Expanding the Roman Numerals

[@kephrekhtheunbroken7510]

Fun fact: sometimes, the romans just added 4 together, so stuff like 14 was written as XIIII instead of XIV, but not always. If you bend this a little you could go slightly higher.

[@JayTemple]

Somewhere in the past few years, I learned that the IV (and IX, etc.) notation didn't come about until the 19th century or so.

[@TheDankBoi69]

Well I think that explains why the 4 in clocks with Roman numerals is written as IIII instead of IV 22/11/22 EDIT: yeah but most clocks STILL display 4 as IIII even if they're made past the 19th century and in my opinion I find this annoying

[@mathguy37]

IIII was wondering if they were going to mention that

[@GetRidOfHandles]

I II III IIII IIIII IIIIIIIIII

[@PopeLando]

@@JayTemple No, although small numbers like 4, 9 and 14 used IIII, the Romans did use the subtractive system, eg XXIX for 29 days in January, April, June etc on their calendars. Edit: I recently saw an old but post-medieval example, the ceiling of St Peter's in the Vatican, which bears the date MDXC for 1590. So it was ancient and also it was used before the 1800s.

[@psimaster]

This felt like a downward spiral and I love it

[@haipingcao2212_.]

xv

[@XorbityXorbGlowbe]

@@haipingcao2212_. indeed, 15

[@liam.28]

welcome to googology

[@tylorcorslan1134]

This video was a trip

[@alphachiu]

is vx just v?

[@IONATVS]

While it is normal nowadays to hear “you can only string a max of 3 Ms, Cs, Xs, or Is together” when discussing Roman Numerals, this is a modern *convention* that exists purely to give each number a “canonical” form like Hindu-Arabic numerals do. When they were actually used regularly in daily life, no such convention existed, IIII and IV were just as valid ways to “spell” the number 4, and IM was just as valid as CMXCIX, the current “standard” way to spell 999 in Roman Numerals. Thus, even sticking purely to the seven universally accepted characters, you can *technically* write any positive whole number, no matter how large, by simply using Ms as tally marks. It’s a brute-force method, but would be allowed. Of course, real people needed to use Roman numerals for real things back in the day, and even back then they had strategies to extend the numbers in a more…useable way for larger whole numbers and simple fractions. The most common being adding S for 1/2 and • for 1/12 fractions (which covers the most common fractions people use in daily life, though 1/5 and 1/7 still had to use another strategy, usually stating the fraction as a ratio) and the two different strategies for extending the system upwards to easier to write multiple thousands (and no, you’re not the first person to come up with using multiple lines to extend it to millions and billions, just by the time regular people were working with numbers above a few million, Hindu-Arabic numerals had firmly replaced Roman ones in most fields, so it was a nonissue. And fields that had used those kinds of numbers for a long time (government accounting, grain shipments, and military inventories being the main ones), they just worked in accounting units that were large in-and-of-themselves to avoid using such large numbers (eg “V Legions of MMMMMM men each” instead of “30000 men” or “M pounds of sterling silver” instead of “240,000 pence,” where pence were a currency seen in daily life, but pounds sterling, while eventually being debased to the point the modern British Pound is comparable to a dollar, was originally worth exactly what it says on the tin, a pound of silver, which would be over $300 today, and basically only existed in noble account books as a way to deal with large numbers).

[@cycrothelargeplanet]

​@Bogus2023coolBecause that's not how roman numerals work?

[@SUPABROS]

​@@cycrothelargeplanetwell it would be much more practical, why is CMXCIX the standard for 999 when you can do IM? Or XXXXII for 42 would be much more easily read than XLII

[@cycrothelargeplanet]

@@SUPABROS I was replying to his first reply

[@IONATVS]

@flipaclip401lover That would be a standard way to spell 9000 in the version of extended Roman Numerals that uses overbars for x1000, yes.

[@IONATVS]

@flipaclip401lover it’s a less common extension of Roman Numerals, but yes, you could spell 9,000,000 that way.

[@stephenweigel]

Another application of Roman numerals is labeling chords in western music theory! Capital letters are major, while lower case are minor, and each chord has functional significance.

[@wyattstevens8574]

Lc could be diminished too! (vii in major, ii in minor)

[@tripleoof8159]

I absolutely love your videos, it's always so interesting! From Vibri to anything math. Keep it up!

[@brinleyhamer729]

X|X|X = 10*10^10,000,000,000 THAS INSANE

[@felipevasconcelos6736]

8:57 That looks nothing like fractions. In Roman numerals, fractions are represented much like integers, with their own symbols, though only additively without the 3-symbol limit. The symbol for a twelfth is · (a dot) and for half its S. These are the main ones, but there’s also Σ or Є for half a twelfth, and a few obscure ones that don’t even show up on my device. I don’t think most Romans were very concerned with small or precise fractions, as the other fractions appear to be only used by apothecaries. Edit: timestamp

[@Swagpion]

8:54

[@felipevasconcelos6736]

@@Swagpion thanks!

[@windowsmercurySP13]

4 quinagintillion

[@jan_Eten]

Σ is 1/24, Ƨ(backwards s) is 1/72, Ƨ wiþ a line is 1/144, Ɔ(backwards c) is 1/48, and Э is 1/288

[@tookiecar1]

@@jan_Eten WHY DID YOU INCLUDE THE THORN😭😭😭

[@Xcyiterr]

I am absolutely smitten with both the adorably unique animation style and the sheer aura of smugness that this video emanates instant sub love the glace

[@KarlsGB]

I like how he says that he can't stack 50 lines on top of each other because vertical space and then he proceeds to stack 50 fractions on top of each other

[@theamazinghippopotomonstro9942]

😂😂

[@LaurencePaulUy]

⁠

[@PepeChadIsDaGoat]

At least bro made that number more compact, sticking the lines would be way longer

[@cheeseburgermonkey7104]

The art you've created is super cute and cool, also this was a really good video for a first impression on me. You've gained another subscriber!

[@Ed1414One]

The Romans actually came up with a system for fraction and it did not look like arabic numeral fractions but instead s for 1/2 or 6/12 and a dot was 1/12

[@atanvardecunambiel8917]

To a lesser extent, T was used for 4/12 (a third) and Q for 3/12 (a quarter)

[@mathismt1222]

@@atanvardecunambiel8917 i didnt know that i only knew the dots

[@SuperWindows78]

@@atanvardecunambiel8917 ah maybe that’s how quarter and third got their name

[@tfan2222]

@@SuperWindows78 Third is an English word, not romance. So, no. Quarter *is* however, and is synonymous with fourth.

[@НиколайАтанасов-г4м]

. : Q T .S? S S. S: SQ ST .I? I

[@amojc3573]

If I remember correctly, there was a QI episode which showed that several units of (I)(I)(I)(I)(I)... chained together and written on a gravestone represented several million victims of a war.

[@Moralsiz]

Your videos are always interesting and entertaining

[@slushiiwoman]

Yo? Okay, I'm going to be completely real with you here, I am probably the biggest nerd when it comes to large numbers, even if I don't really delve to much into the true abyss of large numbers. And, this idea is just, simply beautiful. I love every single bit about it. If we could, d'you mind if we could have a chat, and maybe extend this entire system? I already know of a couple ways this system could be made even better, and to- Well, make it easier to write out. If we could, my robotic heart would be more than happy. Anyways, lovely video my dude, I hope to see more!

[@-minushyphen1two379]

How about extending it to the rationals with continued fractions? I just don’t know how negative numbers would be represented without just using the convenient modern -.

[@slushiiwoman]

@@-minushyphen1two379 Actually, thinking about rational numbers, I feel as if it would be a cool idea if scientific notation was included when representing them. As, since the idea of "Power Towers" was indirectly alluded to in the video (by the idea of having a smaller number being on top of a line to represent how many lines should be topped to the larger number on the bottom), it would only make sense to continue that notion for the sake of consistency (and elagance). How I'd imagine it would work is, Say you want to represent the number 133.7 as a roman numeral. Then, you'd represent that as, NI | I | CXXXIII | DCC | N in this case would be used to denote that you are meant to divide whatever is below the line (or in this case, in-between the lines) by 1,000 Sure, it's a little messy. However, that's because I couldn't be bothered actually having lines with numbers on top of them. But, you get my point. Still, it's a little clunky. However, as of right now, I can't really think of anything else to substitute with. So, this'll do for now. Anyways, thanks for responding to my comment! I do enjoy nerding about this kind of stuff. And ye, baiiii-

[@superlevigaming8521]

What if you go even bigger than hyper-extended roman numerals, using lines UNDER the number? For example, M with a line under it is a giant tower of a thousand lines with M's over them.

[@Unofficial2048tiles]

@@superlevigaming8521 maybe like um II _ = X _ _ X X

[@superlevigaming8521]

@@Unofficial2048tiles I was thinking something more like this: X _ X _ X _ X = X _ IV

[@cjrm15macpherson20]

bro invented tetration using roman numerals

[@zdohvívêkgüptäçøūčh2048]

there were some symbols for 1000, 5000, 10000, 50000, 100000. there was apostrophus, If you were to write 10 thousand, you'd simply write two sets of two C's, one being mirrored, with an I in between them. Each additional set of C's raises the value by a factor of 10. Though no one ever really wrote anything above 100,000. It was also possible to write numbers such as 5000 by using I and two backwards C's. 500 was just one backwards C which is most likely where we get the modern D in standard Roman numerals. There were also some variations of this which linked the C's.

[@WilliamWizer]

OBJECTION!!! that tower at the end has the same problem as having multiple lines. it requires you to write vertically. best way could be to use conway chained arrow notation. or some sort of variation. that allows you to reach numbers so big you can't call them big.

[@FarzanaFathima-t4e]

Or use [_] for vinculum and [_]_ for Roman Expansion.

[@NicoPlayZ9002]

@FarzanaFathima-t4ewhat does that mean?

[@ħydrogen]

Another way to fix this is to add 2 lines on top instead of one, and like the extension in the video, it has a smaller number on top of it. The smaller number represents the amount of X's with lines on top in the power tower. You can stack the 2 lines, then 3 lines, 4, 5, and so on. You still run into the same problem though, and I'm still trying to think of a solution for it.

[@the_vine_queen]

This is honestly a really cool concept! It's reminiscent of the exponent system in Arabic numerals, but it actually takes us further since each line is one thousand instead of just ten. Anyways, I really like this, and I am tempted to turn in actual homework using entirely extended roman numerals.

[@anwaruwid]

You need a century to make sure how to count in roman numerals 😂 I like ours (Arabic) numerals

[@HexStudios2763]

Now, we need fractions!

[@azadanzans5359]

This is amazing. Roman numerals are so impractical, but I kinda of loved the chaotic nature of them, but this just takes it to a new level. Great video, I'd love to even extend it further, which should be easily possible.

[@SingGuang742]

Interesting idea, I also have the same thought as you while I’m looking for number that exceed 4e+06 (4 million) in Roman numerals. Since we are writing bunch of bars, I’ll shorten down the bars count into exponential places, such as: V^2 = V with 2 bars = 5 million X^3 = X with 3 bars = 10 billion And so on

[@diaryofanasdkid9752]

Theoretically you could just add more lines forever

[@vii-ka]

The recursive stacking of number bar number bar number etc brings up the same problem mentioned earlier in the video: lack of vertical space. Is there some way to solve this, too?

[@myrjavi]

simple naive solution: bracket of another roman numeral on the side to denote how many expansions? X [ IV [ __ IV it would repeat IV_IV_IV... X times...

[@TheDoubleTea]

Maybe, to show repetition of one bar stacks, we could have something like IV - - X which equals X - X - X - X. And then we could have three bars being two bars repetitions and four bars being three bars repetitions and so on and so on. Eventually that would bring the same problem, but right now there is no practical use for numbers so big, so it doesn’t really matter. But I’ll continue anyways. To denote the number of horizontal bars, we could have vertical bars, so X | X Would be two tens with ten bars between. Do you know how vertical bars control the amount of horizontal bars? Well, after classifying horizontal bars as (1) and vertical bars as (2), we can make (x) control the amount of (x-1). This basically means from here on out, we can make brackets inside brackets, and make a different bracket to control that, like (4(4(4(4)4)4)4) can be {4}. Then we can have [] controlling {} and then to control the “level” of brackets, we can have more numbers, like X(X)X being ten copies of IX(X)IX inside itself, along with other numbers to make the recursive iterations actual numbers. At this point, we are waaaaaaayyyyy past Graham’s Number and way past numbers used on any basis, so I’ll leave the rest of the notations for googologists to solve. (Googologists are people who study large numbers. Not large numbers as in a million, large numbers that are bigger than Googol, many many bigger than Graham’s Number.)

[@Lemony123]

Fraction had the same problem, but it not even feel like a problem.

[@sponge1234ify]

Basically the same "solution" as fractions: If you need larger than one stack, vertical space should be around the least of your problems, so we ignore it. But bracketing could also work, better than the division ( ~:~ ) symbol, at least.

[@superlevigaming8521]

@@TheDoubleTea "...there is no practical use for numbers so big, so it doesn’t really matter." Well actually, a number like XVI _ IV is 4*1000^16, which grows similar to scientific notation. This means that the limit of hyper-expanded roman numerals only grows as fast as tetration, which is nowhere near the magnitude of Graham's number, which is actually a very important number.

[@stardy8131]

that intro is god damn awesome, and the rest of the video was very good too

[@soapycanthandle]

I had an idea similar to this. Start with X, X line, X line line, X line line line, then what's next? X. LINE. >. that's right, we're going meta. 😎 you can take that sideways number and put more sideways numbers on top of that, until you've gone all the way around the circle 360°, and eventually you get this crazy quadruple-X throwing star lookin' thing.

[@lawrencfgsdfg]

sick video, and criminally underrated channel!

[@katie-ampersand]

"yea i'm going there" i love that you know that you are insane and need help. this is my favorite video

[@Cessated]

if extended further could make a decent googological notation

[@RudyHHOfficial]

n - - n will be n - n ... - n with n ns

[@Cessated]

@@RudyHHOfficial I was more thinking n|n instead of the notation you use, but still good, and same definition

[@RudyHHOfficial]

@@Cessated k i wil use n|n

[@Memeened]

I did EHERNN

[@PilotMiaoumi]

This is so weird and i love it

[@Thegoodgamerpro72jdq]

bruh this is the best crossover in history

[@ND62511]

On the topic of numbers in different languages: I’m currently studying Japanese, and while more common to just use the Arabic numeral symbols, there is a set of symbols corresponding to numbers in Japanese. 一 = 1 ニ = 2 三 = 3 四 = 4 五 = 5 六 = 6 七 = 7 八 = 8 九 = 9 十 = 10 百 = 100 千 = 1,000 一万 = 10,000 Now, 一万 is interesting because it has the 一 (1) symbol in it; that’s because the 万 symbol is odd as it doesn’t _really_ represent 10,000, but rather more of a vague idea of multiplying something by 10,000. In order to get numbers outside these, you just arrange the symbols of the digits next to the appropriate power of 10. So… 五千百二 = 5,102 四百七十一 = 471 九千一 = 9,001 Very interesting to see how other cultures’ number systems work. Some are quite similar to what’s most common while others are quite disconnected.

[@EHMM]

That's chinese..

[@aofrog]

@@EHMM Not exactly, it can be either one.

[@haru-bun]

@@EHMM you see, in chinese, 九千一 means 9,100

[@atanvardecunambiel8917]

@@EHMM Kanji are just hanzi the Japanese yoinked. There are hanzi/kanji beyond wàn/man, each one a myriad times the previous one.

[@alexwang982]

@@atanvardecunambiel8917 亿 and 兆

[@_Guigui]

we have successfully avoided the year 4K bug, as well as the 4M, 4B, and so on

[@pr0hobo]

this is very similar to certain versions of myriad notation in Greek numerals.

[@Warsaquoack]

You are going to become big one day, I just know it

[@pallavij520]

8:58 U Suni!Thats what a fraction looks like

[@AstaryuuGaming]

Just saying, numerals beyond M probably had existed, since units over 1,000 were used somewhat regularly in Roman society, for example military divisions. It's just likely that the common man wouldn't need such high numerals regularly, given that modern uses of big numbers either didn't exist back then, or existed but only a small fraction of Romans would need them, so they were never standardized, hence why there aren't any in the "modern" Roman system, since they're based on whatever got standardized. Also, IX is more common than IV (IV often got represented as IIII), so if we take that into consideration even vanilla Roman numerals can reach a little higher (MMMMCMXCIX, 4,999) without breaking the system. But, with this hyperextended variant, M is only ever needed to be subtracted from or to end a number anyway, so not a huge deal.

[@amitakler4710]

Im so glad i was on youtube in 2:48 am. on the 25.05.2023 ,Thursday Amidiatly subbed

[@deyanatodorova8459]

One small problem. At 5:34 it's notated MV̄ and not ĪV̄. That's because when you think about it, every "4" is just "5 - 1". IV is 4 (5 - 1), XL is 40 (50 - 10), CD is 400 (500 - 100) and MV̄ is 4000 (5000 - 1000), and so on. If you decide to notate it as ĪV̄, that's completely fine. It's just your choice and I can't control it.

[@senorqupal4701]

If the classical Romans stuck around for 2000 years more they might have discovered tetration, which exactly what the video is all about.

[@YeahEsCereal]

I like how we can understand even tho even tho it’s some lines

[@madisonm1310]

Or a googol could just be X^C. My idea was to use unused letters. So A is 5,000, B is 10,000, E is 50,000, etc. But there would still be a limit at IZZZZ which would be 4,999,999,999,999 if I did my math right.

[@TheDankBoi69]

If there is a number "n" above the number "a", then the formula is a•10³ⁿ. Easy.

[@chanyy6838]

11:11 one hundred and sixty four 31414-illion

[@ElFrighet]

At the point 6:22, the 2 lines or the *1000000, can be also represented as the letter but it has a line on top and 2 vertical lines at sides, like a draw of a house without the roof. Anyways really interesting

[@adsbegon8405]

You just unlocked... The Scientific Notation!!!

[@adsbegon8405]

Also, instead of using the horizontal fraction you could use the diagonal fraction ( / ) to signify if its a fraction. And you could use a half line or a line with another small perpendicular line in it to signify the power of a different number such as 4 or 3.

[@LexiLex421]

Ok, I get that you want to extend it, but, what if, we added something like a à,ã,å,á,â,ā, or ä

[@M2000-hi]

4:47 for perspective, Candy Crush has over 20,000 levels

[@caelu.m]

you reinvented the scientific notation for the roman numerals

[@MrLajko]

he inveted roman numeral powers

[@Dolph1nVR]

11:15 3.1415 are the digits of piπ 31415927 (the 5 would be 6 though because of averaging)

[@xE3x]

Exponential Towers... hmm. You could very much well expand this further making something similar to Knuth's up-arrow notation.

[@1e1001]

roman numeral scientific notation, now just add a way to do negative numbers (i imagine either a "-" prefix or a "zero" digit to subtract from (so M0IX = -991, or something)) and you can represent any decimal number

[@syndere6755]

Might I suggest changing things slightly, by making the upper numerals be on a separate plain attached with an underline to an overline. That way, and stacking of numerals can be moved to horizontal space or vertical space

[@animaniacsfan2]

Near the end, you rediscovered "hereditary base notation" for base 1000. Hereditary base-n notation is where you write a number m = a_k*n^k + a_{k-1}*n^(k-1) + ... + a_0*n^0 (just like you normally would in base-n), remove the 0 coefficients, and repeat the same process on the exponents, recursively, until all exponents become 0. In your case, the Roman numeral under each bar is a value of a_k, and the exponent k is above the bar. In fact, hereditary base-n notation is related to Goodstein sequences, which are mathematical sequences whose length grows *way* faster than exponential, even faster than tetrational or other hyper-operators. In fact, Goodstein sequences grow so fast that the "standard" axioms of arithmetic can't prove that the process to generate them always works; you need stronger axioms. en.wikipedia.org/wiki/Goodstein%27s_theorem#Hereditary_base-n_notation

[@musica00-7z]

Oh, you mean how 1337 can be written as 2^(2^(2^1 + 1) + 2^1) + 2^(2^(2^1 + 1)) + 2^(2^(2^1) + 1) + 2^(2^(2^1)) + 2^(2^1 + 1) + 1 Or, alternatively, if "f x" represents 2^x, 1337 takes the form: f (f (f 1 + 1) + f 1) + ff (f 1 + 1) + f (ff 1 + 1) + fff 1 + f (f 1 + 1) + 1

[@animaniacsfan2]

@@musica00-7z Exactly. That's what this video is doing except for base 1000, where the number k over a bar represents multiplication by 1000^k.

[@lythd]

i like how it ends up mapping into decimal because its based on thousands, neat

[@ZoroniteTech]

I coincidentally invented the same system not too long ago. I did however extend it further Having 2 lines with a numeral above it signifies that there is a "numeral-line-numeral" stack that many numerals high. this can be done with 3 lines where it is a stack of "numeral - 2 lines - numeral" and so on and then you get to having too many lines again

[@memerboi69.0]

just make the number of lines another numeral

[@SHIN2025_official]

8:57 "YoU lOoNeY! tHaT iS wHaT a FrAcTiOn LoOkS lIkE!"

[@BryndanMeyerholtTheRealDeal]

There was proto writing, like the hieroglyphs and other ancient scripts, long ago.

[@randomtuba7079]

As a large number enthusiast myself, I tried to extend the Roman Numeral system a couple years ago as well, and got even larger results. The notation begins by lazily creating new Roman Numerals after M: N = 5,000 O = 10,000 P = 50,000 Q = 100,000 R = 500,000 S = 1,000,000 Using these new numerals, you can create numbers like 1,412,421 (SQROMMCDXXI). The current notation allows you to go to 3,999,999 (SSSQSOQMOCMXCIX). However, this is inefficient, since there are a limited amount of letters in the alphabet. So, I decided to travel a different route of extension. Consider S as 1,000,000, like before. Then, create another numeral in the sequence, IS, equaling 5,000,000. Consider IIS as 10,000,000, then IIIS as 50,000,000, then IVS as 100,000,000, and so on. These post-S numerals act like additional letters being created. For example, SIIS would equal 9,000,000, because it represents IIS (10,000,000) minus S (1,000,000). This notation has a limit numeral of QSOQMOCMXCIXS (the 999,999th post-S numeral), since the 1,000,000th post-S numeral would be SS, which is already declared to be 2,000,000. To avoid this, the numeral T is invented, equaling the 1,000,000th post-S numeral, which is 10^500,006. The next logical step is to do the same thing for T, making IT 5*10^500,006, IIT 10^500,007, and so on. The 1,000,000th post-T numeral can be expressed as ST (10^1,000,006). The limit of this sequence would be the 10^500,006th post-T numeral, coined as U (around 10^(5*10^500,005)), since TT is already defined. You can see where this is going. W ≈ 10^10^10^500,006, Y (skipping X) ≈ 10^10^10^10^500,006, Z ≈ 10^10^10^10^10^500,006, and so on, until ultimately reaching K, equaling about 10↑↑15, and we have now run out of letters. To combat this, we invent a bracket notation, where the number inside represents a unique numeral. So, [I] = I, [II] = V, [III] = X, [IV] = L, [V] = C, [VI] = D, [VII] = M, etc. How is this different from our S notation? Well, [XIII] = S, [XIV] = T, [XV] = U, and so on, with [XXVI] = K, our original last numeral. The brackets kind of represent a tetrational operator, and equal the growth rate of Hyper-Extended Roman Numeral Notation. However, we are not stopping here, because we can have [L] (~10↑↑39), [M] (~10↑↑989), [S] (~10↑↑999,989), [K] (~10↑↑10↑↑15), and even [[S]] (~10↑↑10↑↑999,989). By repeatedly nesting brackets, such as in [[[[S]]]] (~10↑↑10↑↑10↑↑10↑↑999,989), we can reach a pentational growth rate. With this breakthrough, we can transcend over bracket nesting by creating a roman numeral array notation. Yes, you heard that right. Our first entry in the array can represent the regular bracket entry, and the second entry in the array can represent the amount of bracket pairs. So, [S,S] would equal [[[...[[[S]]]...]]] with 1,000,000 bracket pairs, equaling about 10↑↑↑1,000,001! Our third entry in the entry is a bit more complicated. With our two-argument array, we can currently reach a limit of [K,[K,[K,[K,[...[K,K]...]]]...]]], where we have a lot of nesting in the second entry. So, let's make the third entry determine the number of nests in the second entry! We can even make the fourth entry determine the number of nests in the third entry, and so on. Now that we have the foundation for our array notation, let's define the array process. ROMAN NUMERAL ARRAY NOTATION: I. (Tailing Rule) Remove all tailing Is. (Ex: [S,S,I,X,I,I] = [S,S,I,X]) II. (Simplifying Rule) If the array has two entries, replace [a,b] with [[...[[a]]...]] w/ b bracket pairs. (Ex: [S,V] becomes [[[[[S]]]]]) III. (Catastrophic Rule) Set the second-to-last entry to the current array with the last entry - 1, and remove the last entry. (Ex: [S,S,I,X] becomes [S,S,[S,S,I,IX]]) That's it! Now we have a fully working array notation that reaches a limit of f_ω in the Fast-Growing Hierarchy, which means that its limit dominates all primitive-recursive functions. For comparison, Hyper-Extended Roman Numeral Notation reaches a limit of f_3 in the Fast-Growing Hierarchy. To better understand the array notation, let's use an example of [X,II,III,I]: [X,II,III] (Rule 1 removes all tailing Is) [X,[X,II,II]] (Rule 3 replaces second-to-last entry with the current array with the last entry - 1, the last entry is removed) [X,[X,[X,II,I]]] (Rule 3 again) [X,[X,[X,II]]] (Rule 1) [X,[X,[[X]]]] (Rule 2 simplifies [X,II] to [[X]]) [X,[[...[[X]]...]]] (Rule 2 simplifies [X,[[X]]] to [[...[[X]]...]] with [[X]] bracket pairs) The array then decomposes to a numeral with an even larger amount of bracket pairs! And to think that this is only with 3 entries. Imagine what 4 entries, 5 entries, even 100 entries would output! While this array notation is impressive, the notation as a whole is much sloppier than Hyper-Extended Roman Numeral Notation, and there are lots of other non-roman numeral notations that go far far beyond this point. However, for a starting point of 3,999, this extension is somewhat impressive. I hope you enjoyed reading! I might create a follow-up extension in the future.

[@idonothavealife]

The problem is S already has a meaning in Roman numerals, it means 0.5 (technically not 1/2 but 6/12)

[@randomtuba7079]

@@idonothavealife Ah that's true, I guess when using the notation, you can just make the decimal S look fancier

[@FunnyTimersChannel]

I have the same idea as this comment!

[@FunnyTimersChannel]

3,999,999 would be SSSQSOQMOCMXCIX

[@Salv2137]

That feels like base 1000positional system with extra steps

[@v88box21]

Looks pretty nice.

[@TrevorCouillard]

If you take the horizontal lines too far you will discover a veritcal line

[@le9038]

How to create the worlds biggest roman number in HERNN In Psudocode: While(true): print(X) print(--) run until it's big enough

[@musica00-7z]

You have it backwards though: repeat (n) {   print(-);   print(M); # use M for maximum effect }

[@frostplays155]

as with all math beyond what they teach you in algebra 2, i can feel the insanity begin to come forth as the explanation continues

[@Tartarus4567]

Here is something interesting. Once, there is a roman number called S. "S" represents a half. And then, there are more fractions using dots. A single unit of dots is equivelant to 1/12 "." So: . -> 1/12 .. -> 1/6 ... -> 1/4 .... -> 1/3 ..... -> 5/12 S -> 1/2 And keep adding the dots and you'll end up with: SS = I

[@qtc178]

I thought of my own similar expansion for Roman numerals when typing them. What I do is put any Roman numeral from 1 to 3,999 between parentheses. For example, (CXXV)=125,000. For larger numbers, I would simply put a Roman numeral in between two opening parentheses and again in between two closing parentheses. For example, (IX(M)IX)=one nonillion.

[@BryndanMeyerholtTheRealDeal]

IV was historically written as IIII

[@Memeened]

I extended it to my version expanded hyper extended Roman numeral notation, (E.H.E.R.N.N). Note: not to be confused with extended Roman numeral notation. Basically instead of writing a large stack of Roman expansion, you can put 2 lines on top of each other and put the Roman numeral on top, that Roman numeral shows how many layers there are. If you want it more precise then put a comma in front of that and write the Roman numeral on the layers. Then the Roman numeral on the bottom can be put in parentheses to indicate that it is a regular Roman numeral that is not in extended Roman numeral notation. For example, X|X|X|X|C can be written as IV,X||C). And don’t ask me why it is horizontal

[@iambic-kilometer]

How about adding a line below (let's agree to not add lines above and below simultaneously, shall we?) to represent division by 1000? Another oddball effect is with the small numbers above the line, you have an instant rudimentary path to log base 10, except for the large roman numeral beneath the lines.

[@z-ro1641]

I think we can use - like we you , for numbers for example 10,001 can be X-I but we might need a letter for 0 but it can be either O, N, Z or just a space

[@Applebapple-qm2pk]

Next step: zero, fractions and square roots 😂

[@zahrz44]

8:57 You looney! That’s what a fraction looks like! *makes the numeral on top on the line smaller*

[@pallavij520]

The Roman numerals are I V X L C D M

[@AdamAshrafkhan]

10:57 we can't do that forever. We need a way to make them smaller!

[@quartzcyanis]

finally, rome can measure it debt in roman numerals lol

[@SovietUnionCCCP]

8:58 “THAT'S WHAT A FRACTION LOOKS LIKE”

[@penguincute3564]

Very good idea I really approve and appreciate it!

[@terminusfinity009]

11:22 this number is named *aghem*: one hundred and sixty-four untrigintimilliduodeciquadringentillion

[@andreibaciu7518]

since a bar on top means x1000 and a bar is just a sideways I; you can put a sideways V to signify x5000 (or any numeral to signify it times 1000 times the entire number) and since that bar on top is a numeral in of itself you can add a bar to itself, or more precisely to its right (or left depending on which way the V points towards) and have _it_ be multiplied by a factor of a thousand and since that is a numeral in of itself you can repeat this cycle again, and have an ever growing spiral of multiplication

[@shaevor5680]

I was expecting that after stacking a bunch of horizontal lines, you would interpret the lines as the letter I rotated by 90 degrees, so I was thinking you'd be ending up having one roman numeral rotated by 90 degrees on top of another, and then if you repeat this process, you end up with some kind of spiral xD

[@WilliametcCook]

HERNN is an awesome acronym, I'm considering using it for that alone

[@Vitrivius]

perhaps, instead of infinitely expanding the number of overbafs, one could use a double line to denote something like titration

[@Baldi_1]

You are the best content creator i've ever seen! Your editing skill is awesome and i love the quality of your videos, you will have a lot of subscribers in a short time and you have potential! also, your character is cute

[@kales901]

yes

[@Chrischi3TutorialLPs]

All thos 9s had me going NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN at one point.

[@Stistreal]

This video is very well made

[@Wasabialt]

Underrated channel

[@jasonwibowo-qt6gq]

Congratulations we just reinvented scientific notation

[@randomperson5579]

if we're using the same ? over base over and over we could represent it as ? line, line base number. but after this rule runs into the same problem as before. I say we just introduce conway chains if you wanna go even higher.

[@cythism8106]

Tetration in Roman numerals. How practical.

[@NachoManWithThePlan]

cant we just do X^C for a googol since thats 10^100 8:59 Also fractions dont look like that in roman numerals, they use dozenal fractions, (1/12) , it starts as • for 1/12 and adds a dot each time until 6/12 or ½, 1 half is S. Then after they just keep adding dots so 7/12 is S• and so on, this is practical because it can represent halfs (S) Thirds (••••) Quarters (•••) And sixths (••).

[@antoncabotta5364]

You could represent the amount of numbers in the tower with another roman numeral like you did with the lines, that would give you an equivalent of exponentiation for roman numerals. Then comes tetration, which is the same thing for towers of powers.

[@SpaceSoups]

Surely the Romans needed to count 4000.

[@yalta5060]

looks like a glaceon

[@sys128]

maybe cuz i am a glaceon

[@yalta5060]

​@@sys128 holy cow

[@commandblockperson46]

A line over a number means that it’s multiply by 1000

[@portal6347]

This! And a box around the top and sides means 10,000

[@borzydar1196]

Homework: Express Graham number in Roman Expansion

[@braydonnguyen9356]

This guy has said 9 more times than I wrote a letter

[@Alex-ff8si]

On the 4th frame of 8:39 for 1 frame( or 16 msec ) it said gee Captain Oblivious

[@MineZapp]

*I* like numbers... ): "tough crowd"...

[@Sb129]

I absolutely love this concept~

[@NEWTBTRILLION]

The INTRO IS FUN BOT

[@pallavib1515]

9:00 THAT IS A FRACTION!

[@julesharris6383]

/M|III|M/ is M with (M with M lines on top of it) on top of it