The Universe of Discourse


Fri, 28 Aug 2020

Zucchinis and Eggplants

This morning Katara asked me why we call these vegetables “zucchini” and “eggplant” but the British call them “courgette” and “aubergine”.

I have only partial answers, and the more I look, the more complicated they get.

Zucchini

The zucchini is a kind of squash, which means that in Europe it is a post-Columbian import from the Americas.

“Squash” itself is from Narragansett, and is not related to the verb “to squash”. So I speculate that what happened here was:

  • American colonists had some name for the zucchini, perhaps derived from an Narragansett or another Algonquian language, or perhaps just “green squash” or “little gourd” or something like that. A squash is not exactly a gourd, but it's not exactly not a gourd either, and the Europeans seem to have accepted it as a gourd (see below).

  • When the vegetable arrived in France, the French named it courgette, which means “little gourd”. (Courge = “gourd”.) Then the Brits borrowed “courgette” from the French.

  • Sometime much later, the Americans changed the name to “zucchini”, which also means “little gourd”, this time in Italian. (Zucca = “gourd”.)

The Big Dictionary has citations for “zucchini” only back to 1929, and “courgette” to 1931. What was this vegetable called before that? Why did the Americans start calling it “zucchini” instead of whatever they called it before, and why “zucchini” and not “courgette”? If it was brought in by Italian immigrants, one might expect to the word to have appeared earlier; the mass immigration of Italians into the U.S. was over by 1920.

Following up on this thought, I found a mention of it in Cuniberti, J. Lovejoy., Herndon, J. B. (1918). Practical Italian recipes for American kitchens, p. 18: “Zucchini are a kind of small squash for sale in groceries and markets of the Italian neighborhoods of our large cities.” Note that Cuniberti explains what a zucchini is, rather than saying something like “the zucchini is sometimes known as a green summer squash” or whatever, which suggests that she thinks it will not already be familiar to the readers. It looks as though the story is: Colonial Europeans in North America stopped eating the zucchini at some point, and forgot about it, until it was re-introduced in the early 20th century by Italian immigrants.

When did the French start calling it courgette? When did the Italians start calling it zucchini? Is the Italian term a calque of the French, or vice versa? Or neither? And since courge (and gourd) are evidently descended from Latin cucurbita, where did the Italians get zucca?

So many mysteries.

Eggplant

Here I was able to get better answers. Unlike squash, the eggplant is native to Eurasia and has been cultivated in western Asia for thousands of years.

The puzzling name “eggplant” is because the fruit, in some varieties, is round, white, and egg-sized.

closeup of
an eggplant with several of its  round, white, egg-sized  fruits that
do indeed look just like eggs

The term “eggplant” was then adopted for other varieties of the same plant where the fruit is entirely un-egglike.

“Eggplant” in English goes back only to 1767. What was it called before that? Here the OED was more help. It gives this quotation, from 1785:

When this [sc. its fruit] is white, it has the name of Egg-Plant.

I inferred that the preceding text described it under a better-known name, so, thanks to the Wonders of the Internet, I looked up the original source:

Melongena or Mad Apple is also of this genus [solanum]; it is cultivated as a curiosity for the largeness and shape of its fruit; and when this is white, it has the name of Egg Plant; and indeed it then perfectly resembles a hen's egg in size, shape, and colour.

(Jean-Jacques Rosseau, Letters on the Elements of Botany, tr. Thos. Martyn 1785. Page 202. (Wikipedia))

The most common term I've found that was used before “egg-plant” itself is “mad apple”. The OED has cites from the late 1500s that also refer to it as a “rage apple”, which is a calque of French pomme de rage. I don't know how long it was called that in French. I also found “Malum Insanam” in the 1736 Lexicon technicum of John Harris, entry “Bacciferous Plants”.

Melongena was used as a scientific genus name around 1700 and later adopted by Linnaeus in 1753. I can't find any sign that it was used in English colloquial, non-scientific writing. Its etymology is a whirlwind trip across the globe. Here's what the OED says about it:

  • The neo-Latin scientific term is from medieval Latin melongena

  • Latin melongena is from medieval Greek μελιντζάνα (/melintzána/), a variant of Byzantine Greek ματιζάνιον (/matizánion/) probably inspired by the common Greek prefix μελανο- (/melano-/) “dark-colored”. (Akin to “melanin” for example.)

  • Greek ματιζάνιον is from Arabic bāḏinjān (بَاذِنْجَان). (The -ιον suffix is a diminutive.)

  • Arabic bāḏinjān is from Persian bādingān (بادنگان)

  • Persian bādingān is from Sanskrit and Pali vātiṅgaṇa (भण्टाकी)

  • Sanskrit vātiṅgaṇa is from Dravidian (for example, Malayalam is vaḻutana (വഴുതന); the OED says “compare… Tamil vaṟutuṇai”, which I could not verify.)

Wowzers.

Okay, now how do we get to “aubergine”? The list above includes Arabic bāḏinjān, and this, like many Arabic words was borrowed into Spanish, as berengena or alberingena. (The “al-” prefix is Arabic for “the” and is attached to many such borrowings, for example “alcohol” and “alcove”.)

From alberingena it's a short step to French aubergine. The OED entry for aubergine doesn't mention this. It claims that aubergine is from “Spanish alberchigo, alverchiga, ‘an apricocke’”. I think it's clear that the OED blew it here, and I think this must be the first time I've ever been confident enough to say that. Even the OED itself supports me on this: the note at the entry for brinjal says: “cognate with the Spanish alberengena is the French aubergine”. Okay then. (Brinjal, of course, is a contraction of berengena, via Portuguese bringella.)

Sanskrit vātiṅgaṇa is also the ultimate source of modern Hindi baingan, as in baingan bharta.

(Wasn't there a classical Latin word for eggplant? If so, what was it? Didn't the Romans eat eggplant? How do you conquer the world without any eggplants?)

[ Addendum: My search for antedatings of “zucchini” turned up some surprises. For example, I found what seemed to be many mentions in an 1896 history of Sicily. These turned out not to be about zucchini at all, but rather the computer's pathetic attempts at recognizing the word Σικελίαν. ]

[ Addendum 20200831: Another surprise: Google Books and Hathi Trust report that “zucchini” appears in the 1905 Collier Modern Eclectic Dictionary of the English Langauge, but it's an incredible OCR failure for the word “acclamation”. ]

[ Addendum 20200911: A reader, Lydia, sent me a beautiful map showing the evolution of the many words for ‘eggplant’. Check it out. ]


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Mon, 24 Aug 2020

Conyngus in gravé

Ripta Pasay brought to my attention the English cookbook Liber Cure Cocorum, published sometime between 1420 and 1440. The recipes are conveyed as poems:

Conyngus in gravé.

Sethe welle þy conyngus in water clere,
  After, in water colde þou wasshe hom sere,
Take mylke of almondes, lay hit anone
  With myed bred or amydone;
Fors hit with cloves or gode gyngere;
  Boyle hit over þo fyre,
Hew þo conyngus, do hom þer to,
  Seson hit with wyn or sugur þo.

(Original plus translation by Cindy Renfrow)

“Conyngus” is a rabbit; English has the cognate “coney”.

If you have read my article on how to read Middle English you won't have much trouble with this. There are a few obsolete words: sere means “separately”; myed bread is bread crumbs, and amydone is starch.

I translate it (very freely) as follows:

Rabbit in gravy.

Boil well your rabbits in clear water,
  then wash them separately in cold water.
Take almond milk, put it on them
  with grated bread or starch;
stuff them with cloves or good ginger;
  boil them over the fire,
cut them up,
  and season with wine or sugar.

Thanks, Ripta!


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Fri, 21 Aug 2020

Mixed-radix fractions in Bengali

[ Previously, Base-4 fractions in Telugu. ]

I was really not expecting to revisit this topic, but a couple of weeks ago, looking for something else, I happened upon the following curiously-named Unicode characters:

    U+09F4 (e0 a7 b4): BENGALI CURRENCY NUMERATOR ONE [৴]
    U+09F5 (e0 a7 b5): BENGALI CURRENCY NUMERATOR TWO [৵]
    U+09F6 (e0 a7 b6): BENGALI CURRENCY NUMERATOR THREE [৶]
    U+09F7 (e0 a7 b7): BENGALI CURRENCY NUMERATOR FOUR [৷]
    U+09F8 (e0 a7 b8): BENGALI CURRENCY NUMERATOR ONE LESS THAN THE DENOMINATOR [৸]
    U+09F9 (e0 a7 b9): BENGALI CURRENCY DENOMINATOR SIXTEEN [৹]

Oh boy, more base-four fractions! What on earth does “NUMERATOR ONE LESS THAN THE DENOMINATOR” mean and how is it used?

An explanation appears in the Unicode proposal to add the related “ganda” sign:

     U+09FB (e0 a7 bb): BENGALI GANDA MARK [৻]

(Anshuman Pandey, “Proposal to Encode the Ganda Currency Mark for Bengali in the BMP of the UCS”, 2007.)

Pandey explains: prior to decimalization, the Bengali rupee (rupayā) was divided into sixteen ānā. Standard Bengali numerals were used to write rupee amounts, but there was a special notation for ānā. The sign ৹ always appears, and means sixteenths. Then. Prefixed to this is a numerator symbol, which goes ৴, ৵, ৶, ৷ for 1, 2, 3, 4. So for example, 3 ānā is written ৶৹, which means !!\frac3{16}!!.

The larger fractions are made by adding the numerators, grouping by 4's:

 1, 2, 3
৷৴ ৷৵ ৷৶ 4, 5, 6, 7
৷৷৷৷৴ ৷৷৵ ৷৷৶ 8, 9, 10, 11
৸৴ ৸৵ ৸৶ 12, 13, 14, 15

except that three fours (৷৷৷) is too many, and is abbreviated by the intriguing NUMERATOR ONE LESS THAN THE DENOMINATOR sign ৸ when more than 11 ānā are being written.

Historically, the ānā was divided into 20 gaṇḍā; the gaṇḍā amounts are written with standard (Benagli decimal) numerals instead of the special-purpose base-4 numerals just described. The gaṇḍā sign ৻ precedes the numeral, so 4 gaṇḍā (!!\frac15!! ānā) is wrtten as ৻৪. (The ৪ is not an 8, it is a four.)

What if you want to write 17 rupees plus !!9\frac15!! ānā? That is 17 rupees plus 9 ānā plus 4 gaṇḍā. If I am reading this report correctly, you write it this way:

১৭৷৷৴৻৪

This breaks down into three parts as ১৭ ৷৷৴ ৻৪. The ১৭ is a 17, for 17 rupees; the ৷৷৴ means 9 ānā (the denominator ৹ is left implicit) and the ৻৪ means 4 gaṇḍā, as before. There is no separator between the rupees and the ānā. But there doesn't need to be, because different numerals are used! An analogous imaginary system in Latin script would be to write the amount as

17dda¢4

where the ‘17’ means 17 rupees, the ‘dda’ means 4+4+1=9 ānā, and the ¢4 means 4 gaṇḍā. There is no trouble seeing where the ‘17’ ends and the ‘dda’ begins.

Pandey says there was an even smaller unit, the kaṛi. It was worth ¼ of a gaṇḍā and was again written with the special base-4 numerals, but as if the gaṇḍā had been divided into 16. A complete amount might be written with decimal numerals for the rupees, base-4 numerals for the ānā, decimal numerals again for the gaṇḍā, and base-4 numerals again for the kaṛi. No separators are needed, because each section is written symbols that are different from the ones in the adjoining sections.


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Thu, 06 Aug 2020

Recommended reading: Matt Levine’s Money Stuff

Lately my favorite read has been Matt Levine’s Money Stuff articles from Bloomberg News. Bloomberg's web site requires a subscription but you can also get the Money Stuff articles as an occasional email. It arrives at most once per day.

Almost every issue teaches me something interesting I didn't know, and almost every issue makes me laugh.

Example of something interesting: a while back it was all over the news that oil prices were negative. Levine was there to explain what was really going on and why. Some people manage index funds. They are not trying to beat the market, they are trying to match the index. So they buy derivatives that give them the right to buy oil futures contracts at whatever the day's closing price is. But say they already own a bunch of oil contracts. If they can get the close-of-day price to dip, then their buy-at-the-end-of-the-day contracts will all be worth more because the counterparties have contracted to buy at the dip price. How can you get the price to dip by the end of the day? Easy, unload 20% of your contracts at a bizarre low price, to make the value of the other 80% spike… it makes my head swim.

But there are weird second- and third-order effects too. Normally if you invest fifty million dollars in oil futures speculation, there is a worst-case: the price of oil goes to zero and you lose your fifty million dollars. But for these derivative futures, the price could in theory become negative, and for short time in April, it did:

If the ETF’s oil futures go to -$37.63 a barrel, as some futures did recently, the ETF investors lose $20—their entire investment—and, uh, oops? The ETF runs out of money when the futures hit zero; someone else has to come up with the other $37.63 per barrel.

One article I particularly remember discussed the kerfuffle a while back concerning whether Kelly Loeffler improperly traded stocks on classified coronavirus-related intelligence that she received in her capacity as a U.S. senator. I found Levine's argument persuasive:

“I didn’t dump stocks, I am a well-advised rich person, someone else manages my stocks, and they dumped stocks without any input from me” … is a good defense! It’s not insider trading if you don’t trade; if your investment manager sold your stocks without input from you then you’re fine. Of course they could be lying, but in context the defense seems pretty plausible. (Kelly Loeffler, for instance, controversially dumped about 0.6% of her portfolio at around the same time, which sure seems like the sort of thing an investment adviser would do without any input from her? You could call your adviser and say “a disaster is coming, sell everything!,” but calling them to say “a disaster is coming, sell a tiny bit!” seems pointless.)

He contrasted this case with that of Richard Burr, who, unlike Loeffler, remains under investigation. The discussion was factual and informative, unlike what you would get from, say, Twitter, or even Metafilter, where the response was mostly limited to variations on “string them up” and “eat the rich”.

Money Stuff is also very funny. Today’s letter discusses a disclosure filed recently by Nikola Corporation:

More impressive is that Nikola’s revenue for the second quarter was very small, just $36,000. Most impressive, though, is how they earned that revenue:

During the three months ended June 30, 2020 and 2019 the Company recorded solar revenues of $0.03 million and $0.04 million, respectively, for the provision of solar installation services to the Executive Chairman, which are billed on time and materials basis. …

“Solar installation projects are not related to our primary operations and are expected to be discontinued,” says Nikola, but I guess they are doing one last job, specifically installing solar panels at founder and executive chairman Trevor Milton’s house? It is a $13 billion company whose only business so far is doing odd jobs around its founder’s house.

A couple of recent articles that cracked me up discussed clueless day-traders pushing up the price of Hertz stock after Hertz had declared bankruptcy, and how Hertz diffidently attempted to get the SEC to approve a new stock issue to cater to these idiots. (The SEC said no.)

One recurring theme in the newsletter is “Everything is Securities Fraud”. This week, Levine asks:

Is it securities fraud for a public company to pay bribes to public officials in exchange for lucrative public benefits?

Of course you'd expect that the executives would be criminally charged, as they have been. But is there a cause for the company’s shareholders to sue? If you follow the newsletter, you know what the answer will be:

Oh absolutely…

because Everything is Securities Fraud.

Still it is a little weird. Paying bribes to get public benefits is, you might think, the sort of activity that benefits shareholders. Sure they were deceived, and sure the stock price was too high because investors thought the company’s good performance was more legitimate and sustainable than it was, etc., but the shareholders are strange victims. In effect, executives broke the law in order to steal money for the shareholders, and when the shareholders found out they sued? It seems a little ungrateful?

I recommend it.

Levine also has a Twitter account but it is mostly just links to his newsletter articles.

[ Addendum 20200821: Unfortunately, just a few days after I posted this, Matt Levin announced that his newletter would be on hiatus for a few months, as he would be on paternity leave. Sorry! ]


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Wed, 05 Aug 2020

A maybe-interesting number trick?

I'm not sure if this is interesting, trivial, or both. You decide.

Let's divide the numbers from 1 to 30 into the following six groups:

A 1 2 4 8 16
B 3 6 12 17 24
C 5 9 10 18 20
D 7 14 19 25 28
E 11 13 21 22 26
F 15 23 27 29 30

Choose any two rows. Chose a number from each row, and multiply them mod 31. (That is, multiply them, and if the product is 31 or larger, divide it by 31 and keep the remainder.)

Regardless of which two numbers you chose, the result will always be in the same row. For example, any two numbers chosen from rows B and D will multiply to yield a number in row E. If both numbers are chosen from row F, their product will always appear in row A.


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Sun, 02 Aug 2020

Flappers

Gulliver's Travels (1726), Part III, chapter 2:

I observed, here and there, many in the habit of servants, with a blown bladder, fastened like a flail to the end of a stick, which they carried in their hands. In each bladder was a small quantity of dried peas, or little pebbles, as I was afterwards informed. With these bladders, they now and then flapped the mouths and ears of those who stood near them, of which practice I could not then conceive the meaning. It seems the minds of these people are so taken up with intense speculations, that they neither can speak, nor attend to the discourses of others, without being roused by some external action upon the organs of speech and hearing… . This flapper is likewise employed diligently to attend his master in his walks, and upon occasion to give him a soft flap on his eyes; because he is always so wrapped up in cogitation, that he is in manifest danger of falling down every precipice, and bouncing his head against every post; and in the streets, of justling others, or being justled himself into the kennel.

When I first told Katara about this, several years ago, instead of “the minds of these people are so taken up with intense speculations” I said they were obsessed with their phones.

Now the phones themselves have become the flappers:

Y. Tung and K. G. Shin, "Use of Phone Sensors to Enhance Distracted Pedestrians’ Safety," in IEEE Transactions on Mobile Computing, vol. 17, no. 6, pp. 1469–1482, 1 June 2018, doi: 10.1109/TMC.2017.2764909.

Our minds are not even taken up with intense speculations, but with Instagram. Dean Swift would no doubt be disgusted.


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Sat, 01 Aug 2020

How are finite fields constructed?

Here's another recent Math Stack Exchange answer I'm pleased with.

OP asked:

I know this question has been asked many times and there is good information out there which has clarified a lot for me but I still do not understand how the addition and multiplication tables for !!GF(4)!! is constructed?

I've seen [links] but none explicity explain the construction and I'm too new to be told "its an extension of !!GF(2)!!"

The only “reasonable” answer here is “get an undergraduate abstract algebra text and read the chapter on finite fields”. Because come on, you can't expect some random stranger to appear and write up a detailed but short explanation at your exact level of knowledge.

But sometimes Internet Magic Lightning strikes  and that's what you do get! And OP set themselves up to be struck by magic lightning, because you can't get a detailed but short explanation at your exact level of knowledge if you don't provide a detailed but short explanation of your exact level of knowledge — and this person did just that. They understand finite fields of prime order, but not how to construct the extension fields. No problem, I can explain that!

I had special fun writing this answer because I just love constructing extensions of finite fields. (Previously: [1] [2])


For any given !!n!!, there is at most one field with !!n!! elements: only one, if !!n!! is a power of a prime number (!!2, 3, 2^2, 5, 7, 2^3, 3^2, 11, 13, \ldots!!) and none otherwise (!!6, 10, 12, 14\ldots!!). This field with !!n!! elements is written as !!\Bbb F_n!! or as !!GF(n)!!.

Suppose we want to construct !!\Bbb F_n!! where !!n=p^k!!. When !!k=1!!, this is easy-peasy: take the !!n!! elements to be the integers !!0, 1, 2\ldots p-1!!, and the addition and multiplication are done modulo !!n!!.

When !!k>1!! it is more interesting. One possible construction goes like this:

  1. The elements of !!\Bbb F_{p^k}!! are the polynomials $$a_{k-1}x^{k-1} + a_{k-2}x^{k-2} + \ldots + a_1x+a_0$$ where the coefficients !!a_i!! are elements of !!\Bbb F_p!!. That is, the coefficients are just integers in !!{0, 1, \ldots p-1}!!, but with the understanding that the addition and multiplication will be done modulo !!p!!. Note that there are !!p^k!! of these polynomials in total.

  2. Addition of polynomials is done exactly as usual: combine like terms, but remember that the the coefficients are added modulo !!p!! because they are elements of !!\Bbb F_p!!.

  3. Multiplication is more interesting:

    a. Pick an irreducible polynomial !!P!! of degree !!k!!. “Irreducible” means that it does not factor into a product of smaller polynomials. How to actually locate an irreducible polynomial is an interesting question; here we will mostly ignore it.

    b. To multiply two elements, multiply them normally, remembering that the coefficients are in !!\Bbb F_p!!. Divide the product by !!P!! and keep the remainder. Since !!P!! has degree !!k!!, the remainder must have degree at most !!k-1!!, and this is your answer.


Now we will see an example: we will construct !!\Bbb F_{2^2}!!. Here !!k=2!! and !!p=2!!. The elements will be polynomials of degree at most 1, with coefficients in !!\Bbb F_2!!. There are four elements: !!0x+0, 0x+1, 1x+0, !! and !!1x+1!!. As usual we will write these as !!0, 1, x, x+1!!. This will not be misleading.

Addition is straightforward: combine like terms, remembering that !!1+1=0!! because the coefficients are in !!\Bbb F_2!!:

$$\begin{array}{c|cccc} + & 0 & 1 & x & x+1 \\ \hline 0 & 0 & 1 & x & x+1 \\ 1 & 1 & 0 & x+1 & x \\ x & x & x+1 & 0 & 1 \\ x+1 & x+1 & x & 1 & 0 \end{array} $$

The multiplication as always is more interesting. We need to find an irreducible polynomial !!P!!. It so happens that !!P=x^2+x+1!! is the only one that works. (If you didn't know this, you could find out easily: a reducible polynomial of degree 2 factors into two linear factors. So the reducible polynomials are !!x^2, x·(x+1) = x^2+x!!, and !!(x+1)^2 = x^2+2x+1 = x^2+1!!. That leaves only !!x^2+x+1!!.)

To multiply two polynomials, we multiply them normally, then divide by !!x^2+x+1!! and keep the remainder. For example, what is !!(x+1)(x+1)!!? It's !!x^2+2x+1 = x^2 + 1!!. There is a theorem from elementary algebra (the “division theorem”) that we can find a unique quotient !!Q!! and remainder !!R!!, with the degree of !!R!! less than 2, such that !!PQ+R = x^2+1!!. In this case, !!Q=1, R=x!! works. (You should check this.) Since !!R=x!! this is our answer: !!(x+1)(x+1) = x!!.

Let's try !!x·x = x^2!!. We want !!PQ+R = x^2!!, and it happens that !!Q=1, R=x+1!! works. So !!x·x = x+1!!.

I strongly recommend that you calculate the multiplication table yourself. But here it is if you want to check:

$$\begin{array}{c|cccc} · & 0 & 1 & x & x+1 \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & x & x+1 \\ x & 0 & x & x+1 & 1 \\ x+1 & 0 & x+1 & 1 & x \end{array} $$

To calculate the unique field !!\Bbb F_{2^3}!! of order 8, you let the elements be the 8 second-degree polynomials !!0, 1, x, \ldots, x^2+x, x^2+x+1!! and instead of reducing by !!x^2+x+1!!, you reduce by !!x^3+x+1!!. (Not by !!x^3+x^2+x+1!!, because that factors as !!(x^2+1)(x+1)!!.) To calculate the unique field !!\Bbb F_{3^2}!! of order 27, you start with the 27 third-degree polynomials with coefficients in !!{0,1,2}!!, and you reduce by !!x^3+2x+1!! (I think).


The special notation !!\Bbb F_p[x]!! means the ring of all polynomials with coefficients from !!\Bbb F_p!!. !!\langle P \rangle!! means the ring of all multiples of polynomial !!P!!. (A ring is a set with an addition, subtraction, and multiplication defined.)

When we write !!\Bbb F_p[x] / \langle P\rangle!! we are constructing a thing called a “quotient” structure. This is a generalization of the process that turns the ordinary integers !!\Bbb Z!! into the modular-arithmetic integers we have been calling !!\Bbb F_p!!. To construct !!\Bbb F_p!!, we start with !!\Bbb Z!! and then agree that two elements of !!\Bbb Z!! will be considered equivalent if they differ by a multiple of !!p!!.

To get !!\Bbb F_p[x] / \langle P \rangle!! we start with !!\Bbb F_p[x]!!, and then agree that elements of !!\Bbb F_p[x]!! will be considered equivalent if they differ by a multiple of !!P!!. The division theorem guarantees that of all the equivalent polynomials in a class, exactly one of them will have degree less than that of !!P!!, and that is the one we choose as a representative of its class and write into the multiplication table. This is what we are doing when we “divide by !!P!! and keep the remainder”.


A particularly important example of this construction is !!\Bbb R[x] / \langle x^2 + 1\rangle!!. That is, we take the set of polynomials with real coefficients, but we consider two polynomials equivalent if they differ by a multiple of !!x^2 + 1!!. By the division theorem, each polynomial is then equivalent to some first-degree polynomial !!ax+b!!.

Let's multiply $$(ax+b)(cx+d).$$ As usual we obtain $$acx^2 + (ad+bc)x + bd.$$ From this we can subtract !!ac(x^2 + 1)!! to obtain the equivalent first-degree polynomial $$(ad+bc) x + (bd-ac).$$

Now recall that in the complex numbers, !!(b+ai)(d + ci) = (bd-ac) + (ad+bc)i!!. We have just constructed the complex numbers,with the polynomial !!x!! playing the role of !!i!!.


[ Note to self: maybe write a separate article about what makes this a good answer, and how it is structured. ]


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