen when the Russian state rose again from its ashes. Russia’s weakness, they pointed out, was likely to be only temporary. They urged the West to fix a canopy of security above them before it was too late.
It was not just history that made them frightened. As a diplomat from one of those states told me in 2017, the language that Russian leaders used in talks with them away from public microphones was different from the way in which they talked to ministers and officials from North America and western Europe. When Russia began to recover its sense of might and self-worth in the early 2000s, the bullying tone returned with menace.
I maintain my position that Russia-Ukraine is Europe’s issue, not ours; that we should have quit NATO when the Warsaw Pact dissolved in 1991, leaving NATO as the framework of an all-European defense alliance.
At the same time, I also maintain my February 18th point: that a key factor driving events in that neck of the woods is the very, very strong desire of nations that have once experienced Russian rule not to experience it again.
The Shelley problem
By way of commemorating the 20th anniversary of my U.S. citizenship, I signed off my April 22nd podcast with the USAF Band and their Singing Sergeants performing “This Land Is Your Land.”
That brought in a reproof, although a very mild and polite one, from a listener. Did I not know that “This Land Is Your Land” was written by Woody Guthrie, an actual communist?
Yes I did. (Although NB: Guthrie was certainly a communist in his sympathies, but I don’t think he was a Party member.) I figured that if “This Land Is Your Land” was good enough for the Singing Sergeants, it’s good enough for Radio Derb.
There is a nontrivial issue here, though: the issue of creations versus the personality of the creators. A common statement of the issue goes something like: If you saw a picture that you really liked, wanted, and could afford to buy, but which was drawn by Adolf Hitler, would you buy it to hang in your living-room?
In my opinion we hear far too much about Hitler. I prefer to think of the issue here as the Shelley problem.
Percy Bysshe Shelley (1792-1822) was an English poet, a great English poet. He was also a raging lefty, and in fact a simply terrible human being all round. I don’t know which he treated more badly, his women or his creditors; but it was a close-run thing.
Should the knowledge of all that subtract from our enjoyment of Shelley’s verse? That’s the Shelley problem.
It seems to me it’s one of those things that each of us has to make up his own mind about. There is no right answer.
For myself, I’m out on the extreme-tolerance wing. If you go to the “Readings” pages on my personal website you can hear me recite a poem by Mao Tse-tung, who was a mass murderer (and a lousy human being in thirty-seven other ways, too). Was I wrong to include it? Obviously I don’t think so, especially since I preceded the reading with some scathing remarks about the poet.
Going back to the case of Woody Guthrie: A great many people of his generation — Guthrie was born in 1912, whence “Woody” — took to leftist politics, including many much smarter than Woody: George Orwell and Arthur Koestler, to name two that I hold in high esteem. There were good reasons to do so. They felt that working-class people were hard done by, and they were right.
Leftism in 1940 was not at all the same as the fashion-statement leftism of today. Guthrie would, I am sure, have been appalled by wokery.
Sure, imagining that Stalin’s U.S.S.R. was a workers’ paradise was pretty dumb. All sorts of people are dumb in all sorts of ways about things remote from them in space or time, though; I think some of the points elsewhere in this Diary illustrate that. I’m inclined to forgive it.
There aren’t so many good songs in the world that we can afford to exclude from our lives songs written by people with daft ideas about politics.
Oh: I’m a big Bob Dylan fan, too, and Dylan worshipped Guthrie … Sorry!
Hyperpolyglottery
On April 5th The Washington Post ran a story about a hyperpolyglot, defined by one expert to be a person who can speak eleven languages or more. The actual hyperpolyglot they’re writing about is 46-year-old Vaughn Smith, a modest, self-effacing fellow who cleans carpets for a living in D.C.
Smith has full conversational ability in at least 24 languages, and can get along decently well in several more. He can read and write in eight alphabets and scripts.
It seems incredible, also depressing to a foreign-language duffer like me, but I believe it. I have met two characters like this on my travels, both as it happens in Hong Kong.
One was a young Frenchman of obviously high intelligence, touring the world for amusement. After a few days in Hong Kong, from a standing start, he could engage the locals in good Cantonese — better Cantonese than I had mastered after several months in the place.
The other was more a Vaughan Smith type, not at all intellectual, nor even, so far as I could judge, of much-above-average intelligence. An overseas-Chinese from Indonesia, he was in some kind of low-level trading — “import-export,” was all I could get out of him. His English was fluent, only slightly accented. He chattered cheerfully in Cantonese without pauses. I tried him out in Mandarin, which I was attempting to learn: he sounded like a Peking TV newsreader.
He assured me he could also speak Hakka, Teochew, Indonesian, Malay, Thai, and half a dozen other languages. I was skeptical until he took me to dinner one evening at an Indian restaurant in lower Kowloon. He got into a long, obviously fluent conversation with one of the staff, in a language I couldn’t place at all.
“What the hell language was that?” I asked when the staff member had gone. He replied: “That? Oh, Sindhi. A lot of the local Indians speak it.” (Which would make them technically Pakistanis … but Hong Kongers referred to all subcontinentals as “Indians.”)
George MacDonald Fraser, author of the Flashman historical novels, must have encountered a hyperpolyglot at some point: he makes his hero one.
“Each language has a rhythm for me, and my ear catches and holds the sounds; I seem to know what a man is saying even when I don’t understand the words” says Sir Harry Flashman, although he would rather understand what a local woman is saying, and would occasionally engage one to instruct him in the, er, vernacular.
(Sir Harry gets his comeuppance from Otto von Bismarck in one of the books when Bismarck, apprised of Flashy’s linguistic ability, replies sniffily:“A useful talent in head-waiters”
Bismarck really did say that in some context; it’s in von Bülow’s Memoirs.)
My own favorite hyperpolyglot was a real person, in fact a novelist, whose stories about life among Britain’s gypsies enlivened my childhood. This was George Borrow (1803-1881).
An English scholar named Ann Ridler wrote a 546-page book, George Borrow As a Linguist of which, I feel pretty sure, I am the only owner in the Western Hemisphere. It’s beautifully done. Ms Ridler did prodigies of research into Borrow’s life, work, learning, and acquaintances and summarized the results in a table at the end of the book.
Ms Ridler’s book is too large and unwieldy for my desktop copier, but I did image the end of the final table here. It shows Borrow having had reading competence in 51 languages, speaking competence in 20, extant translations in 47.
The 20 in which he had speaking competence were: German, Danish, Latin, Italian, French, Spanish, Portuguese, Modern Welsh, Irish, Scots Gaelic, Modern Greek, Russian, Armenian, Romani (in English, Hungarian, and Spanish variants), Hungarian, Turkish, “Tartar,” Manchu, Moorish Arabic, and “Arabic of the East.” The 51 in which he had reading competence included Icelandic, Cornish, Old Church Slavonic, Sanskrit, Finnish, Basque, and Classical Hebrew.
The language that gave him the most trouble was Manchu. He set himself to learn it for the British and Foreign Bible Society, which was planning a translation of the Scriptures. That was in January of 1833. Six months later he told the Society: “I have mastered Mandchou.” This, without there being any Manchu grammar textbooks available.
Six months after that, however, he admitted:
It is one of those deceitful tongues, the seeming simplicity of whose structure induces you to suppose, after applying to them for a month or two, that little more remains to be learned, but which, should you continue to study a year, as I have studied this, show themselves to you in their veritable colours, amazing you with their copiousness, puzzling with their idioms. In a word Mandchou is equally as difficult as Sanscrit or Persian, neither of which languages has ever been thoroughly acquired by any European, though at first they flatter the student with their deceitful simplicity.
And how are you today, Count Bismarck?
Math Corner
Here’s a wee brainteaser.
This is another one from Dr Peter Winkler, the puzzle master at MoMath. It was posted to the MoMath subscription list in March last year, so I figure by this point they won’t mind my reproducing it.
Brainteaser: What is the first digit after the decimal point in the number you get by raising the square root of 2 plus the square root of 3 to the billionth power?
John Derbyshire [email him] writes an incredible amount on all sorts of subjects for all kinds of outlets. (This no longer includes National Review, whose editors had some kind of tantrum and fired him.) He is the author of We Are Doomed: Reclaiming Conservative Pessimism and several other books. He has had two books published by VDARE.com com: FROM THE DISSIDENT RIGHT (also available in Kindle) and FROM THE DISSIDENT RIGHT II: ESSAYS 2013.
English
Arabic
Chinese (Simplified)
Dutch
French
German
Hebrew
Hindi
Italian
Japanese
Korean
Persian
Polish
Portuguese
Russian
Spanish
Swedish
Turkish
Vietnamese
The Shelley Problem:
To Sinatra’s detractors, he was a boor, a drunk, a womanizer, a consort with organized criminals, petty, vindictive, loutish….
To Sinatra’s admirers, he was a boor, a drunk, a womanizer, a consort with organized criminals, petty, vindictive, loutish….BUT…. the greatest interpreter and performer of popular music in the twentieth century (or any century for that matter).
I lived in Mongolia eighteen months almost twenty years ago. To most Mongolians, “the Socialist era” was the time of modernism. They got by a lucky chance the best the Soviet Union could offer. You could say Russian culture and Soviet Union science without the Russians. Their previous culture was Dalai Lama and under “Manchu” control. They have public holiday on their liberation day from the Manchus. A people as backward as pre Chinese invasion Tibet, became in a life time, a people whose elite were fluent in three languages. The fall of the Soviet Union brought them common destitution. However their former Communist elite quickly regained their privileges by becoming capitalist stake holders. The rest of the population remained impoverished but have progressed through development of natural resources in the last twenty years.
You need to do something with those sorry looking plants also!
I enjoyed this ramble.
Derbyshire Writes
That statement is rubbish, especially as the Baltic States, Belarus, Ukraine, the Central Asian States and all the rest were able to leave the Soviet Union, ie Russian control, peacefully and without hindrance. Under the Soviet Union, and previously, these peoples had even been able to learn their own languages in school and university, this despite all the atrocities committed, especially under Stalin.
Derbyshire writes
The Russian Government permits self rule for nearly all its minorities, where they can learn their own languages in state schools, in many cases. This is particularly evident in Siberia, where large native minorities are still evident. So, no, the Russian Government is not awful.
If you are a minority, the Chinese Government is certainly awful, as long term aims are the obliteration of the native culture by massive migration of Han Chinese into these areas.
The United States is nearer the Chinese than the Russian model. Over the vast bulk of the United States, native populations and cultures have been obliterated by settlers and later migrants. There are still large Reservations in States like New Mexico and Arizona, but their status and resources in no way match comparable areas in the Russian Federation.
So, if you are comparing the 3 states: Russia has a good record, the United States has a poor one, and China a very poor one. An old Russophobe like Derbyshire would never admit this, though.
Indeed, that’s the case, in the same way that that lots of Chinese people skipped the whole Alexander Graham Bell phase and went straight from no phone to mobile phones.
As late as 2009, cash was still King in China. It’s not like one couldn’t use a credit card, but one could use cash for about anything. I, for one, appreciated that more than most, because I understand the problem forecasted in Revelation 13. Not solely because of the “Cash is King” thing, I thought of China as the “Wild, wild East” back in those days, less than 1 1/2 decades back. I was enamored by the place, but extremely disappointed by 12 years later.
The government has gone full Orwell over there now, using the (nothingburger, at this point) excuse of the Flu Manchu to screw with the people of Shanghai and other cities. Shanghai, the gleaming crowning jewel of China, has people being marked for quarantine camp entry based on what their phones tell the authorities, people locked in their apartments for 50 days now, and various indignities.
I notice that not a one of the Unz Review writers has written WORD ONE about the goings on in China right now! It’s not so much what the regular writers write, but that there are no special articles about this from anyone.
How about you, Mr. Derbyshire? Will you write something about it soon? Right now, people have to go to Peak Stupidity to learn anything, as fed to us from our China sources. (You know how that can be, typos, cusswords, politically incorrect jokes, that whole thing …)
The Bangor joke reminds me of this one
A: My wife’s gone to the West Indies.
B: Jamaica?
A: No, she went of her own accord!
that inspired the title of a reggae-influenced Led Zeppelin song, oddly rendered by them as “D’yer Mak’er.” No wonder hard-r-sounding American radio DJs pronounced it “dyer maker.” A better rendering, in my opinion, would’ve been “D’ja Make ’Er”, maybe with the question mark, “D’ja Make ’Er?”.
Re: Borrow and his languages…
Scots Gaelic is not a language. It’s Irish with an accent and a different orthography. It’s at most a dialect which I can understand fairly well (I’m reasonably proficient in Irish). The same is true of Cornish in relation to Welsh.
I am on the side of “shut up and sing” while at a concert, but, for rock music in general, if you’re going to let politics get in the way, there won’t be a whole lot you can enjoy. Even if songs are themselves political, lyrics are dead last in making a good song. I knew the old 1980s Bruce Hornsby songs, for example, were lefty clap-trap, but one doesn’t even need to think about the lyrics when enjoying the music.
Woody Guthrie’s This Land is My Land was an original American folk song, known well enough to have its 10 y/o kid knock-offs (something about getting off MY land, cause “I got a shotgun, and you don’t got one” – the property rights version – who knew we were Libertarians already at such a young age?) His son Arlo was a very good entertainer. I went to one of his concerts. I didn’t know then, but that famous Spirit of New Orleans, a very favorite of mine, was written by the great songwriter Steve Goodman.
Finally, I’d missed putting in the 6 Peak Stupidity posts so far on the latest Chinese Flu Manchu Totalitarianism. We’ll have another short post on this today or tomorrow. For now, see:
It’s baaaacck! The Kung Flu in China.
Stupidity does not stop at the California coastline
Chinese Covid-testing Craziness – quarantining babies and the testing video
Covid Zero, Politics, and Totalitarianism in the Orwellian envy of the World
The latest on the Covid-Zero Shanghai Shitshow
Escape from Suzhou – the motion picture
Oh, right. Of course, Bangkok goes without saying, doesn’t it? (Not without lewd thoughts, but without saying…)
Maybe you people don’t care about our neighbor to the north, but there is that city – with an international airport even – up in Saskatchewan with the Seinfeldian name that rhymes with a female body part. No, not Pits-burg, dummy!
I used to know a hyperpolyglot: by trade he was an academic lawyer. In each year’s Long Vacation he’d trot off somewhere exotic and return fluent in the local lingo.
A typical autumn conversation: me – “You’re back , John. Which language this time?”
John – “Hungarian.”
We’d lost touch by then but the fall of the USSR must have been a blessing for him.
My point is that a man in academic life won’t really be able to fake mastery: there will always be people around – at least in any decent university – who would spot such fakery in an instant.
This guy is a celebrity among Chinese immigrants with his YouTube videos. Chinese people are so good-natured.
There is an old English word which suits Derbyshire to the T. It is twit. Rabbiting on about Russia when the excesses were Soviet excesses, do a little research the soviet leadership was as follows’
Lenin Russian
Stalin Georgian
Malenkov Geographically a Khazak
Kruschev Russian
Brezhnev Ukrainian
Gorbachev Ukrainian
The question of whether something is a language or a dialect is arbitrary. As someone said long ago, a language is a dialect with an army and a navy. This is exemplified by the current usage of the words Serbian and Croatian as opposed to the former term Serbo-Croat.
You are correct about competence in Irish conferring almost equal competence in Scottish Gaelic, but my competence in Swedish grants me a similar privilege in regard to Norwegian.
I don’t think knowledge of Welsh is much help with Cornish, or vice versa. They are too different.
Lol, “Russian” rule.
Here’s Russia in 1896, more than two decades before (((you know who))) would bring their destructive revolutionary spirit and enslave a people for 70 years. Only to be followed up by an additional decade of rape of the country by the same group until the second coming of Peter the Great* showed up.
*Putin
I found a copy of Mein Kempf at the Goodwill, with an intro from Abe Foxman. I bet Greenblatt would just call for it’s being banned.
The Communists did unto Mongolia a lot of what they are now doing to Tibet. Unfortunately for the Mongols, they had no easy way to attract world attention, and this was in the 1930s when Communism was hip and modish among “modern” minded people.
I’m not going to say that all they did was bad. But a lot of babies were thrown out with that bath water.
Godflea is a ChiCom propagandist and Larry Romanoff is no better. Andrew Anglin is now at the dailystormer.in ( “white taoism now” ). So none of them is going to deal critically with the latest shennanigans in China. John Derbyshire has been back twice in 25 years and does nothing other than regurgitate Neocon propaganda. So good luck to you and your website.
PS
Mobile phones….. You’re not an English immigrant like the Derb and Pistol Pete Brimelow ( How many 75 year-olds have 3 daughters under the age of twelve )? Nothing to be ashamed of, but I suggest you keep that photo of Prince Philip well hidden.
Thanks for the reply, VMA. You are quite right about the 3 bloggers/writers you mentioned first.* OTOH, Mr. Derbyshire has a Chinese wife. I imagine she’s heard something of what’s going on there, and Mr. D. has a big advantage in knowing Chinese. (Since he discussed people knowing lots of languages in this very “Diary”, very modestly, I might add, I’ll state here that I am amazed that ANYONE not growing up with it can learn Chinese! What a clusterf__k the written language is!)
Still, Mr. Unz puts all kinds of articles up here, in keeping with his motto up top. It’s very weird there’s been absolutely nothing about the crazy goings-on right there, right now.
Thanks for your wishes. Peak Stupidity has its sources, and we have ways of making them talk. (Unfortunately, some of the time, that’s in Chinese!)
I really don’t know why you have a problem with the use of “mobile phones” though. Yeah, I know the term “cell”, but that’s more a technical term that does not do a good job describing their purpose. You know what, I take back a little bit of what I said about the Chinese language. One thing they do is just put together the very simple terms to make up new tech terminology. A cell phone is a dian hua, literally “electronic talk”.
PS: I’m sorry, it’s probably me, but that Prince Phillip joke went, whooosh, right over my head. Is he a current prince, some guy from history, or a guy on a tobacco pouch?
.
* What it looks to the unaided eye, is that Anglin is in a pissing contest with Steve Sailer on whom Tucker Carlson gets his material from. ;-}
Agreed. It is very hard to make sense of 20th century Russian history without reference to (((you know who)))
Aleksandr Solzhenitsyn tried to explain this in his book “Two Hundred Years Together.” Today one attempts in vain to get an American to even react to the fact that every book Aleksandr Solzhenitsyn wrote is available to Americans…except this one.
“Stanley and the Women” is perhaps the best Kingsley Amis novel. Very close second best would be “Girl, 20”, and “Difficulties with Girls”.
The last guy who could actually write and not be a cunt about it.
apparently 25% Jewish
Rumors persist that he was of what is known as a Georgian Jew.
Russian, both by blood and geography
Russian. Mother may have been an Ukrainian.
50% Russian, 50% Ukrainian.
China is deliberately sabotaging exports to the West under cover of lockdown. On behalf of WWIII.
You are a trifle naive. Take not everything at face value.
City of New Orleans. Name of the train.
I always liked that song, too.
Nearly forgot: We’re now advised not to use checks, particularly if we send them through the mails. Too many criminals in today’s America.
https://www.msn.com/en-us/health/medical/don-t-waste-your-money-man-loses-93-000-after-check-is-stolen-and-washed/vi-AAWPrnJ
https://www.unz.com/isteve/asserted-alito-supreme-court-abortion-decision-draft-leaked/#comment-5321551
Have to admit, I write perhaps one check a month at this point. Sic transit..
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Shelley’s young wife, Harriet, was pregnant, and he abandoned her to run off to Europe with two women. I don’t see how any treatment of creditors could possibly be “close” to this sociopathic cruelty to Harriet. Shelley, btw, had a history of inventing hoaxes that he had been attacked by ruffians..
Shel Silverstein wrote the poignant abandonment song “Sylvia’s Mother.”
Sorry, my fingers just up and typed that, HammerJack. I knew the name perfectly well with it being in my head an all! Thanks for the correction.
I’ve read this theory, but I find a lack of decent motivation for that. What would it be? Do you we need all the Cheap China-made Crap that badly?
China still has export-based manufacturing, so wouldn’t this just be hurting the big SE manufacturing cities themselves more than the West?
Peak Stupidity will have more on the story tomorrow.
I’m in total agreement there. On a similar matter, I did ask him once why UR didn’t have a specialist military columnist, but didn’t get a reply. Given the situation in thr Ukraine, the views of a genuine, independent military commentator would be interesting to read. Anyway, there’s a lot of wars out there, needing to be covered. Lots of UR columnists cover these wars, but from a political, social and economic perspective: none from a military perspective.
The only guys at UR who acted as military columnists didn’t last long. Patrick Lang found the atmosphere ( ie we commenters) uncongenial. Andrei Martyanov and Mr Unz seem to have fallen out terminally. As regards the military situation in Ukraine, all Mr Unz does is to recommend that we watch Scott Ritter videos. Though, to be fair, he did rustle up an article by Mr Ritter a few weeks ago.
I don’t. I’m not American or English, but find the American insistence on “cell phone” completely ridiculous. It’s so undescriptive. So I genuinely thought you might be of recent English ancestry. If you’re not, then I must commend your independence of mind in ditching the “cell phone.”
Instead of apologising for the sloppiness of his anti-Russian campaign so far, Derbyshire has stepped it up a notch. Unfortunately, his determination does not pay off in terms of the quality of his mini-crusade.
He began by referring us to material on this malodorous Magnitsky affair. Then he came up with impressionistic figures about corruption in Russia – not a very strategic move for a citizen of a country which is widely supposed to have a bought and paid for parliament, not to mention its media.
Now he tells us that countries once occupied by the Russians do not crave their return.
Well John, today happens to be the very day that my home country celebrates the liberation of the Germans in 1945 and commemorates the victims of those harrowing years. The German General Blaskowitz signed the surrender of the German troops in Holland on 4 May of that year.
I was only nine at the time, but I have not forgotten the explosion of joy on that occasion. The celebrations lasted for months. Is that a reflection on the Germans of today and their beautiful country? Of course not. People just don’t like to live under the heels of foreign troops. Even in 1913, the Dutch celebrated the fact that the French occupation of their country had ended exactly one century ago (with the help of the Russians, by the way – as was the case with the German occupation).
The fact that you quoted Robert Service to support your views does not make your case any stronger. In the American Historical Review, his biography of Trotsky was characterised as little more than a lampoon. The reviewer, Bertrand Patenaude, is quoted on the World Socialist Website as saying:
No doubt he is doing the same with Putin today. I tried to listen to one of his lectures, but quickly decided that I was wasting my time. I noticed that one of the commentators who had sat through more of it protested against him having said that Russia had “obviously” interfered in Trump’s election.
Next time, pick a more reliable witness, John (are you the “Jonathan Derbyshire” who interviewed this “historian” for the New Statesman a while back – yes, I assume you are).
It’s easy, I live in Shanghai and the problem Shanghai is facing right now is that there are a lot of civil servants in the local government from the bottom to the middle who are trying to extract the country’s wealth. Because they were the ones most affected by the Idea of the Jewish banker’s self-interest. Shanghai, after all, was a platform for communication with Jews.
I don’t deny that there is a problem with being too strict or rigid in enforcing the law, but the problem is that there are some people who are acting in their own interests at the expense of others.
When this idea conflicts with national policy, a large number of Jewish media such as CNN and the CIA create and exaggerate fake news.
Dei Lu,
No offense, but after all the information I’ve seen, your comment sounds like a load of crap. Are you going to deny that Shanghai (and other cities) have apartment buildings under LOCKDOWN, that there are quarantine camps, and that the authorities are swabbing everything but the kitchen sink to find that nasty “positivity”?
It’s quite the reach to blame Jewish bankers for this one! That corruption you refer to has been around for 3,500 years there. Has it been the Jews’ fault the whole time? Maybe that Western explorer started the extraction of wealth from China back in the 13th century AD, what was his name, Shlomo Polostein?
Face it, Mr. Dei, with its massive economic power and hardworking people, China could be a good place to live right now. Instead that Xi guy had to fuck it up, and, as seems to be a regular thing there, the place is needlessly becoming a hot mess.
不,谢谢
Math Corner: The digit is ‘9’. I don’t have a full proof, but note that one billion is an even number. Then looking at powers of (sqrt(2)+sqrt(3))^2 = (5+2*sqrt(6)) we see that they take the form (n+d*sqrt(6)) where n/d is a convergent of the continued fraction expansion of sqrt(6). Since n/d converges from above to sqrt(6), the expression (n+d*sqrt(6)) “converges” from below to (n+d*(n/d)) = 2n. In other words, each successive power of (5+2*sqrt(6)) is closer to the next larger integer than was the previous power. Already by the second power (5+2*sqrt(6))^2 the distance to the next larger integer is already less than 1/10, so by the five-hundred-millionth power (5+2*sqrt(6))^500000000 = (sqrt(2)+sqrt(3))^1000000000 the distance remains less than 1/10, hence the digit in question is ‘9’.
Surely there’s a better way! Some nifty property of the binomial coefficients, perhaps?
I read in the book Dreadnought that Chancellor Bernhard von Bulow had stipulated that his memoirs be published only after his demise. Perhaps it was a wise choice, as his gossipy and self-serving observations ruined his earthly reputation. The Kaiser, a guy not known for his sense of humor even by Teutonic standards, remarked that it was his only known instance of a man who had died and then committed suicide.
I live in Beijing, some people across the street tested positive, some other people live in my building is a contact to the positive case, not a close contact obviously since he is locked down at home in this building, otherwise he will be send to medic-camp.
As one of the neighbors that share the stairs, elevator, gate and pipeline with this contact in the building, I am also locked down in my home for 6 days covering the whole Labor’s Day vocation. Medical team in white gown knocked my door on day 2 give me a test. Man in white gown sit at my building gate, help us to dump the trash. Delivery guy left the goods at the gate so I go downstairs to pick them myself. My refrigerator and kitchen all fulfilled as I saw what happening in Shanghai, I prepared myself like most Beijinger did.
It is informed that I am not locked down anymore since nobody in my building is positive. I can go anywhere if I get two Negatives within 3 days.
Four nephews live in Shanghai. Their grandma, my aunt, is very worried how they living the life. Since we saw a lot news what a mess in Shanghai, and the grandma in hometown lost contact with the grandchildren recently after the children told her that since there are too many disinformation in Shanghai so there are also communication controls for them. That’s strange, I see many message everyday online from Shanghai. Are my nephews positive? Are they in the medic-camp? Or do they prefer to mute themselves with such excuses? I don’t know. But the family is not very worried for this temporary lost of communication. They are young person live in big city alone, they can handle even if they are positive and live in medic-camp, they won’t infect the family. My aunt worries about the covid sequela for sure.
What the family and most of my friends think about Zero COVID policy and Shanghai?
Zero policy is the right way for China. We do not want to handle the other-way situation. There is the China way (dynamic Zero), and the other-way (let it be). We can not take it the other-way: medical resource run out and mass death of the old weak diseased person, large scale sequela of the general population, economic crash down, print money inflation and big wealth transfer to the rich during big crisis, waves after waves, variants after variants.
Give us more time we will solve this problem by our own way, new vaccines, new medicines, new solutions, etc. We will not take it the other-way since we still have the chance to finally work out the better solution, no matter how rigid, how awkward it is, no matter how costly and destructive it is, no matter how the international/even internal media laugh at us smear us, no matter how all the other counties have to take it the other-way already. As long as we see it better than the other-way, we go our own way.
Shanghai is the Capital city of Capitalism China. Greed, coward is in human nature.
Shanghai hesitated to lock down and mass test during the very early stage, it developed into large scale crisis. Shanghai people pay the high price, China learn the lesson. Yet, it’s not too late to actually do something.
The daily new case 4th May 2022 in Shanghai: 261 case +4390 asymptoms. Compare to the peak: 5407 case 28th Apr, 25173 asymptoms 10th Apr. The big picture is: Shanghai is getting back to normal in June/July.
Many others cities of China has been infected by Shanghai. But with the lesson learned, it’s unlikely we have another capitalism Capital city within mainland China.
Stop pretend like ‘you’ really care Chinese life and well beings. ‘You’ are cursing and blaming just to feel good about yourselves. As long as ‘you’ give no constructive suggestions or solutions, ‘you’ are disgusting blaming us. Look into the mirror, how ‘you’ sick people deal with your own life and well beings. Shameless or brainless, ‘you’ are one of them, or both.
Lenin was a quarter Jewish, and a quarter to a half Chuvash. In American terms, he would probably he considered 1/4th Asian.
“An English scholar named Ann Ridler wrote a 546-page book, George Borrow As a Linguist of which, I feel pretty sure, I am the only owner in the Western Hemisphere.”
Thank you for the recommendation. I now have my copy, and since I live on about the third meridian west of Greenwich you are no longer the only owner in the Western Hemisphere, unless your reference datum, like that of the ancients, is in the Azores.
It cost me about £14, including postage, from AbeBooks, which I think is a bargain.
This is as stupid as saying Tom Brokaw, Geographically a Lakota Indian
I can’t tell for sure if this comment was meant to be directed in reply to me, but I will answer either way, because the mindset seen here is fascinating. Yes, that’s written in the same sense as Dr. Spock’s words to Captain Kirk as he sees a life form so alien that it can never be truly understood by the human mind.
I do appreciate your anecdotes from your dealings with the Covid-19 Panic-Redux over there in Peking, along with your info. from Shanghai. I give you much credit for honesty, #369.
I am fascinated that you are truly still scared of this virus. I want to know if you have a real fear of it, and if you did in ’20. Could you tell me if you have ANY family, friends or acquaintances who died or were in very bad shape at any point directly due to Covid-19? Is it the case that you know any such people now? Do you know the common-sense knowledge about mutation of viruses? They do mutate to less deadly strains, just as Influenza has, even though there will still be bad years here and there as strains the body is just not used to fighting come and go.
What do those case numbers mean, especially compared to denominators of a couple dozen million people in your biggest cities there (incl. yours and Shanghai)? Are these all deadly ill people? Are they even sick?
Could you tell me what this state of ZERO COVID means? Do you know? Is it the goal that this particular Coronavirus be made nonexistent in any Chinese human body? What will be the case then for travelers into and out of China?
I’ll write more… just fascinating, Captain.
I see that you feel the people of Shanghai should be punished with the dreaded Flu Manchu because they have been too Capitalist, greedy, and cowardly. Regarding the former, you do know that even the most retarded people in the Chinese gov’t in Peking know better than to go back to Communism, right? Nobody, but nobody, there wants to go back to the Chairman Mao times economically.
Greed, well, yeah, that makes the world go round, and if you let folks make their own choices with their own hard-earned money, you will be surprised how charitable they can be. They just don’t like doing that with guns to their heads.
As for cowardly, anybody who is so pussified as to agree with a Totalitarian government locking you in your quarters at will and deciding when to send you away for weeks at a time based on a swab up your nose has no right to call anyone else cowardly. Is it possible the people of Shanghai just have more common sense than Pekingers and aren’t quite so susceptible to being scared shitless by their government? It’s something to think about, right?
Do you know how many Chinese people live in America, counting at least a million or two illegal aliens? I estimate from 5 to 10 million. Even below my low end, it would be 1% of the population. (Can you imagine 14 million Westerners in China?)
What I’m getting at is that these Chinese people know what’s going on from living in America, as people DO NOT drop dead in front of them and people are living normally now (albeit having let governments here set big Totalitarian precedents). One would think that you could get SOME news of the situation here and in all of the West, as people have put this piece of stupidity behind them.
As for me, out of 5,000 people at a Big-Biz concern I well know, only 3 people were said to have died of the Covid-19. After wearing the face diaper only at the point of near firing and being around thousands of people for 2 years with no concerns about social distancing bullshit, I didn’t catch that original Covid~Classic. I and my son likely did catch the Omicron variant recently (we had a sore throat, a dry cough for a day, lack of smell for a week, etc.), but I can’t be sure, as we weren’t going to bother getting tested.
It wasn’t the end of the world. Yes, we are “taking it the other way” here. That way is sanity. Your way is insanity. The fact that you want that … means something … fascinating …
I know quite a few Chinese people, and I do care about some of them. It’s not like it’s my job to care about what happens to China, but it’s just all so disappointing. See, my country is heading in the wrong direction financially, culturally, demographically and via loss of liberties. 15 years ago, I thought of China as the “Wild, wild East”, and I mean that in a good way. Now, as far as a beacon of hope for the world, shit, forget it, not after this latest business.
You say I have no constructive suggestions, but my whole argument against Totalitarianism IS my suggestion. You had a good thing going, and you people are just letting the CCP screw it all up again*. I don’t get what you mean by “blaming”. All that stuff going on there is something I think Americans should know about as a warning to not let it happen here.
The video that Peak Stupidity has up here has these “Big Whites”, cops in ridiculous, hot-as-hell, biohazard suits kicking down the door, should be a warning to Americans to hold onto their guns. Do that at my house, and you will all have holes in your lungs. Covid-19 will be the least of your problems!
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* I was heartened to see a video recently of Chinese people being stopped from leaving the train station where they were being “held”. They all just made a rush across the turnstiles. As I alway say, you can’t fire them all, and you can’t arrest them all… there’s great power in numbers!
I always smile when liberal leftwipes rhapsodize about Woody Guthrie. Not sure why he hasn’t been “outed, othered, and cancelled” yet (sounds like what they do to hashbrowns at Waffle House, but ol’ Woody was a big time racist. He hated blacks. I mean, that just makes me like the guy more, but others would probably differ with that. He also hated those folks whose breath smells like smoked fish.
Sounds like the Woodman “had it all sorted” , as the Brits might say. Good show!
Across the strait from Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch.
The pronunciation here is less amenable to the pun. Most Americans would say “bang gore”– more suitable for political humor! Though stationed in Maine years ago, I can’t remember how Maineacs said it. “Banga”? “Bang-gaw”? Anyone know?
Maine has starkly different accents along the coast and in the interior. Two of my shipmates grew up five miles apart. One had one, the other the other.
And a Calais, also in Maine. That is pronounced like the foot problem. Calais, Maine, is the easternmost community in the US, unless you count Alaskan islands across the Date Line.
“Wogs begin at Calais” works in the US as well, or would if we used the term wog.
Cleftwipe is a better-placed term. (Also, these people are in no way liberal.)
Son Arlo said that when his dad first arrived in Greenwich Village, the local “theorists” held him in awe. He was what they all wished they were, the genuine article.
Macumazahn wrote:
Yeah, there is.
(Spoiler alert!)
Let A = (√3 + √2) ^ 1,000,000,000
Let B= (√3 – √2) ^ 1,000,000,000
If you write out the (beginning of) the binomial expansion for each, you will see that the terms on which they have the same signs are necessarily all integers (having √3 and √2 each to even powers). I’m also using the obvious fact that all of the binomial coefficients are integers, of course.)
The other terms have opposite signs between A and B.
Therefore, A+B is an integer.
But, it is obvious that B is a very, very small number! (Because √3 – √2 is a lot less than one.)
And also A = (A+B) – B.
Therefore, you are subtracting a very small number from an integer to get A, which is the number we want.
It follows that the first digits after the decimal point (in fact, the first few hundred million after the decimal point!) are all 9. (This follows just from the standard subtraction algorithm where you keep “borrowing” to do the subtraction.)
Once you see this, you see that the same phenomena occurs for
(√m+ √(m+1)) ^ 1,000,000,000 for any positive integer m.
It is fun to play with a calculator for the first few even powers and see the 9’s start piling up behind the decimal point.
Of course, it is quite impossible to answer the initial brainteaser with a calculator: you would need a crazy high level of precision.
This is a simple example of how brains beat computational brute force.
Thanks. This argument gives even more – your approach will work whenever |√m- √n| < 1 which is kind of cool: if |√m- √n| < 1 then as k increases (√m+ √n)^2k draws ever-closer to the integers.
Macumazahn wrote to me:
Yeah, that’s right: I actually did play around with that but I didn’t want to go into greater generality.
I find how this works to be very strange, by the way. I bet there are a whole slew of things like this that even those of us who are pretty good at math are ignorant of.
Of course, I also suspect that there are mathematicians who know a huge amount of this kind of stuff and who would find it amusing that I don’t know all of it!
And now I need to think through your continued fraction argument — one of those many subjects about which I know just a little but wish I knew more.
Any suggestions as to where I should look to learn a bit more about continued fractions?
Now, I do.
The following fills the gap in my earlier post.
First, the powers of (5+2√6)^k clearly take the form (a(k)+b(k)√6) where both a(k) and b(k) are integers.
I claim that the fractions a(k)/b(k) are the odd convergents of the CF expansion of √6.
Proof:
Consider the expression
(s+t√6)*(a(1)+b(1)√6)
where s, t, a(1) and b(1) are real numbers. This expands to
(s+t√6)*(a(1)+b(1)√6) = s*a(1)+6*t*b(1) + (s*b(1)+t*a(1))*√6
If in particular we substitute s=a(k) and t=b(k) then we have
s*a(1)+6*t*b(1) + (s*b(1)+t*a(1))*√6 = a(k)*a(1)+6*b(k)*b(1) + (a(k)*b(1)+b(k)*a(1))*√6 = a(k+1) + b(k+1)√6
so we get two recurrences
a(k+1) = a(k)*a(1)+6*b(k)*b(1)
b(k+1) = a(k)*b(1)+b(k)*a(1)
If we define a(1)=5 and b(1)=2 we obtain two recurrence relations
a(k+1) = 5a(k)+12b(k)
b(k+1) = 2a(k)+5b(k)
which together generate coefficients a(k) and b(k) such that (5+2√6)^k = a(k)+b(k)√6 for k>0.
Turning now to the CF expansion √6 = [2;2,4,2,4,2,4,2,4,2,4,2,4,…] consider an odd convergent of the form
C(2i+1) = 2+1/(2+1/(4+1/(2+1/(4+1/(2+1/(4+1/(2+1/(4+1/(2+1/(4+1/(2+1/(4+1/2…))))))))))))
recall that 2+2=4 so we can substitute as follows
C(2i+1) = 2+1/(2+1/(2+2+1/(2+1/(4+1/(2+1/(4+1/(2+1/(4+1/(2+1/(4+1/(2+1/(4+1/2…))))))))))))
but note that this can be written in terms of the previous odd convergent, i.e.
C(2i+1) = 2+1/(2+1/(2+2+1/(2+1/(4+1/(2+1/(4+1/(2+1/(4+1/(2+1/(4+1/(2+1/(4+1/2…))))))))))))
C(2i+1) = 2+1/(2+1/(2+C(2i-1))) = 2+1/(2+1/(2+C(2(i-1)+1)))
This gives rise to a pair of recurrence relations, one for the numerators and another for the denominators.
Now suppose that j=2k-1 so that n(j)/d(j) = C(j) = C(2k-1) is an odd convergent of the CF expansion of √6.
Then the next odd convergent is at index 2(k+1)-1 = 2k+1 = j+2 so
n(j+2)/d(j+2) = C(j+2) = C(2k+1) = C(2(k+1)-1) = n(2(k+1)-1)/d(2(k+1)-1) and we have
n(j+2)/d(j+2) = C(j+2) = C(2k+1) = 2+1/(2+1/(2+C(j))) = 2+1/(2+1/(2+C(2k-1))) = 2+1/(2+1/(2+n(j)/d(j)))
There’s nothing for it but to grind it out…
n(j+2)/d(j+2) =
2+1/(2+1/(2+n(j)/d(j)))
2+1/(2+1/(2+n(j)/d(j)))
2+1/(2+1/((2d(j)+n(j))/d(j)))
2+1/(2+1/((2d(j)+n(j))/d(j)))
2+1/(2+1/((2d(j)+n(j))/d(j)))
2+1/(2+d(j)/((2d(j)+n(j))))
2+1/(2+d(j)/(2d(j)+n(j)))
2+1/(2+d(j)/(2d(j)+n(j)))
2+1/((2(2d(j)+n(j))+d(j))/(2d(j)+n(j)))
2+1/((2(2d(j)+n(j))+d(j))/(2d(j)+n(j)))
2+1/((2(2d(j)+n(j))+d(j))/(2d(j)+n(j)))
2+1/((4d(j)+2n(j)+d(j))/(2d(j)+n(j)))
2+1/((4d(j)+2n(j)+d(j))/(2d(j)+n(j)))
2+1/((4d(j)+2n(j)+d(j))/(2d(j)+n(j)))
2+1/((5d(j)+2n(j))/(2d(j)+n(j)))
2+1/((5d(j)+2n(j))/(2d(j)+n(j)))
2+1/((5d(j)+2n(j))/(2d(j)+n(j)))
2+((2d(j)+n(j))/(5d(j)+2n(j)))
2+(2d(j)+n(j))/(5d(j)+2n(j))
2+(2d(j)+n(j))/(5d(j)+2n(j))
(2(5d(j)+2n(j))+(2d(j)+n(j)))/(5d(j)+2n(j))
(2(5d(j)+2n(j))+(2d(j)+n(j)))/(5d(j)+2n(j))
(2(5d(j)+2n(j))+(2d(j)+n(j)))/(5d(j)+2n(j))
(10d(j)+4n(j)+(2d(j)+n(j)))/(5d(j)+2n(j))
(10d(j)+4n(j)+2d(j)+n(j))/(5d(j)+2n(j))
(10d(j)+4n(j)+2d(j)+n(j))/(5d(j)+2n(j))
(12d(j)+5n(j))/(5d(j)+2n(j))
(12d(j)+5n(j))/(5d(j)+2n(j))
That is to say
n(j+2)/d(j+2) = 2+1/(2+1/(2+n(j)/d(j))) = (12d(j)+5n(j))/(5d(j)+2n(j))
so we get two recurrences
n(j+2) = 12d(j)+5n(j)
d(j+2) = 5d(j)+2n(j)
The first odd convergent C(1) = [2;2] = 2+1/2 = 5/2 = n(1)/d(1) so we have
n(1)=5, d(1)=2
n(j+2) = 12d(j)+5n(j)
d(j+2) = 5d(j)+2n(j)
Returning to the powers of 5+2√6, they take the form a(k)+b(k)√6 where
a(1)=5, b(1)=2
a(k+1) = 12b(k)+5a(k)
b(k+1) = 5b(k)+2a(k)
Observe that the seeds are identical: n(1)=a(1)=5 and d(1)=b(1)=2
and the recurrences are identical, so
the sequence of odd convergents C(1), C(3), C(5), … is identical with
the sequence of ratios a(1)/b(1), a(2)/b(2), a(3)/b(3), … which was to be proved.
With the above result in hand, my earlier post is made rigorous.
PhysicistDave wrote:
Yes, it‘s puzzling. I don’t think I’ve ever seen anything quite like this before.
As slick and effective as the √3-√2 approach is, the continued fraction also appears very naturally and might help reveal what’s behind it. It is a property of the even powers of the original expression, but looking instead at 5+2√6 it applies to every power.
The coefficient-pairs in the powers, and the numerator/denominator pairs in the odd convergents of the CF, are both governed by the same recurrence relations and have the same initial conditions. This means that the ratios a(i)/b(i) converge to √6 which in turn forces the powers (and in particular the expressions b(i)√6) ever-closer to the integers. So my question becomes, why the same recurrences and conditions? I don’t see the answer yet. Maybe there’s a way to relate the odd powers of the original expression (√3+√2) to the even convergents of the CF and thereby complete the correspondence between the powers and the convergents. I’d love to see similar analyses for √3+√5 and √5+√7 et cetera, but who has the time?
Sorry I can’t recommend a reference on continued fractions – it’s been 45 years since I brushed across the subject in my undergraduate number theory class. Of course there’s Khinchin’s classic Continued Fractions, but as I recall that’s rough sledding. Still, it leads to an awareness of Khinchin’s Constant – which I find to be one of the most astonishing results in all mathematics.