How Did Anyone Do Math in Roman Numerals?
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I mean, its the only one that is both not confusing for beginners (because it doesn't trick them into the "cool, let's `simplify` the dx at the denominator with the next one" mindset) and it also translates easily to code (or other 1D encoding), like you can write "second derivative of f with respect to x" as D[f, x, 2] and "integral of a with respect to t" as "D[a, t, -1]".
My point is mainly that there is nothing rational at all in how humans choose notations... without some obscure historic events, we'd probably still be using some derivative of roman numerals or maybe even sexagesimals! (https://en.wikipedia.org/wiki/Sexagesimal)
(I imagine the reason is because "highly performing" individuals have their own "internal" language to think in, so general language is just a for communication, so a political decision... unfortunately for education :()
I don't know if higher-order functions are really that tough to teach/understand but I think it would actually simplify and demistify many things.
Currently, of the widespread notations I like this one best:
f(x0) = d/dx (sin(x)*cos(x)+x^2) | x=x0 (with the last part in subscript)
I also miss variable scoping from math writing and it disturbs me that variable names often carry semantics, like p(x) and p(y) can be the probability density functions of random variables X and Y (so p is a different function depending on the name of the input variable that you substitute so it doesn't actually operate on real numbers, but (string, number) pairs). I'd prefer to explicitly mark the functions as p_X(x) and p_Y(y).
Similar things come up a lot with differential equations where you don't really know whether something (like y or u) is supposed to be a function (of x or t) or "just" a variable.
Despite the general opinion among laypeople that math notation is very precise and unambiguous, I find that it's often very sloppy and unless you already understand the context very well, it can easily be misleading. Math notation is somewhere between normal natural language and programming languages, and depending on the writer it may be closer to one or the other.
One can argue that this is necessary for compactness.
A lot of that is familiarity but I don't think all of it is.
Part of it is the brevity, and part of it is the shortcuts. E.g. when I did my masters, one thing I quickly realised was that papers that expressed an algorithm using mathematical notation almost always lacked essential details.
My impression is that it's too obvious when there are too large leaps in code, whereas in mathematical notation everyone accepts leaps that can obscure that essential details have been left out.
E.g. you'd have papers on thresholding of images for OCR (deciding what is background and what is foreground) where it turned out the results were highly dependent on certain values represented by certain variables that were never defined, for example, putting in a situation of reconstructing parameters by trial and error if you wanted to reproduce the results.
Today I'm immediately suspicious if results are presented as maths outside of fields where the maths is essential (and sometimes even then) as I see it as having a tendency to be used to gloss over sloppy work or save space by leaving out essential details.
I'm sure this is not the case in all fields, and that people with a more extensive maths background will be able to fill in more of those leaps without much effort, and so it might very well be acceptable in some fields. But to me a notation that makes it that easy to hide missing details is a liability.
I'd love to see this beauty or clarity or whatever that people find in mathematics, but I've never caught even a hint of it. Seems like it needs a good IDE to make up for deficiencies in its language.
That allows you to use Python inside a mathematical environment and it's amazing.
Saved my bacon a few times when I need to translate maths to code and I need to poke it a bit to get an understanding of how it works.
(Linguists would want to murder me for saying this, I know.)
That's why some programming languages can even be defined by implementation. (Though as a programmer I try my best to avoid these languages...)
https://en.wikipedia.org/wiki/Total_derivative#The_total_der...
By the way, most of that article is terrible, but the linear map definition is the one we want. There's also the directional (Gâteaux) derivative:
https://en.wikipedia.org/wiki/Directional_derivative
Candidly, unless you really know your problem, we virtually always want the total derivative since it gives rise to things like gradients and Hessians, which are useful objects that we can store in memory.
Now, the reason that I bring these two up is that their spaces, or really their types, are different. Given a function f:X->Y, the total derivative is a linear operator from X to Y:
(total) f'(x) \in L(X,Y)
The directional derivative is an element in the space Y:
(dir) f'(x;dx) \in Y
Now, at this point, the notation is screwed up since we used Lagrange notation for both. The reason that we can get away with this is that under certain assumptions, that are mostly satisfied in the things we care about, we have that:
f'(x)dx = f'(x;dx)
Alright, so why should we care? Leibniz and Newton notation do a terrible job at capturing this information. Lagrange and Euler notation do a good job at this. For your example:
f(x0) = d/dx (sin(x)cos(x)+x^2) | x=x0
The types don't line up because sin(x)cos(x)+x^2 is value, literally a real number, not a function. Using the above, I would write this as:
(x \in R |-> sin(x)cos(x)+x^2)'(x0)
In LaTeX |-> would be \mapsto. This types correctly in the definitions above since
x \in R |-> sin(x)cos(x)+x^2 \in [R -> R]
and
(x \in R |-> sin(x)cos(x)+x^2)'(x0) \in L(R,R)
Of course, you probably wanted the value and not the function, which explains why we cheat in 1-D. So, we really should write:
(x \in R |-> sin(x)cos(x)+x^2)'(x0) 1 \in R
where we feed it the direction 1. And, yes, this is slightly more cumbersome that we may want, which is why there's a huge number of different notations. However, I do assert that the above generalizes properly all the way into infinite dimensions (Hilbert spaces) and provides a good foundation for typing out mathematical codes.
By the way, if anyone is looking for a book that does this right in my opinion, Rudin's "Principles of Mathematical Analysis" is amazing and his notation is good. For infinite dimensions, I prefer Zeidler's "Nonlinear Functional Analysis and Its Applications." Personally, what I look for is what I call properly typed notation that gives us easy access to useful tools like gradients, Taylor series, chain rule, and implicit and inverse function theorems. Again, most engineering and applied math work requires these theorems everywhere, so I find it best to keep them clean.
Is there any sensible rationale for sin^(-1)(x) meaning arcsin(x) while sin^2(x) means (sin(x))^2?
cf.
f(x) = x^2 + 1
x := 2
f(2) = 5It is confusing to beginners, but it's very useful once you understand it.
It's really not much different everywhere else in society. Different programming languages/frameworks/etc. are doing essentially the same thing (if you ignore the speed of execution). All the languages are Turing complete and can do more or less the same IO. But it's still much easier for people to use certain tools for certain problems than others.
The right notation allows you to focus on what's important and forget what is not.
---
[1] and yet, I guess I'll just babble on adding more words anyway...
[2] be they mathematical concepts, programming paradigms, or cultural norms and quirks.
This allows you to do all sort of calculations with wave functions without constantly grinding to a halt bogged down by integrals and conjugates all over the place.
The ladder operators (aka raising & lowering), really put this notation to good use[2]. It would be tremendously tedious to manipulate such expressions repeatedly without the simplifying properties of Bra-Kets. Once you are done manipulating them you then use the rules of the bra-kets to transform it back into a plain old integral, evaluate, and you're done.
I used to think they were very mysterious until I realised they were (essentially) notational convenience.
[1]: https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation
[2]: http://www.tcm.phy.cam.ac.uk/~bds10/aqp/lec3_compressed.pdf
You could see this as an actively distinct concept (Feynman diagrams are graphs which give rise to some algebraic structure, whereas Dirac's notation is an algebra in itself) or you can see them both as just a means of abstracting away many operations (e.g. in Dirac's notation, this would be multiplication by operators and inner products, both of which are integrals in some sense, as just a non-commutative type of multiplication) in really powerful notation.
It takes a good amount of intelligence to read it, and it took an amazing genius to write it. But put it in plain notation, and it's a collection of 7th grade algebra problems.
Notation is important.
In the same line, it's great at shining a light on the substitution rule for integration.
Given that your point that it can be obscure at first remains valid, I'd walk the middle line of introducing students to the f'(x) notation first; and after the introduction to integrals introduce this notation to them.
A good self-check is to see if you can convert from Leibniz notation to a more rigorous one at any given step in the computation and understand that step rigorously. Personally, I find that functional notation (using D as an operator on the space of functions, etc.) to be as simple to use and much more likely to alert me when I'm about to confuse myself.
Isn't Δ(x^2) = 2xΔx ≠ (Δx)^2 ? The object Δ(x^2) has one infinitesimals while (Δx)^2 has two, and the number of infinitesimals is conserved. (You can only get finite quantities by taking the ratio of equal numbers of infinitesimals.)
If you mean the infinitesimal difference, then Δ(x^2) = 2Δx still isn't (Δx)^2.
But I see how it can be confusing with printed characters. I guess Leibniz just took ligatures for granted when he came up with his stuff.
The derivative w.r.t. x of f(x) is D_x f in Lagrange notation. It looks a bit like matrix multiplication for a good reason—a matrix is just a representation of a linear operator on a finite dimensional vector space.
To the extent that this is true, how much do you think that is due to the notation itself, and how much is due to someone identifying the essential underlying concepts and then making those the basis for a good notation?
-- Whitehead
I think the interesting part is perceiving language as a tool.
With Arabic numbers, I have to learn that 1 is one thing, 2 represents 2 things, 3 represents 3 things and so on. Then I have to remember that 2 + 2 = 4, and 3 + 2 = 5, there is more memorization required.
Where Roman Numerals for someone with little or no education could get by in the market square with some simple rules and even use twigs as a primitive calculator. It is not until you get to much more complex ideas that the Arabic notation wins out.
So perhaps, different notations lend themselves better or worse depending on the application ?
Just some random thoughts that this very interesting post brought to mind.
I follow you thus far.
>Then I have to remember that 2 + 2 = 4, and 3 + 2 = 5
No you don't.
You have to remember that 1 + 1 = 2; 2 + 1 = 3; ...; 9 + 1 = 10; and then the rules repeat themselves, respecting columns for addition. All mathematics between 1 and 10 like 4 + 5 are already known at this point.
Roman numerals, on the other hand, give you no easily repeatable pattern to follow as the order of magnitude increases. I + I = II, III + II = V, V + V = X, X + X + X + X + X = L, LL = C, and I only got this far because of what you said. What comes next? C * 5 = x_1, x_1 * 2 = x_2, x_2 * 5 = x_3, ..., x_n * 5 = x_n+1, x_n+1 * 2 = x_n+3
The complexity is unbounded. Sure, if you constrain yourself to "getting by in the market place in Rome", that's one thing, but even then I would imagine arguments around arithmetic could go either way.
If you spent several decades working with a counting board and had only occasionally seen Hindu–Arabic arithmetic, you would likely feel the opposite (as, indeed, people did for the first few centuries after written arithmetic was introduced to Europe – for example our word cipher, meaning secret code, comes from the word for the Arabic 0, reflecting people’s early confusion about pen and paper arithmetic).
The Romans (and others in the Roman empire) were the premier engineers, merchants, bureaucrats, astronomers, etc. of their era and region. They didn’t have any problem doing extremely complex computations.
As for your specific concerns: the easy pattern is that the letter for a group of five literally looks like half of the letter for a group of ten.
V = X/2, L = C/2, D = ↀ/2, ↁ = ↂ/2, etc.
So you need to remember the meanings and relations for the symbols for I, X, C, ↀ, ↂ and then just count them. The patterns that: IIIII = V; XXXXX = L; CCCCC = D, ↀↀↀↀↀ = ↁ are really not that hard to remember. The groups of five are mostly there as a shorthand, because writing nine of the same symbol in a row is harder to count and takes up more space.
These patterns are certainly no harder than remembering the English words ten, hundred, thousand (or Latin words unus, decem, centum, mille), which are also arbitrary symbols.
Since the numbers are always written in order, you can learn to separate them by digits. People would have “chunked” a long string of these symbols into the word for each each digit, and pronounced them using words pretty much like modern languages. Just like in our natural languages, the system is not strictly positional – you just skip writing/pronouncing any lines on the counting board with no pebbles.
So if you see DCCCXXXXVII you think of it / read it as “eight hundred forty seven”, first split into the groups DCCC XXXX VII, with the digit represented by the pattern and the order of magnitude represented by the symbols used. Or alternately, you would have seen them as the visual patterns “three pebbles on the hundred line and one in the space above; four pebbles on the tens line; two pebbles on the one line and one in the space above”.
When you’re thinking of the meaning of these symbols, you’re going to be fluently translating them in your head back and forth between three representations: verbal, counting board, and written. Once these have all been worked with extensively, there’s not much friction. It’s just like learning to read, or learning music notation. Someone experienced can read a musical phrase on a score and hear the sound of the whole expression in their head, rather than trying to count which line each note is on, count out the tempo, etc.
Objectively, the Roman system is easier to teach up to a basic level, especially to someone illiterate. Basic calculations on a counting board are straight forward and easy to explain and motivate. Multi-digit multiplication gets a bit annoying in both cases because it involves the summation of many partial partial products. Long division and square roots get nasty in both systems.
Where the Hindu–Arabic system really shines is when people need to frequently work with very large numbers, very precise numbers (though remember there were no decimal fractions per se in Europe until >1600), or numbers of different orders of magnitude, have access to cheap and abundant paper, and can spend years training to do basic arithmetic. The biggest advantages of pen and paper methods for basic arithmetic are that it’s easy to see the whole work process, and therefore more easily check for mistakes, and that writing the final answer doesn’t take as much space. It’s also much easier to write down and explain pen and paper arithmetic methods in a printed book. The counting board methods are often faster to perform.
But more importantly still, pen and paper arithmetic is easy to generalize to more sophisticated mathematical notation for fractions, algebraic equations, etc.
I made a fairly concise point, but you're right about one thing
>we all do this sometimes
Because you most certainly just did
>As for your specific concerns: the easy pattern is that the letter for a group of five literally looks like half of the letter for a group of ten.
That's not the issue with the "easily repeatable pattern". The issue is this:
>V = X/2, L = C/2, D = ↀ/2, ↁ = ↂ/2, etc.
Do you see that "etc" you post? That's because, to represent 10^n, you need to know 2*n + 1 characters.
This is an obvious downfall that Hindu-Arabic numerals obviously do not have. Even if you ignore the shorthand (which would be foolish, parsing 5-9 characters in a row can take a serious moment of focus), you still need to know an unbounded amount of characters to represent the integers alone, and let's not even discuss the obvious fallacy of fractions etc..
Also,
>Objectively, the Roman system is easier to teach up to a basic level
Even this is an unfounded statement. Make an argument. Prove it. Objectivity of teachability can't possibly be something you just throw out casually in an argument, let alone an argument where you told someone they were biased and ignorant for, as far as I can tell, no reason at all.
It’s very seldom that anyone in the Roman empire – or medieval Europe – would need bigger numbers than, say, one million, especially for writing final answers down. But at least one counting board we know about from 300 BCE probably had lines for 10 orders of magnitude, and it’s not like more couldn’t easily be added if necessary. http://www.akg-images.fr/archive/-2UMDHU23V18G.html
As for fractions: Roman fractions were mostly twelfths (think inches and ounces).
Roman numerals and counting boards are obviously not an ideal system for writing down very large or very precise numbers, as I said. I’m not here claiming that they are a better basis for society’s numeration than more strictly positional Hindu–Arabic numerals.
My point is that we shouldn’t be so hasty to dismiss them out of hand or exaggerate their flaws. They were a highly effective system for doing complex calculations: precisely estimating the positions of stars and planets centuries into the past/future, running a bureaucracy overlooking an empire of 60+ million people, building large-scale engineering projects, and so on.
When it comes to teaching, I can only go by my own anecdotal experience trying to teach young children about numbers, the writings of various elementary math teachers, and the fragmentary remnants of debates in medieval Europe. I don’t know of any modern peer-reviewed research about Roman or medieval European counting boards, sorry.
(There is lots of evidence that learning place value using Hindu–Arabic numerals is very difficult for children, compared to other concepts and skills, requiring several years of study before primary school students really figure it out.)
Seems like counting boards might make a partial comeback sometime though: https://vimeo.com/204368634
I mean, all of the examples we used in school to learn basic arithmetic had things like "(picture of 4 apples) + (picture of 2 apples) = (picture of 6 apples)". Roman numerals are kind of a pictograph in a way.
To each their own, but I'm unsure of the relative ease of roman numerals
I must be below average in that regard; I never have trouble being able to distinguish the number of things when there are three or fewer, but it's somewhat difficult for me to tell four and five apart, and I essentially have to resort to manually counting for five and six.
II + II = IIII
2 + 2 = 4
Don't see how the latter problem lends itself to any more memorization beyond symbols
What is 2 + 5? Well 2 is II and 5 is V which is IIIII. So then we have IIIIIII which is (IIIII)II or VII which is 7.
(There is more help; it's not as simple as people have been portraying, though...)
Decimal arithmetic becomes quite a bit easier if you normalize your numbers to the digits from –5 to 5, and it would help prepare students for algebra.
For example, if you do base ten with digits EDCBA012345 (E=-5, D=-4, etc.) -- that's 11 digits -- then five can be written as 5 or 1E; fifteen is 15 or 2E, etc.
So you either:
- Accept that some numbers can be written in more than one way, which seems a huge disadvantage over what we have today.
- Rule out certain sequences of digits (E can only follow a negative digit, 5 can only follow a positive digit, if 5 follows a 0, then it depends on the digit before the 0, etc.). Not only is this inelegant, but it adds seemingly arbitrary rules for people to memorize when they're first learning how to write numbers.
- Remove one of the digits (e.g., remove E). Then things aren't symmetric around zero -- e.g. five is 5, but minus five is A5.
Writing numbers in multiple possible ways is really not that big a problem for most uses (for record-keeping in business transactions maybe). Personally I recommend becoming familiar with two different normalizations: all same-signed digits (the form we use now) or [–5, 5] with digits before a terminal 5 rounded away from zero. But in general for personal scratch work numbers don’t need to be normalized unless you feel like it. As long as you understand that 20–3 is the same as 10+7, it doesn’t really matter which one you think of as primary.
We use multiple number representations all the time, with e.g. the vulgar fraction 11/8 alternately representable as the “mixed number” 1 + 3/8, or as the decimal fraction 1.375, the simple continued fraction [1; 2, 1, 2], the percentage 137.5%, or the common logarithm ~10^(0.1383). Figuring out that 1.375, 1.385̅, 1.42̅5̅, 1.43̅5, and 2.6̅2̅5̅ are the same number isn’t that hard, with some practice.
You just have to put the larger number first for the addition.
It's not enough of the story, though - consider:
VI + V = VIV
V + IV = VIVThe problem is IV isn't 6, it's 4.
I'm not sure having to look at all the possible conversions of a context free grammar to reduce the state to its minimum is actually easier than some simple arithmetic in our current system.
In base 10 math there are 18 outcomes for adding two digits, and you have to do carry operations for 9 of them. With carries there are 20 and you have special cases for 10. In Roman numerals there are 10 outcomes, you have to do carries for 5 of them, special casing for two (4 and 9), the numbers are at least one digit longer and non uniform, so it's harder to line up the columns to do an addition in the first place.
No wonder algebra was invented in Arabic.
This doesn't seem quite accurate. Roman numerals are organised around multiples of 5, whereas a base-5 system should be organised around powers of 5.
XX - III =
XVIIIII
- III =
XVII
It's literally just addition in reverse. Instead of having reduction passes where you convert IIIII to V, etc., you'd have an expansion passes to convert V to IIIII.
And that still isn't event touching multiplication or division, which are very common in simple commerce. 100 coins for 8 chickens, or 60 coins for 5 chicken. What's the better deal? I want 7 of these 3 coin breads. You say that's 30 coins? Guess I should check that...
Multiplication doesn't in general require an abacus if you're trained in roman numerals. Let's take your "more complicated numbers", 42 * 13.
Expand XLII * XIII = X * XLII + XLII + XLII + XLII, the first requires you to know a times table to see that it's CDXX, the rest you'd add mentally, LLL - XXX = CXX, so it's CDXX + CXXVI = DXLVI. Five hundred forty six.
Like, I'm not saying it's as clean as 42 * 13 = 420 + 126 = 546, it's not. But it's not educationally prohibitive either.
From my example, which was meant for subtraction, not addition or multiplications, we get the following:
XLII - XIII
so we convert to an optimized form for arithmetic (which is, as I've noted a second format you need to know and keep track of, since it's not valid Roman numeral format
XXXXII - XIII
then we note that we have to borrow from the X "place", so let's do that
XXXVIIIIIII - XIII
now we can actually subtract
(XXXX - X) + V + (IIIIIII - III) = XXX + V + IIII = XXXVIIII
Now we have to convert back to valid Roman numerals
XXXVIV = XXXIX
This is using common current arithmetic techniques. I'm not sure whether this is the method they would have employed, or whether it would have been a different method that could have been easier or harder depending on how it combines with the attributes of this number system. Important things to note about the method shown are a pre and post conversion step (or steps, if you don't get every sub-conversion right initially or don't go to a lowest common format), and then the arithmetic itself requires more borrows.
Subtraction is as easy as addition (unless you go into negative numbers). The stategy is to walk the subtrahend left-to-right, eliminating from both as you go. If you run into a value that you don't have, then you expand the minuend from the right until you do.
For your "difficult" example of 42 - 13:
XXXXII - XIII
1 Eliminate an X from both.
Now you have XXXII - III.
2 Eliminate I from both
Now you have XXXI - II. Repeating (2) gives you XXX - I.
3 Since there is no I to eliminate, expand the minuend from the right to create one.
Now you have XXVV - I, repeat since you still don't have a I, repeat (3) which gives you XXVIIIII - I
Eliminate the from I both, giving the final result: XXVIIII.
Optional, you can "reduce" this to XXVIV, like you would if you had a fraction, like 3/9, but as stated in the article, this notation was not common place in ancient Rome.
Subtractive notation is confusing.
Seriously I figured they used abacus for everything they just figured out the notation to write it down in at the end some would convert to the if notation while others would not.
The article doesn't say subtractive notation was not used in classic roman times, it says only that it was rare.
Yes, this is exactly right, and should be at the top of this discussion thread. Romans (like the Greeks, the Babylonians, and others in the ancient world) did their calculations using a counting board. Roman numerals were only used for recording the final answers.
“When a Roman wished to settle accounts with someone, he would use the expression vocare aliquem ad calculos - ‘to call them to the pebbles.’” http://mathforum.org/library/drmath/view/57572.html
Our words calculate, calculus, calculator, etc. all come from the Latin word for the pebbles they used as counting-board tokens.
looks at watch
Well, damn, it's IIII. However my watch does use IX over VIIII, what's up with that?
I, II, III, IIII (consisting of I only)
V, VI, VII, VIII (consisting of I and V)
IX, X, XI, XII (consisting of I and X)
[1] https://www.hautehorlogerie.org/en/encyclopaedia/glossary-of...
http://mentalfloss.com/article/24578/why-do-some-clocks-use-...
The reason I always heard is about ease of casting: it requires a more complex mold if 4 is written as IV instead of IIII.
The digits are on the watch are standard arabic digits but the description calls it "indian numerals"!
But of course, there's no way they actually said it that way.
Right? Right??
1: Unus -a -um | primus -a -um
2: duo, duae, duo | secundus -a -um
3: tres, tria | tertius -a -um
4: quattuor | quartus -a -um
5: quinque | quintus -a -um
6: sex | sextus -a -um
7: septem | septimus -a -um
8: octo | octavus -a -um
9: novem | nonus -a -um
10: decem | decimus -a -um
11: undecim | undecimus -a -um
12: duodecim | duodecimus -a -um
13: tredecim | tertius decimus -a -um
14: quattuordecim | quartus decimus -a -um
15: quindecim | quintus decimus -a -um
16: sedecim | sextus decimus -a -um
17: septendecim | septimus decimus -a -um
18: duodeviginti | duodevice(n)simus -a -um
19: undeviginti | undevice(n)simus -a -um
20: viginti | vice(n)simus -a -um
30: triginta | trice(n)simus -a -um
40: quadraginta | quadrage(n)simus -a -um
50: quinquaginta | quinquage(n)simus -a -um
60: sexaginta | sexage(n)simus -a -um
70: septuaginta | septuage(n)simus -a -um
80: octoginta | octoge(n)simus -a -um
90: nonaginta | nonage(n)simus -a -um
100: centum | cente(n)simus -a -um
500: quingenti | quingente(n)simus -a -um
1000: mille | mille(n)simus -a -um
Just like an English speaker sees "1, 2, 3, 4, 5, 6" and pronounces "one, two, three, four, five, six". Sure, the Roman numerals look like letters, but that doesn't mean you pronounce them that way.
There aren’t many (if any?) human languages that duplicate this feature of being a purely positional system.
But reading this article and trying out a few things, I found it was great fun!
My friends lived in France a while and said their landlady could never count their rent (paid in cash) correctly first time. Always stumbled somewhere between 100x+60 and 100x+100 for integer values of x.
The Swiss have corrected this in Swiss French.
Actually it's four-twenty-ten-nine (quatre vingt dix neuf) -- -- which is perfectly logical on a twenty-based numbering system. Of course it's not very helpful that they switched to ten-based, but only up to 60.
http://www.woodwardfrench.com/lesson/numbers-from-1-to-100-i...
I speak a language that does this as well (Georgian), where say 54 is ormotsdatotxmeti (two times twenty and fourteen.)
I didn't find maths particularly different difficulty wise, when thinking about it in Georgian or not. What did trip me up, was the times! Up to x:29 it's 29 minutes past x, but at x:30 it is 30 minutes to ++x. Weird.
Most other European languages at least (I'm sure it's present elsewhere too, but I don't know enough non-European languages to say) have some remnants of more widespread counting in 12's or 20's or both (e.g. in English a "dozen" is 12, a "gross" is a dozen dozen (144), and a "score" is 20, hence "four score and seven years ago" in Lincolns Gettysburg Address - 87 years).
I understand the point about logics in counting, but this is just a new word to learn, like déposition or balafré. One can analyze these words from an ethymology perspective but normally you just learn them by heart.
At least this is what I do in English when I need to remember went or throughout .
It bugs me when people mix our standard numerals with Roman numerals, such as 12MM to mean twelve million. They are different numerals and the meaning is not defined when they are used together.
And Roman numerals are not like SI suffixes, meaning they are not multiplicative; Roman numerals are additive, so MM is two thousand, not one million. Also, M is an SI suffix, so 12M means twelve million and 12MM just looks like a typo.
Obviously people do not use SI suffixes may not feel the same way, this is just my pet peeve because I use SI suffixes in science.
$ perl6 -e 'say Ⅻ / Ⅲ'
4Perhaps you mean "how" rather than "why". Others have answered with the gory "how" details.
Perhaps you mean "why would anyone think to make this work?" To the degree anyone thought "let's make this work", the "this" was adopting the full Unicode vision -- decent digital support for the many native languages humans use (from Arabic to Zulu).
Perhaps you mean "why" as in "why would someone write the underlying language/compiler code that makes Latin math work?" Some old languages, like Latin, just happen to have been automatically enabled too. Some compiler dev would have to write code to stop it working.
Perhaps you mean "why would weird Unicode characters be acceptable in a programming language's source code?" Because P6 defers to the Unicode standard by default for determining what languages and characters may be included in source code.
This in turn allows everyone on the planet to feel welcome to write code in their native language and/or use characters they're familiar with in their line of work. (For example mathematicians can use the appropriate mathematical symbols to express themselves in P6 code.)
That Perl allows them in integer literals is strange, but also logical in a weird way. (Now I wonder what other number systems it supports.)
EDIT: Ok, so I installed perl6 (rakudo) just to test this out, and it apparently doesn't work out of the box. Pity.
https://github.com/rakudo/rakudo/blob/beec02a6fa69e3ac290b4d...
By any chance, was it from system packages that are likely outdated? You can get up-to-date packages [here](https://github.com/nxadm/rakudo-pkg/releases) or for Windows/Mac, use [Rakudo Star](http://rakudo.org/downloads/star/). Anything older than 2017.04 is ancient and anything older than 2015.12 precedes first stable release and is likely broken.
> Now I wonder what other number systems it supports
It goes by Unicode's definition of what's a digit or a number (using Nd and No) properties. Nd chars are "digits" and can be combined. `No` chars are "Numeric other" and can't be combined; can only use them as numeric literals. [Among other things](https://docs.perl6.org/language/unicode_texas.html#Numeric_c...), that includes Unicodey fractions. And as a cherry on top, you can use superscript chars to raise to a power:
$ rakudo -e 'say ½ + 42²'
1764.5
$ rakudo -e 'put (^0x110000).grep(*.uniprop eq any <Nd No>)».chr'
0 1 2 3 4 5 6 7 8 9 ² ³ ¹ ¼ ½ ¾ ٠ ١ ٢ ٣ ٤ ٥ ٦ ٧ ٨ ٩ ۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹ ߀ ߁
߂ ߃ ߄ ߅ ߆ ߇ ߈ ߉ ० १ २ ३ ४ ५ ६ ७ ८ ९ ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯ ৴ ৵ ৶ ৷ ৸ ৹ ੦ ੧ ੨ ੩
੪ ੫ ੬ ੭ ੮ ੯ ૦ ૧ ૨ ૩ ૪ ૫ ૬ ૭ ૮ ૯ ୦ ୧ ୨ ୩ ୪ ୫ ୬ ୭ ୮ ୯ ୲ ୳ ୴
୵ ୶ ୷ ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯ ௰ ௱ ௲ ౦ ౧ ౨ ౩ ౪ ౫ ౬ ౭ ౮ ౯ ౸
౹ ౺ ౻ ౼ ౽ ౾ ೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯ ൘ ൙ ൚ ൛ ൜ ൝ ൞ ൦ ൧ ൨ ൩ ൪
൫ ൬ ൭ ൮ ൯ ൰ ൱ ൲ ൳ ൴ ൵ ൶ ൷ ൸ ෦ ෧ ෨ ෩ ෪ ෫ ෬ ෭ ෮ ෯ ๐ ๑ ๒ ๓ ๔ ๕
๖ ๗ ๘ ๙ ໐ ໑ ໒ ໓ ໔ ໕ ໖ ໗ ໘ ໙ ༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩ ༪ ༫ ༬ ༭ ༮ ༯ ༰
༱ ༲ ༳ ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ ႐ ႑ ႒ ႓ ႔ ႕ ႖ ႗ ႘ ႙ ፩ ፪ ፫ ፬ ፭
፮ ፯ ፰ ፱ ፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ ፻ ፼ ០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩ ៰ ៱ ៲
៳ ៴ ៵ ៶ ៷ ៸ ៹ ᠐ ᠑ ᠒ ᠓ ᠔ ᠕ ᠖ ᠗ ᠘ ᠙ ᥆ ᥇ ᥈ ᥉ ᥊ ᥋ ᥌ ᥍ ᥎ ᥏ ᧐ ᧑
᧒ ᧓ ᧔ ᧕ ᧖ ᧗ ᧘ ᧙ ᧚ ᪀ ᪁ ᪂ ᪃ ᪄ ᪅ ᪆ ᪇ ᪈ ᪉ ᪐ ᪑ ᪒ ᪓ ᪔ ᪕ ᪖ ᪗
᪘ ᪙ ᭐ ᭑ ᭒ ᭓ ᭔ ᭕ ᭖ ᭗ ᭘ ᭙ ᮰ ᮱ ᮲ ᮳ ᮴ ᮵ ᮶ ᮷ ᮸ ᮹ ᱀ ᱁ ᱂ ᱃ ᱄
᱅ ᱆ ᱇ ᱈ ᱉ ᱐ ᱑ ᱒ ᱓ ᱔ ᱕ ᱖ ᱗ ᱘ ᱙ ⁰ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ⅐ ⅑
⅒ ⅓ ⅔ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅛ ⅜ ⅝ ⅞ ⅟ ↉ ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ⑪ ⑫ ⑬ ⑭ ⑮
⑯ ⑰ ⑱ ⑲ ⑳ ⑴ ⑵ ⑶ ⑷ ⑸ ⑹ ⑺ ⑻ ⑼ ⑽ ⑾ ⑿ ⒀ ⒁ ⒂ ⒃ ⒄ ⒅ ⒆ ⒇ ⒈ ⒉
⒊ ⒋ ⒌ ⒍ ⒎ ⒏ ⒐ ⒑ ⒒ ⒓ ⒔ ⒕ ⒖ ⒗ ⒘ ⒙ ⒚ ⒛ ⓪ ⓫ ⓬ ⓭ ⓮ ⓯ ⓰ ⓱ ⓲
⓳ ⓴ ⓵ ⓶ ⓷ ⓸ ⓹ ⓺ ⓻ ⓼ ⓽ ⓾ ⓿
⳽ ㆒ ㆓ ㆔ ㆕ ㈠ ㈡ ㈢ ㈣ ㈤ ㈥ ㈦ ㈧ ㈨
㈩ ㉈ ㉉ ㉊ ㉋ ㉌ ㉍ ㉎ ㉏ ㉑ ㉒ ㉓ ㉔ ㉕ ㉖ ㉗ ㉘ ㉙ ㉚ ㉛ ㉜ ㉝ ㉞ ㉟ ㊀ ㊁ ㊂
㊃ ㊄ ㊅ ㊆ ㊇ ㊈ ㊉ ㊱ ㊲ ㊳ ㊴ ㊵ ㊶ ㊷ ㊸ ㊹ ㊺ ㊻ ㊼ ㊽ ㊾ ㊿ ꘠ ꘡ ꘢ ꘣ ꘤ ꘥
꘦ ꘧ ꘨ ꘩ ꠰ ꠱ ꠲ ꠳ ꠴ ꠵ ꣐ ꣑ ꣒ ꣓ ꣔ ꣕ ꣖ ꣗ ꣘ ꣙ ꤀ ꤁ ꤂ ꤃ ꤄ ꤅ ꤆
꤇ ꤈ ꤉ ꧐ ꧑ ꧒ ꧓ ꧔ ꧕ ꧖ ꧗ ꧘ ꧙ ꧰ ꧱ ꧲ ꧳ ꧴ ꧵ ꧶ ꧷ ꧸ ꧹ ꩐ ꩑ ꩒
꩓ ꩔ ꩕ ꩖ ꩗ ꩘ ꩙ ꯰ ꯱ ꯲ ꯳ ꯴ ꯵ ꯶ ꯷ ꯸ ꯹ ０ １ ２ ３ ４ ５ ６ ７ ８
９ 𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏 𐄐 𐄑 𐄒 𐄓 𐄔 𐄕 𐄖 𐄗 𐄘 𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡 𐄢 𐄣 𐄤 𐄥 𐄦 𐄧
𐄨 𐄩 𐄪 𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳 𐅵 𐅶 𐅷 𐅸 𐆊 𐆋 𐋡 𐋢 𐋣 𐋤 𐋥 𐋦 𐋧 𐋨 𐋩 𐋪 𐋫 𐋬 𐋭 𐋮 𐋯 𐋰 𐋱
𐋲 𐋳 𐋴 𐋵 𐋶 𐋷 𐋸 𐋹 𐋺 𐋻 𐌠 𐌡 𐌢 𐌣 𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩 𐡘 𐡙 𐡚 𐡛 𐡜 𐡝 𐡞 𐡟 𐡹 𐡺 𐡻
𐡼 𐡽 𐡾 𐡿 𐢧 𐢨 𐢩 𐢪 𐢫 𐢬 𐢭 𐢮 𐢯 𐣻 𐣼 𐣽 𐣾 𐣿 𐤖 𐤗 𐤘 𐤙 𐤚 𐤛 𐦼 𐦽 𐧀 𐧁 𐧂 𐧃 𐧄 𐧅 𐧆 𐧇 𐧈 𐧉
𐧊 𐧋 𐧌 𐧍 𐧎 𐧏 𐧒 𐧓 𐧔 𐧕 𐧖 𐧗 𐧘 𐧙 𐧚 𐧛 𐧜 𐧝 𐧞 𐧟 𐧠 𐧡 𐧢 𐧣 𐧤 𐧥 𐧦 𐧧 𐧨 𐧩 𐧪 𐧫 𐧬 𐧭 𐧮
𐧯 𐧰 𐧱 𐧲 𐧳 𐧴 𐧵 𐧶 𐧷 𐧸 𐧹 𐧺 𐧻 𐧼 𐧽 𐧾 𐧿 𐩀 𐩁 𐩂 𐩃 𐩄 𐩅 𐩆 𐩇 𐩽 𐩾 𐪝 𐪞 𐪟 𐫫 𐫬 𐫭 𐫮 𐫯
𐭘 𐭙 𐭚 𐭛 𐭜 𐭝 𐭞 𐭟 𐭸 𐭹 𐭺 𐭻 𐭼 𐭽 𐭾 𐭿 𐮩 𐮪 𐮫 𐮬 𐮭 𐮮 𐮯 𐳺 𐳻 𐳼 𐳽 𐳾 𐳿 𐹠 𐹡 𐹢 𐹣 𐹤 𐹥
𐹦 𐹧 𐹨 𐹩 𐹪 𐹫 𐹬 𐹭 𐹮 𐹯 𐹰 𐹱 𐹲 𐹳 𐹴 𐹵 𐹶 𐹷 𐹸 𐹹 𐹺 𐹻 𐹼 𐹽 𐹾 𑁒 𑁓 𑁔 𑁕 𑁖 𑁗 𑁘 𑁙 𑁚 𑁛
𑁜 𑁝 𑁞 𑁟 𑁠 𑁡 𑁢 𑁣 𑁤 𑁥 𑁦 𑁧 𑁨 𑁩 𑁪 𑁫 𑁬 𑁭 𑁮 𑁯 𑃰 𑃱 𑃲 𑃳 𑃴 𑃵 𑃶 𑃷 𑃸 𑃹 𑄶 𑄷 𑄸 𑄹 𑄺
𑄻 𑄼 𑄽 𑄾 𑄿 𑇐 𑇑 𑇒 𑇓 𑇔 𑇕 𑇖 𑇗 𑇘 𑇙 𑇡 𑇢 𑇣 𑇤 𑇥 𑇦 𑇧 𑇨 𑇩 𑇪 𑇫 𑇬 𑇭 𑇮 𑇯 𑇰 𑇱 𑇲 𑇳 𑇴
𑋰 𑋱 𑋲 𑋳 𑋴 𑋵 𑋶 𑋷 𑋸 𑋹 𑑐 𑑑 𑑒 𑑓 𑑔 𑑕 𑑖 𑑗 𑑘 𑑙 𑓐 𑓑 𑓒 𑓓 𑓔 𑓕 𑓖 𑓗 𑓘 𑓙 𑙐 𑙑 𑙒 𑙓 𑙔
𑙕 𑙖 𑙗 𑙘 𑙙 𑛀 𑛁 𑛂 𑛃 𑛄 𑛅 𑛆 𑛇 𑛈 𑛉 𑜰 𑜱 𑜲 𑜳 𑜴 𑜵 𑜶 𑜷 𑜸 𑜹 𑜺 𑜻 𑣠 𑣡 𑣢 𑣣 𑣤 𑣥 𑣦 𑣧
𑣨 𑣩 𑣪 𑣫 𑣬 𑣭 𑣮 𑣯 𑣰 𑣱 𑣲 𑱐 𑱑 𑱒 𑱓 𑱔 𑱕 𑱖 𑱗 𑱘 𑱙 𑱚 𑱛 𑱜 𑱝 𑱞 𑱟 𑱠 𑱡 𑱢 𑱣 𑱤 𑱥 𑱦 𑱧
𑱨 𑱩 𑱪 𑱫 𑱬 𖩠 𖩡 𖩢 𖩣 𖩤 𖩥 𖩦 𖩧 𖩨 𖩩 𖭐 𖭑 𖭒 𖭓 𖭔 𖭕 𖭖 𖭗 𖭘 𖭙 𖭛 𖭜 𖭝 𖭞 𖭟 𖭠 𖭡 𝍠 𝍡 𝍢
𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 𝍪 𝍫 𝍬 𝍭 𝍮 𝍯 𝍰 𝍱 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗 𝟘 𝟙 𝟚 𝟛 𝟜 𝟝 𝟞 𝟟 𝟠 𝟡 𝟢
𝟣 𝟤 𝟥 𝟦 𝟧 𝟨 𝟩 𝟪 𝟫 𝟬 𝟭 𝟮 𝟯 𝟰 𝟱 𝟲 𝟳 𝟴 𝟵 𝟶 𝟷 𝟸 𝟹 𝟺 𝟻 𝟼 𝟽 𝟾 𝟿 𞣇 𞣈 𞣉 𞣊 𞣋
𞣌 𞣍 𞣎 𞣏 𞥐 𞥑 𞥒 𞥓 𞥔 𞥕 𞥖 𞥗 𞥘 𞥙 🄀 🄁 🄂 🄃 🄄 🄅 🄆 🄇 🄈 🄉 🄊 🄋 🄌http://www.dozenal.org/drupal/content/brief-introduction-doz...
These people believe that base 12 would be superior to base 10. And they do make good points, expressing fractions like those in 3rds become easier. Is it superior in the grand scheme? I'll leave that debate to actual experts. Will people see the utility and switch? I'm going to keep my base 10 skills up, you know, just in case duodecimal doesn't catch on. ;-)
10 (decimal) is not divisible by 3, and rewriting it in base 12 as A won't change that. You simply get 29.4 instead of 33.3...
Other applications, like documenting how much oil to put in a car or whatever, there would be more options for picking memorable numbers. Like, my motorcycle takes 1.6L of oil. It could be 1 represented as 1 2/3 L, which is more visually intuitive. Not sure if I explained that well enough, maybe someone else has got a better metaphor handy...
It's just hardwired in the brain how much 0.6, or 60%, represents, I think internally my brain finds the closest easy references like 10%, 25%, 33%, 50% or 75% and then interpolates or adds or removes chunks of 10%, the rest is rounding errors. I mean, i dont have to think consciously if 0.69 is more or less than 75% and i know exactly where both 66.66% and 75% is and can place it somewhere in between.
On the other hand, to divide by 3 you basically need to do long division.
As to your question specifically regarding base-16 instead of base-12, it depends.
Decimal itself is just a bizarre choice, most likely due to humans having literally ten digits. In decimal we can represent exact fractions of 1/2, 1/5, and 1/10 (without repeated decimals like 0.33333 for 1/3). Counting by fives (and twos) is very easy.
But choosing prime factors of 2 and 5 is a strange choice in itself. Why skip 3? Why is it more useful to easily represent fraction 1/5th as 0.2 instead of 1/3rd? How often do we use fifths?
Hexadecimal in one sense is easier, all prime factors are two. So we can represent 1/2, 1/4, 1/8, and 1/16 exactly.
Duodecimal (Base 12) is very convenient for having a high proportion of exact fractions. Eg - 1/12, 1/6, 1/4, 1/3, and 1/2 can all be represented exactly. I'd argue in everyday use we're more likely to consider 1/3rd of something than 1/5th. Counting by twos, threes, fours, and sixes is easy. Watch, let's count to 20 (24 in decimal) by 3's : 3, 6, 9, 10, 13, 16, 19, 20. In 4's : 4, 8, 10, 14, 18, 20. By 6 : 6, 10, 16, 20.
Base-12 offers four handy subdivisions (excluding 1) instead of two for decimal or three for hexadecimal. That beats hexadecimal using fewer unique digits. It beats decimal by two using only two extra unique digits.
And I think it's these reasons it was chosen for various historical subdivisional units (inches per foot, pence per shilling).
The other item to consider is the relative number of unique values per digit. I'm not sure of the utility of having 10, 12, or 16 here.
At one extreme, while binary is useful for discretising signals in digital logic, using only zeroes and ones becomes cumbersome for daily use at higher numbers.
Once we're at base 10 and higher, I'm not sure how much here extra digits help or hurt.
Because, I'd think, many things follow power laws which makes values closer to 1 occur more frequently.
Six-fi? Typo or genre I'm not aware of?
Luckily I'm not yet at the copy-editing stage ;-)
But humans are kinda complex, and don't trade well long term learning for boring tasks. We seem to be great exactly around base 10 (probably 12 would be perfect, like people already said).
But what would it even mean to construct a numeral-system with a non-integral radix?
I speak a language (Mende) that has no formal mathematics but counting is done in base 10; there are 10 individual words for 1 to 10 , the word for one is "yaella", the word for 10 is "poo", the word (really expression) for 11 is "poo mahun yaella", which literally translates to 10 on top of 1 ie 10+1...I can only think they stopped at 10 words because there are ten fingers to do elementary arithmetic?? granted there are other words for high magnitudes...there are apparently other languages with other bases besides 10 so who knows...possibly due to using other cultural artifacts to serve as digits.
Yeah, basically. There are other ways to count on your hands though. I seem to recall there are places that count on their hands using the 12 finger joints (or segments), and it works just as well.
Edit: To clarify, they use their thumb to point to the joints or segments on each of their fingers, in turn.
That would mean 1/2 doesn't have a finite expansion, but why would it? Most numbers don't.
The alternative later way the Greeks (e.g. Euclid, Archimedes) wrote numbers was using their alphabet for 1–9, using separate letters for 10–90 and 100–900, then writing 1000–9000 with a “thousand” symbol plus the letter for 1–9 on top. Overall also a big pain in the butt.
Here is a more complete one: https://thonyc.wordpress.com/2017/02/10/the-widespread-and-p...
(Surfaced on HN with https://news.ycombinator.com/item?id=13636277 )
60x 60x 60x 60
The are
6x 10x 6x 10
Much simpler than it sounds from "base 60".
In fact they did this in ancient Egypt too!
http://www.mathnstuff.com/math/spoken/here/2class/60/egyptm....
In conversation, it's just the I-M/0-1000 range for instance, learning how to notate that number is a different issue.
Notation is somewhat straight-forward: When you get to 4, the number undergoes a state change to 'subtract from the next number, dropping everything behind it, and we're in a new state so "the" number is changed to 5' so III becomes IV. Otherwise, just keep adding 1. That should work, adding to the previous rules as the state changes. Going to X leaves the V rules in place.
So it would be a XXXII-bit computer, for example.