Ordinals vs Cardinals (and how many algebraic numbers are there?)

[@Dunkle0steus]

"first" comes from "forest" meaning "the most fore" or "the closest to the front". second comes from Latin "secundus" meaning "to come after", relating to "sequential". "third" is such due to metathesis, which is the process by which two phonemes are swapped in a word. It's relatively common in a lot of languages. It's why we have crocodile but many Romance languages have something like "cockadrill". The original "crocodilus" was metathesized into the romance form, and sometime in the 16th or 17th century in England, someone looked up the word in Latin and decided we should all pronounce it the Latin way. Metathesis often enters language in which people may pronounce a word incorrectly. Examples include: "calvary" instead of "cavalry", "perscription" instead of "prescription", "nucular" instead of "nuclear", "comfterble" instead of "comfortable". Sometimes in rhotic accents like in North America, non-existent letters may be added, such as "fermiliar" instead of "familiar". The case where "fifth" is not "fiveth" is an event in which many short vowel sounds became longer. This is the same reason "four" has a 'u' in it and forty does not. Four had a long vowel sound that forty didn't take on. We see the same vowel lengthening in words like "make", "take", etc. which might have sounded closer to "mack(e)" or "tack(e)" originally. Another (but different) example of vowel lengthening is why we have "child" but "children", "long" and "strong", but "length" and "strength".

[@TGC40401]

cool

[@scottgoodson8295]

I don't think "information" and its pronunciation as you've described it here are an example of metathesis

[@Dunkle0steus]

@@scottgoodson8295 I think you're right

[@TheNoHatCat]

The reason we have a difference between five/fifth is because in Middle English f becomes v when between vowels

[@Dunkle0steus]

LadyLillith In Old English, 'f' was pronounced 'v' anywhere but at the front of a word

[@sharkinahat]

There are also programmer ordinals - 0th, 1st, 2nd... UINT_MAX-th

[@oreole9608]

3th?

[@fzigunov]

0th, 1st, 10th, 11st, 100th, 101st...

[@DiamondSane]

So if array has 1 element we call it with index 0. This is kind of mess in numerating. Name single element array as array having 0 elements, redefining 0(cardinal), or finally start to begin with 1 as it should be.

[@OMGclueless]

I think it's better to think of array indices as cardinal numbers: the number of prior elements in the array. Thinking that way explains why 0-based indexing is so useful for computer implementations: It maps directly to addresses in memory (or more abstractly, maps subarray indices to parent indices). i.e. address of element i = address of array start + i * size of an element. Exactly one multiplication by a constant, and one addition operation, with no off-by-one adjustments. That simplicity matters a lot when it is in practice the single most common operation performed by a computer.

[@ainsleyphoenixquinn4534]

Yes, because in programming there's no ordinals, only index number, which basically means memory offset of the number in the array. Offset 0(1st number) means, you don't need to "skip" any number in an array to get the number you want, index 1 (2nd number) means, you need to skip one number and so on. That's why programmers instinctively make m = n-1 when using order number as index. There are programming languages that changes this, but the most used languages define index number as offset of the variable. Source: my academic studies in computer science. To be honest, it shouldn't be called ordinals, because it's not an order, just an offset. UINT_MAX (for our non-programmer friends) means maximum obtainable positive integer number represented by n binary digits. Commonly used integers have 4,8,16,32 bits per number. And because we can only use that much digits, we can't write certain numbers that require more bits. And thus any integer is less or equal UINT_MAX. Mathematicians would use something like 0,1,2,...,N, where N is highest number of a set.

[@EPShouse]

Matt, you forgot to mention the "wholy" trigonometric function: the cardinal sin :)

[@nanamacapagal8342]

Ah, Cardinal Sin. The man who helped and took some part in leading the People Power Revolution of the Philippines, the first bloodless revolution in Asia.

[@Unelith]

- Name at least 3 sins - Sin of wrath, sin of pride, sin of pi/2

[@eliasmazhukin2009]

@@Unelith So 1 is a sin?

[@johnchessant3012]

The same idea was used for proving that the (positive) rationals are countable, if you call the 'height' of m/n as m + n. Also, the fact that almost all numbers are transcendental immediately follows from this video, whereupon it could be mentioned that it is very non-trivial to prove that a given number is transcendental.

[@gnramires]

Do note however that most (real) numbers are also algorithmically incompressible, so they are effectively singularly indescribable (computable). Numbers like e and pi are of course describable, so in a way they're very special as well. There are only countably many computable numbers because (blank tape) Turing machines are countable.

[@liborkundrat185]

10th comment! (I'll always be right with this comment in one base, which is good enough for me.)

[@Gastel]

Clever.

[@JahMusicTube]

Unless you were first!

[@Gastel]

Which was verifiable at the time of writing.

[@JahMusicTube]

Gastel Off course! :)

[@paulfoss5385]

I stumbled upon an ordering of the algebraic numbers once. You take the prime factorization of whole numbers greater than 1 as a power M of the Nth prime, the Nth prime becomes X^(N-1) and its power M becomes a coefficient equal to the Mth number in the list 1, -1, 2, -2, 3, -3, 4... You scoop up any new algebraic numbers that are roots of each polynomial. Mind you this method is terrible in practice, but I like the aesthetics of it. 2: 1=0 😔 3: x=0 Zero is the first algebraic number on the list. 4: -1=0 😭 5: x^2=0 Zero is already on the list 6: x+1=0 Negative one is the second number on the list. 7: x^3=0 Zero is already on the list 8: 2=0 🤬 9: -x=0 Zero is already on the list 10: 1+x^2=0 Plus/minus i added to the list 11: x^4=0 Zero is already on the list 12: x-1=0 One is added to the list 13: x^5=0 14: x^3+1=0 And so (1 +/- i*root(3))/2 are added to the list 15: x^2+x=0 Factors into earlier polynomials 16: -2=0 😭🤬 Whatever, can't be helped 17: x^6=0 18: -x+1=0 One is already on the list Etc.

[@sethspears1630]

In the polynomial definition used at 3:40 and other spots, the second term should have a (n-1) as an exponent.

[@SunroseStudios]

that peek at the ordinal infinities followed by Matt shooing them away gave us a good laugh! funny we only learned about them a few days ago haha

[@miguelash886]

In Spanish happens something similar: uno - primer/primero/primera (1rst); dos - segundo/segunda; tres - tercer/tercero/tercera; cuatro - cuarto/cuarta... etc.

[@tompov227]

First comes from a protoindoeuropean compound of two morphemes, esentially you can think of it as really being "fore-est" as in the most fore or foremost. The word that gives English "fore" is the "fir" part and the suffix "-est" that makes adjectives superlatives is responsible for the "st" in "first" Second, as far as I know just comes from latin but i don't know why it replaced the original English word for 2nd

[@felixswadel5017]

Thomas P It comes from secundus (meaning following, and favourable by extension) an archaic form of sequor, sequi, secutus sum, the Latin verb to follow. Unfortunately I forget which form it was. I think it was either the past participle secutus, or the gerund, sequendum.

[@bluewales73]

Which mean "First and foremost" is redundant.

[@z-beeblebrox]

That's cool! Now we can work out that the exact opposite of "first" isn't "last" but in fact "aftest" Also, apparently before "second" was adopted, the 2nd was called "other". Which kind of makes sense since "another" is a way to order things without counting them.

[@hixeeno6730]

In the Scandinavian languages the word for 2nd is still “other” (andra)

[@magnusbarse]

@@hixeeno6730 Now you have reached me about my own language (Swedish). Thank you!

[@hsmptg]

Portuguese: 1: um - 1st: primeiro 2: dois - 2nd: segundo 3: três - 3rd: terceiro 4: quatro - 4th: quarto 5: cinco - 5th quinto 6: seis - 6th: sexto 7: sete - 7th: sétimo 8: oito - 8th: oitavo 9: nove - 9th nono 10: dez - 10th décimo That is, in Portuguese cardinals and ordinals are also similar except 1 and 2!

[@vitormelomedeiros]

Terceiro is also a bit different, but I'd say the same "amount of different-ness"? Like, it still ends in a "-ro", while the rest ends with "-to", but yeah, weird pattern! E pô, legal ver os BR aqui no canal do Matt hahahah

[@hsmptg]

Torilovem Interwebs I'm afraid I am PT (not BR). Yeah, in Portugal we also use Portuguese! :D

[@vitormelomedeiros]

@@hsmptg Ah sim, eu assumi que fosse brasileiro porque somos bem mais numerosos, mas sempre bom ver qualquer pessoa lusófona independentemente de ser do meu país ou não, hahahah

[@RafaelB.M.]

Torilovem Interwebs hahahahah toma essa invertida! Foi engraçado... Apesar de não ter motivo pra isso eu também assumi que o Hélio fosse brasileiro. Pode-se dizer que demos um Parker Guess sobre a nacionalidade dele.

[@vitormelomedeiros]

@@RafaelB.M. HAHAHAHAH um Parker Guess sobre a nacionalidade dele, SIM

[@macronencer]

You know I've always wondered... if the Pope is 'Number 1', then what's a cardinal's ordinal?

[@TSANOOvlogs]

120th! Keep in mind this exclamation mark is an exclamation mark and has nothing to do with factorials.

[@woodenpotato7550]

The aclaration about the exclamation mark is redundant. If you were trying to use a factorial, you'd use "120!th", causing no confusion at all

[@JayTheYggdrasil]

Snooze123 Who said it was to protect from confusion? I'm sure plenty of people would just bring that up for the sake of boredom and internet superiority.

[@johnchessant3012]

Tomas Noordsij Why not "5!th!"?

[@TSANOOvlogs]

@@johnchessant3012 oooo that's nice

[@Supremebubble]

I think the collest thing about the fact that there are countably many algebraic numbers is that it explains why it is possible that it was known that there are transcendental numbers before anyone had an example of one. Because if there are more real than algebraic numbers then of course there must be non-algebraic real numbers (= transcendental numbers).

[@angelmendez-rivera351]

That doesn't make sense, because it wasn't known that R had larger cardinality than Q or A until centuries after the first transcendental numbers were discovered.

[@Supremebubble]

Angel Mendez-Rivera You are in fact correct, Liouville was the first to prove their existence by a different method. I had jumped to conclusions without knowing the actual history, sorry.

[@OlafDoschke]

As you ask about different languages, in German the similarity already is from 4 onwards (Vier vs Vierter) The suffix only changes with the gender, i.e. there is first as Erster (male), Erste (female) and Erstes (neutral), but that's a separate topic anyway. The typical way to note this is with a point only, not "er" or "te" or "tes" suffix, so the gender plays no role when writing down ordinals, you simply write 1.,2., 3. instead of 1,2,3, the cardinals followed by points are the ordinals, as is usual in numbred lists in English, too. Note: We use the comma as a decimal point, so there is no confusion of 1. with 1,0. Besides, if you wonder, the German word for decimal point is decimal comma.

[@ElchiKing]

Tbh. all except 1 and 3 work the same.

[@OlafDoschke]

Correct, my fault. Zwei-ter/-te/-tes contains Zwei. Only Ers(t).. and Drit(t)... differ from Eins and Drei.

[@quacking.duck.3243]

In Italian we have "primo" and "secondo" for first and second ordinals (compare with "uno" and "due" for 1 and 2). I looked it up and "primo" actually comes from sanskrit "prath-amas" which means "the one before all" (same with Greek "protos"), while "secondo" is a bit more obvious: it comes from the Latin verb "sequi" which means "to follow". An interesting tidbit: in Chinese and Japanese, there exists a whole set of names for ordinals which does not derive from normal numbers. They are called "Heavenly Stems", and they originally indicated days of the week but they have since lost that meaning. An example of modern use is in the military, where they are used to indicate subsequent models of various stuff, for example an aircraft: Ki-84 ko, Ki-84 otsu, etc. Pretty cool if you ask me. en.wikipedia.org/wiki/Heavenly_Stems

[@TaiFerret]

Actually "primo" came from the Latin "primus", which is ultimately related to the Sanskrit word, since both Latin and Sanskrit are Indo-European languages.

[@TaiFerret]

In Japanese there are two different pronunciations for the numbers 1-10 and a few others. Some of these extra forms can also be used in larger numbers. They usually use the Chinese derived pronunciations, like in the names of the months (which are just the number plus the word for moon/month both in the Chinese derived form), but when counting basic objects like apples for instance they use their own native pronunciations, but only up to ten.

[@Anklejbiter]

These are the gems I come across when sorting by new and scrolling

[@ChristianPerrotta]

In Portuguese, Spanish, French, Italian, Latin, English, German, and maybe almost all of the european languages have differences between the names of the numbers in ordinals and cardinals. In Japanese, it's not different at all. You just put a suffix and it transforms into the cardinal.

[@grill-surf-bust]

In Japanese I thought the pronunciation for a number can depend if it's used as a measure (cardinal) vs an ordering. Like hitotsu (one of something) vs ichiban (first of some things) even though they're written with the same kanji 一

[@dr.pjuskebusk2501]

U can add Danish to the list

[@leonardmichlmayr6522]

Grammar knows many more categories of numbers than just ordinals and cardinals (leaving aside other integral numbers in mathematics): there words for iteration, multiplication, proportions. E. g. in latin {semel, bis, ter, quater, ... }, {simplex, duplex, quadruplex, ... }, {simplus, duplus, triplus, quadruplus,...} There are number words referring to the exponent of a direct product: e.g. pair, triple,... In some languages units of measurement are attached to the numbers. In Japanese ten persons are juunin, ten pieces of roughly ball shaped objects are jukko, ten days are touka, and ten minutes are jippun. @GuyG, ordinals are simply derived by appending "me": juuninme is the 10th person. Ichiban is "number one".

[@grill-surf-bust]

Leonard Oh, thanks for correcting me. And sorry if I've mislead anyone. I only have a sense of what's grammatical in one language (American English) so I'm very much an amateur when it comes to expressing, understanding or falsifying linguistic claims.

[@tiikoni8742]

0:30, in Finnish: nolla (0), nollas (0th) yksi (1), ensimmäinen (1st) kaksi (2), toinen (2nd) kolme (3), kolmas (3rd) neljä (4), neljäs (4th) ... So just 1st and 2nd are completely different, other are quite identical.

[@twipameyer1210]

in german: eins (1) erster (1st) zwei (2) zweiter (2nd) drei (3) dritter (3rd) vier (4) vierter (4th) so there are some differences in vowals in "drei - dritter" (and later in "sieben - siebter" (7)) but basically it's only the 1st which is the most common mattern according to WALS: wals.info/feature/53A#2/28.1/144.6 there are even languages which do not differenciate between cardinals and ordinals. I hope even they can enjoy parts of your video.

[@PheliX-Truba]

in latvian: nulle (0), in "proper" latvian there is no such word, but people tend to use: nulltais (0th) viens (1), pirmais (1st) divi (2), otrais (2nd) trīs (3), trešais (3rd) četri (4), ceturtais (4th) so... only 1 and 2 are different

[@gabor6259]

Same in Hungarian. For 1 and 2 it's different, for other numbers it's similar. nulla (0), nulladik(0th) egy (1), első (1st) kettő (2), második (2nd) három (3), harmadik (3rd) négy (4), negyedik (4th) száz (100), századik (100th) ezer (1000), ezredik (1000th) (egy)millió (1 000 000), (egy)milliomodik (1,000,000th)

[@choppergamer]

That omega+1 ordinal peeking from the bottom gave me a vsauce flashback (960th)

[@LasseVågsætherKarlsen]

After the Cardinals got one of their players injured in the second half of the final, the Ordinals took home the victory with a score of 3-2, giving them first place in the tournament.

[@whoscandice8791]

Although this will probably won’t be read, I just wanted to say that I really enjoyed your show today at the theater

[@ObiWanBillKenobi]

Here’s a great question that I’ve never found an answer for: What are the names for series like “primary, secondary, tertiary, quaternary,” “degree, minute, second, ...”, “single, double, triple, quadruple,” “singlet, doublet, triplet,” “only, twin, triplet,” “pair, trinity, ...,” “couple,” “duology, trilogy, quartet, quintet, sextet, ...,” “squared, cubed, quaded, ...,” etc.

[@Blitnock]

Those are (mostly) ordinals and cardinals with restricted contexts. For example, "primary, secondary, tertiary" are words for ordinals restricted to the context of "layers"; "degree, minute, second, ..." are successive units of time or angle measurement going back at least to Babylon or Sumeria-- since units are generally ordered, these can be again be taken as restricted ordinals; "single, double, triple, quadruple" are cardinals indicating the size of a "tuple" and so on. The same holds for "singlet, doublet, triplet", "pair, trinity, ..." and "duology, trilogy, quartet quintet, ..."; these indicate cardinality for restricted contexts. In your last example, I don't think "quaded" as a word. As for "squared and cubed" these are nouns for what results when a quantity is squared or cubed. In mathematics, for higher powers, people say "raised to the fourth power", or "raised to the 3/2 power", or "raised to the 1+i power", etc.. Here the terms squared and cubed are simpler words representing the most common exponentiation operations which are raising to the powers 2 and 3. But exponents are not just restricted to ordinal or cardinals, since 1+i, for example, is neither ordinal nor cardinal, but it is a legitimate exponent.

[@VicvicW]

This weirdly explained a lot of things that those more complex videos couldn’t quite explain to me properly. Like Transcendentals. I just get it now. Why that never made sense to me before, I cannot say.

[@vextorite]

00:30 I'm Glad to say its the same in other languages, IsiZulu is my native language and 1st and 2nd aren't the same as 1 and 2, but the rest flow freely.

[@Scheater5]

It's very similar in Japanese - there are two whole sets of pronunciations for numbers. The one you typically use for counting is also GENERALLY used with ordinals, except for the first two (hito and futa, like "party of one" is hitotsu, where as "ichi" is the typical counting pronunciation). *Been studying Japanese about 3 years, no expert.

[@rwtema]

357th - 357 has a base 3 representation that ends with its base 7 representation.

[@xenontesla122]

I think the most annoying use of ordinal numbers where there should be cardinals is in music. If you play the interval (difference in pitch) of a major 2nd, then add another major 2nd, you get a major third instead of a 4th because the notes start at 1.

[@TheMysteryman32]

Ha, ha, ha. In english there is only two forms ordinals and cardinals. When in polish there is seventeen. Yes, 17. The example: English - Two, Second; Polish - Dwa, Dwie, Dwoje, Dwóch, Dwaj, Dwiema, Dwom, Dwoma, Dwojga, Dwojgu, Dwojgiem, Dwójka, Dwójki, Dwójkę, Dwójką, Dwójce, Dwójko. Take that.

[@AlexBelethe]

What do all these different form even mean?! Is it for different situations or dependable on the particular words in the sentence (like a vs an depending on the first letter in the following word) or what?

[@TheMysteryman32]

Well, to explain as simple as possible. Most of them ar construct because of thre layers of modification. First, by gender (there is three of them - masculine, feminine and neuter). Second, by something that I would translate as gramatical cases. Every noun, numeral and others are changed one of seven, and each could be plural or singular. Finaly there are ordinals and cardinals. To make it easy lets take example. In english: "Hey, you two" is simple way to bring attention of two people. In polish it's little more difficult, because you have take to consider if you calling two male ("Hej, wy dwaj"), two female ("Hej, wy dwie"), or mixed situation ("Hej, wy dwoje"). And this is only one of this gramatical cases. To make it little bit confusing forms sometimes are these same in two difrent condidtions. Hopefuly I helped to understand this confusion.

[@nahco3994]

Slavic languages like to inflect their cardinal numbers according to the object they 'belong' to. The numerals have to agree in gender, can sometimes be plural themselves (!), some are modified in terms of case by any prepositions that apply, and they themselves may dictate number and case of the object they belong to. Not sure about Polish, but in Russian it makes a difference whether you refer to one thing (thing in nominative singular), 2-4 things (thing in genitive singular... yes, it makes no sense), or 5+ things (things in genitive plural). In compound numerals, the last word of the numeral decides (i.e. 21 would be treated the same as 1, because the last word is 'one'). The numerals 1-4 have their own rather irregular inflection tables (there's also a plural form of the numeral 'one', which I always found slightly amusing). Five and onwards follow a generic one. In compound numerals, *every single word* of the numeral has to be inflected (You there in the back row! Inflect 'two thousand eight hundred seventy four' into dative case! Wrong! Next!). Now add adjectives (which also have to be inflected according to the numerals' whims) into the mix and grammatical structures that demand a certain case, and what you're left with is one horrible mess. Oh, and the orthography of the numerals is a literal minefield of course.

[@trequor]

Dear god... and this is why the world doesn't speak Polish hahaha

Just to add what @@TheMysteryman32 didn't explain - the cases. In English sentences, it is enough to use prepositions like "from", "about", etc... So "I went away from two people", "I talked about two people", "I walked with two people", etc. In Slavic languages, the suffix of the word changes based on the "case". Now, I am not Polish but Slovak, so the following are examples in Slovak. "Two" is "dva", "dve", or "dvaja". But "from two" is "od dvoch", "with two" is "s dvomi", "to two" is "dvom", etc... We don't think about it when we talk, it comes naturally.

[@aligator7181]

Hot off the presses : We can pair every positive floating point number using up only about 20% of the integers Algorithm #1 : Convert a float with a zero whole part into an integer 1. Reverse the character sequence representing this float 2. Remove the decimal point to obtain the desired integer Example : Convert the float 0.002743 into its integer equivalent 1. Reverse the float string to obtain 347200.0 2. Remove the decimal point to obtain the integer 3472000 3. Note : All corresponding integers will be terminated by the character “0” Algorithm #2 : Convert a float with a non-zero whole part into an integer 1. Count the number of whole digits or NWD (those preceding the decimal point ) 2. Append a number of “0” digits equal to NWD to the float. 3. Append a digit “1” to the resulting float 4. Remove the decimal point to obtain the desired integer Example : Convert the float 1230.0098 into its integer equivalent 1. Count the number of whole digits : NWD = 4 2. Append NWD = 4 “0” digits to obtain 1230.00980000 3. Append a digit “1” to obtain 1230.009800001 4. Remove the decimal point : this gives us the final integer of 1230009800001 Note : We append a digit “1” to distinguish the integers derived from floats with non-zero whole parts from integers generated by floats with zero whole parts . This can be done for every float [whole.fraction] where whole > 0 !!! AMAZING!!! We just came up with a scheme which pairs each positive float with a positive integer. Reverse Algorithms : Converting valid syntax integers into floats also exist

[@solarmaster3405]

2:00 No, no, stay back Omegas!

[@haniyasu8236]

There's actually yet another way to show that the set of algebraic numbers is countable using prime numbers and the fundamental theorem of arithmetic of all things. First, associate every algebraic number with it's minimal polynomial as in the video. Next, for simplicity, only look at the polynomials with whole number coefficients. (since the integers are the same size as the naturals, we can assume that there are the same number of polynomials with natural coefficients as integer coefficients) Next, we can associate every one of the minimal polynomials to a *unique* natural by associating each degree term with a different prime number and each coefficient with an exponent on that prime. Then, since every natural can be written as a *unique* product of primes, this should give us a unique whole number for every one of the polynomials. For example, if you have 0 = 1 + x^2 + 3x^3, then you would associate x^0 with 2, x^1 with 3, x^3 with 5, and x^4 with 7, getting a final whole number of 2^1 * 3^0 * 5^1 * 7^3 = 3430. And because prime factorizations are unique, every other polynomial will have a completely different whole number associated with it. Thus, the cardinality of the algebraic numbers must be less than or equal to the size of the naturals, and since they are obviously infinite and aleph-null is the smallest cardinal, there must be aleph-null algebraic numbers.

[@roeesi-personal]

Great video, just a tiny correction - Alef is pronunced *A*lef - with the accent on the "a" and not on the "e". As a hebrew speaker it disturbed me a little but apart from this (and from that you didn't start your ordinals from 0 as it should be) everything was good.

[@jonahs92]

חחח, גם אני!

[@magnusbarse]

As James Grime said in a video. I'm English I pronounce other languages wrong.

[@SeLlamanGorillaz]

If a linguistic comment is something you wish! In Japanese there are cardinal and ordinal numbers, but the interesting thing before we get there is that you specify a *counter word* after every number. In English you say "one _sheet_ of paper", "two _rolls_ of tape" etc. In Japanese you have to do this with everything, for example (these go in order of noun-number-counter): ビール三本 - biiru sanbon - three cylindrical-things of beer 毛布二枚 - moufu nimai - two flat-things of blanket ネズミ六匹 - nezumi roppiki - six small-animals of mouse To make an ordinal number, you simply put the ordinal prefix 第 (dai) on the front of the number, or the ordinal suffix 目 (me) after the counter word. The difference between them is pretty nuanced, like the latter is a little less emphatic. There's no other change in pronunciation, it's pretty straightforward.

[@1224chrisng]

n-1th where n is the next comment to be commented

True for everyone.

[@sbares]

My favourite way to show it goes something like this: For each algebraic number r, pick some polynomial P (of degree n, with rational coefficients) that has r as the root with index k. Map each of the coefficients of P to a natural number (using your favourite injection from Q to N) to get a sequence of natural numbers a0, a1, ... an. Then f : A -> N, f(r) = 2^k * 3^a0 * 5^a1 * ... * p(n+1)^an is injective by the fundamental theorem of arithmetic.

[@Tumbolisu]

How have people been doing a flawless job of writing the position of their comments for about a day now, if the video is less than an hour old?

[@ChloeAriT]

Probably early access via Patreon.

[@MiniGirlGeek]

An orderly queue and raffle tickets

[@nivolord]

There are also a countable number of computable reals. This includes real algebraic numbers, but also any other number which is calculable by an algorithm, like pi and e. They are countable, because you can enumerate all algorithms (you can write them in text, and the set of possible texts are finite). Presumably, the same can be said of computable complex numbers.

[@Sam_on_YouTube]

Then Cantor went insane trying to figure out whether or not there is an infinity between aleph 0 and aleph 1. It was later proven that we can't prove their is and we can't prove there isn't. He spent many years trying to answer a question only for it to be proven after his death it had no answer. No wonder he went crazy.

[@HagenvonEitzen]

+Sam There is no infinity cardinality between aleph_0 and aleph_1. However, it is undecidable (on the grounds of the usual axioms) whether 2^aleph_0 (=the cardinality of the continuum) equals aleph_1

[@Sam_on_YouTube]

@@HagenvonEitzen Thanks for the correction.

[@Khazam1992]

00:30 we have the same idea in Arabic Language, For Ordinal, we say: Aual(1st), Thani(2nd), Thalith(3rd), Rabea(4th), Khamis(5th), Sadis(6th), Sabea(7th), Thamin(8th), ... For Cardinal: Wahed(1), Ethnin(2), Thalatha(3), Arbaa(4), Khamsa(5), Sita(6), Saba(7), Thamania(8), ... Though the numbers naming came from/share the same root for Ordinal and Cardinal, the 1st(Aual) and 1(Wahed) came/derived from different roots which is kinda similar to 1st, 2nd, and 3rd compared with 1, 2, and 3 in English.

[@amyshaw893]

0 is an ordinal for programmers. the 0th element of an array

[@Qazqi]

Except it's either "the first element" or "element zero".

[@spacewombat]

Arguably, still an ordinal when calling it "element zero", because it's still indicating its position in an ordered set. If that's not a good enough argument, try the Zeroth Law of Thermodynamics.

[@amyshaw893]

well ive only ever called it the 0th element

[@calrogman]

Correct. The C language specification refers to e.g. v[0] as the first element of v.

[@Qazqi]

And yet every API I've seen that uses a name like this calls it first() or First(). Things get really unclear when you say the first element and mean the second.

[@flikkie72]

This is the first video of yours that contained elements that i couldn't follow. Could have something to do with the celebration of the pretty rare event of all counters of the current datetime changing in one second I did last night.

[@normansmith4166]

Matt Parker. The world most incredible mathematical egg.

[@RomanNumural9]

He's a Parker egg. (Just poking fun though, no insult intended)

[@normansmith4166]

Obviously

[@gurrrn1102]

He’s a mathematical curates egg

[@traktortarik8224]

“First” comes from Proto-Germanic *furistaz, which meant “first” or “foremost”. “One” comes from Old English “ān”, which was used as both the number and as a definite article, much like in French (“un”) or German (“ein”). The word sort of split into two by Modern English. “Second” comes from French “second”, which goes all the way back to the Proto-Italic *sekwontinos, the present active participle of *sekwōr, which meant “to follow”. The explanation is exactly the same for French, and for the German words for 1 and 1st, and probably other languages too.

[@27122712ful]

In Spanish: uno(one),primero(first); dos(two),segundo(second); tres(three),tercero(third)....

[@frmcf]

In Spanish the fun continues into higher numbers too!

[@frmcf]

Well... first, they're not all that easy in English. eighth only has one 't', so you don't add 'th'; ninth loses an 'e', cf. 'nine'; twelve becomes 'twelfth' with 'f'; twenty becomes 'twentieth' with 'ie', following the normal rules for changing 'y' to 'ie' when adding suffixes, same for thiertieth etc; you have to remember 'twenty-first', not 'twenty-oneth' etc. In Spanish: uno -> primero dos -> segundo tres -> tercero cuatro -> cuarto cinco -> quinto seis -> sexto siete -> séptimo ocho -> octavo nueve -> noveno diez -> décimo once -> undécimo doce -> duodécimo trece -> decimotercero (...just when you thought it was getting predictable they went and changed the order) It's really pretty mental, from an English speaker's point of view. EDIT: Full disclosure, I had to look up the ones above eight, because I seldom go above the eighth floor of a building in my daily life, and that's the context that ordinal numbers are most used in. Fortunately, in Spanish you don't need them for dates.

[@frmcf]

I reckon that with the Spanish ordinals it's fairly easy to work 'backwards' from the ordinal to the cadrinal. ie. if you hear 'octavo' you'll recognise it as related to 'ocho' even if you're not used to hearing the ordinals. What's more difficult is when you realise mid-sentence that it's, say, the 14th edition of the festival, and you have no idea if it's 'cuatrodécimo', 'cuartodécimo', 'decimocuarto'. I think it's the last one, but I would honestly have to look it up or ask a native speaker. Spanish also has 'dieciséis', btw. Unlike French 'seize' or Catalan 'setze'. PS. Catalan ordinals are easier! But that's another story for another day.

[@Tomsfilipsons]

But for musicians a "third" is the distance between three and one ("fourth" between four and one, etc). So when a musician, on Tuesday, says that he'll show up for the gig on Wednesday, don't be surprised when he shows up on Thursday. He was merely trying to explain to you that it will take him two days (a "Wednesday") to get there.

[@CharlesGrayScale]

twenty-fifth

0:35 Apropos other languages - in German: Numeral -> Cardinal -> Ordinal 0 -> null -> nullte(r) 1 -> eins -> erste(r) 2 -> zwei -> zweite(r) 3 -> drei -> dritte(r) 4 -> vier -> vierte(r) 5 -> fünf -> fünfte(r) 6 -> sechs -> sechste(r) 7 -> sieben -> siebente(r) / siebte(r) also possible 8 -> acht -> achte(r) 9 -> neun -> neunte(r) Only 1 and 3 are very different. Exept for this you need only the ending "te" (feminine) or "ter" (masculine). For 8, the rule would give achtte(r), but for asthetic reasons, its only achte(r) with one t. For 7 exists two valid variants. I personally use the shorter version "siebter" instead auf "siebenter".

[@koosnaamloos4291]

4:00 shouldn't it be n-1?

[@JDB2552]

I momentarily read this title as Orioles vs. Cardinals, and wondered what baseball math was being discussed.

[@MajorNr01]

As a computer scientist it's obvious that algebraic numbers are countable. Polynomials with integer coefficients are just words over a finite character-set and those words make a countable set.

[@standupmaths]

Yes, it’s obvious when viewed the right way! But there are infinitely many rational coefficients, so that’s not a finite set. And the digits 0 to 9 are a finite set but they can still make the uncountably infinite reals.

[@Sibula]

In Finnish there are two ordinals that differ a lot from the cardinals: "one" is "yksi" and "first" is "ensimmäinen", "two" is "kaksi" and "second" is "toinen".

[@rtpoe]

So when Joseph Ratzinger became Pope Benedict XVI, did he change from a cardinal to an ordinal?

[@piasotski]

n+1-->n-1 on about 3 min 40 sec when you explain to the public the concept of polynomials! Although doing a PhD in theoretical physics and not being needed in down-to-earth explanations of math, really love to watch your stuff. The "how do they really measure the area of countries" is the so-far favorite! Good luck and Glory to Ukraine!)

[@louisetaylor2224]

3rd !

[@PC_Simo]

🇫🇮: 1 = ”Yksi”; 1.(/1st) = ”Ensimmäinen” 🇷🇺: 1 = «Один»; 1st = «Первый» 🇩🇪: 1 = ”Ein”; 1st = ”Erste” Works in at least some other languages ✅.

[@AndersEngerJensen]

We got to the 190th comment before someone broke the streak. Granted we, The Patreons were asked to do this, so... ;)

[@frankharr9466]

I designed a conlang where there are cardinal, ordinal and nominal numbers. Nominal numbers are numbers that are used as the dames of things. Like Room 270. It's the first room on the left as you reach the second floor. There are nine rooms on that floor. I don't think that has anything bearing on the math here, I just thought it was fun.

[@garethwear5651]

I am the first!

[@luxshokk]

Congratulations!

[@harshsrivastava9570]

oh really? neato

[@FredrikLagerstrom]

So you are an ordinal?

[@sharkinahat]

This is probably the first 'first' comment that's actually relevant to the video. Kind of meta really.

[@marceljones7940]

noice

[@chrisg3030]

"They're pretty much the same." So before we ventured off to have fun beyond the finite realm why did we bother to make a distinction? I suggest the clue is in what we call the cardinal points of the compass: north, south, east, west. Their status consists in each being named with one word. Then we get north-west, south-east etc. At this stage the order of the terms in these compounds doesn't seem to matter, though it does make a difference in further subdivsions, such as north-north-east, east-north-east. Similarly cardinal numbers, whether I, V, X, L, D etc or 0,1,2,3,4,5,6,7,8,9, would have simply been those designated with one digit, but the ordinals with more: XV or 534. Again order, hence ordinal, makes a difference at this level. But they're ordinals whether they signify quantity or position.

[@jude1293]

16th!

[@rosepinkskyblue]

In my language, one is “ek” એક but first is “pahelo” પહેલો and it’s very close the word pahel પહેલ, which means to initiate, so in a way pahelo means the one who begins and I think that’s beautiful. I’m from Gujarat in India and the language of course is Gujarati ગુજરાતી

[@TaekiroHeru]

Cent soixante-seizième ! (176th!)

[@derradfahrer5029]

German: (append "te" for female, append "ter" for male, append "tes" for neuter) 1 - Eins erste - 1te, 2 - Zwei zweite - 2te, sometimes (fam.) "zwo" 2 - zwo zwote - 2te 3 - Drei dritte - 3te, 4 - Vier vierte - 4te, 5 - Fünf fünfte - 5te, ... 20 - Zwanzig zwanzigste - 20te (note the additional "s" from now on, called "Binde-S" "binding-S"), but NOT -Zwozig- 21 - Einundzwanzig einundzwanzigste - 21te 100 - (Ein)Hundert (ein)hundertste - 100te 200 - Zweihundert zweihundertste - 200te, or agian but even less common "zwohundert" "zwohundertste"

[@MartiniComedian]

24th

[@PierreVapeur]

In french, something funny about the number 2: cardinal: "deux" ordinal: "deuxième", or "second". The word "second" is rather used if the size of the set is 2.

[@U014B]

2:01 Browsing Facebook after Bowsette became a thing like

[@MrDannyDetail]

This is a good place to ask about one of lifes mysteries, that I ponder on a lot as a genealogist. In the first year of my life I was zero (should my age have not been stated in days or weeks or months, as is common at this point in life), in the second year of my life I was one etc The cardinal and ordinal seemingly out of step. A couple of hundred years ago a gravestone may have said something like 'died in the 65th year of his life' meaning that the person had lived 65 full years, and was now 66 by our method of counting ages. It seems we effectively added 1 to all ages as a collective change at some point.

[@NitronF117]

76th trombones.

[@dammu298]

1036th Note for programmers: 0th is not a thing. 1st array elements is still 1st element in array, you just make reference as foo[0]. 0 in this case is an index which shows the place of element in array. Using index 0 instead of 1 as 1st position in array is just convenient way to handle memory management (if in short).

[@CrimsonEclipse5]

123rd! First three digit series of integers in order.

[@therabbits69]

All these videos recently makes me beyond happy.

[@AlexBate1]

Sixth!

[@Liggliluff]

If I'm not wrong, only 1st and 2nd breaks the trend in Hungarian where you have different words for "first" and "second", but for twenty-first and twenty-second, that's "twenty-oneth" and "twenty-twoth". But I'm not sure how you say 101 and 102.

[@thedolphinbox]

2nd

[@adamdavidson9810]

respectable

[@hovikgasparyan9729]

Second reply to second comment!

[@bookslug2919]

Second reply to the second comment of the second comment!

[@ukko1998]

Second like to the second reply to the second comment of the second comment!

[@Pete-Prolly]

English, "Greek sound," (Greek spelling) 1→first, ena→proto, (ένα→πρώτο) 2→second, dio→deftero, (δύο→δεύτερο) 3→third, tria→trito, (τρία→τρίτο) 4→fourth, tessera→tettarto, (τέσσερα→ τέταρτο),... So, to answer your question, (regarding the Greek language,) only "one" & "first" sound different. The rest begin with same letter and sound. 🎶🎵 1 is the loneliest number...🎵

[@Majoen1998]

ω + 1st

[@robobrain10000]

Ye, this is true in Telugu, a south Indian Language, as well. However, it only happens with one/first. Two and on are similar. One translates to Okatti. First translates to Modhatti. Two translates to Rendu. Second translates to Rendovatti. Three translates to Muudoo. Third translates to Muudovatti. And so on.

[@Jako1987]

69th 😍

[@elvis_mello]

In portuguese it follows the same patern: first is "primeiro", second is "segundo" and third is "terceiro", as one is "um", two is "dois" and three is "três". After these ones, every ordinal number has the same "root" as their corresponding cardinal.

[@TheClaireHazel]

Thirtieth 😊

[@sebastianwesterlund8777]

I like the ordering where you map all the coefficients of the polynomial to powers in the prime factorization - so polynomial 3 is x , polynomial 5 is x^2 and polynomial 15 is x^2+x

[@sebastianwesterlund8777]

@@rosiefay7283 Those are the powers of the prime numbers, so 12 = 2^2 + 3, so that becomes x+2. Basically the prime factorization of a number is just a sequence (a1,a2,a3...) that eventually ends in all 0, which then corresponds to all the powers in the prime factorization - and then you just put those powers as the corresponding coefficients.

[@palmerantony]

Fourth!

[@bookslug2919]

Now you can multiply.

[@quanjano382]

my language teacher always taught me that things that break the pattern are usually the ones that are used the most often, so since 1st-5th are the most used they would be the most unusual

[@NathanRae]

I'm Square Pyramid Number = 140!

[@KyleDavidE]

140! = 1.3462... × 10^241

[@asailijhijr]

*th

[@menachemsalomon]

FYI, I've always heard *א,* _aleph,_ pronounced with the emphasis on the first syllable, so A'-leph rather than a-LEPH'.

[@curiousjames]

10th whooply whoop!

[@krystofdayne]

In German: 1: Eins - 1st: Erster/Erste/Erstes (depending on gender) 2: Zwei - 2nd: Zweiter/Zweite/Zweites 3: Drei - 3rd: Dritter/Dritte/Drittes 4: Vier - 4th: Vierter/Vierte/Viertes ... And then it becomes boring and predictable. So it's sort of special for 1 and 3 but still pretty close. In French: 1: Un/Une - 1st: Premier/Première 2: Deux - 2nd: Deuxième ... And already, after 1, it becomes predictable in French. So those are the languages that I speak, but yeah, something definitely seems to be going on with 1 ;-) which doesn't seem that surprising, in all honesty.

[@cemerson]

Seventh!

[@_WhiteMage]

Economists make use of ordinals when talking about "utility," which you could call satisfaction. Different bundles of goods give different levels of utility, but there is no magnitude information. We might naively assume there's some number of "utils" that can be added together; e.g. if a hot dog gives 70 utils and a bun gives 40, we might assume the two together give 110, but that's not the case. We get more satisfaction from this bundle than from its items separately. Some bundles we get _less_ enjoyment from than their constituent parts; I like chips and soda, but not when they're mixed together. Putting it very simply, when you spend $5, you're indicating you get more utility from what you bought than with any other combination of things you could spend $5 on. You can make all sorts of multi-dimensional graphs and heat maps out of these, with "indifference curves" representing a specific elevation on a mountain, you could say--different combinations that all give equal utility.

[@stevescott9289]

11th

[@scanerang]

The reason the first few ordinals don't follow the rest is because they are used more often. It's like with the most common verbs, they "mutate" more quickly to fit what is prefered most

[@tomp3927]

Seventeenth!

[@andrewmirror4611]

In Russian there are special ordinals for the first, the second, and the hundredth. Don't ask. In Chinese there is no ordinals (they use prefix 第 to define number as an ordinal) but the number 2 as a cardinal is 两, for everything else they use the primal word 二.

[@gurrrn1102]

There’s a special cardinal for 40 in Russian too. PLEASE don’t ask.

[@atrumluminarium]

[10n+1]'st

[@gvigary1]

When I was a freelance system designer, I worked on a financial package called Infinity. So my firm, which was as small as could be (just me), simply had to be named Aleph Null 😎

[@jonniiiii]

Nineteenth!

[@energyーy]

In German, 1 = "eins" and 1te = "erste" (te is the replacement for st in German). 2 (zwei) is how you would expect, but 3 (drei) turns to "dritte." Same reasons for as in English, without the Latin loan word.

[@Kashados]

840th

[@stevethecatcouch6532]

Matt stopped counting before reaching the first irrational algebraic number. If I've counted right, then -sqrt(2), a root of x^2-2 = 0, is the 11th algebraic number and the first irrational one.

[@TheTalkieToaster]

Fifteenth!