It's interesting how the notation used can encourage or retard progress. For example, Leibniz's calculus notation was vastly superior to Newton's, and calculus theory advanced much more quickly where Leibniz's notation was used.
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 think a lot of my confusions when I first learned calculus would have been eliminated with a notation that clearly expressed that derivatives operate on whole functions, not on values. So the derivative of f(x) at x=0 is not some function of f(0), but it is derivative(f)(x). Also, even without derivatives, sometimes the expression f(x) refers to the whole function, sometimes just a particular value of it at a specific x.
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.
It's actually pretty common notation in a lot of fields, and mathematicians frequently switch between whichever of these notations is most convenient for a particular problem. Your point of course still stands for why isn't it the main one used in Calculus 1 courses, but it's common in differential equations courses in my experience (and PDE's often simplifies it further to just have D_x f as f_x). And many times where the "derivative" is not necessarily the "calc 1" derivative, you use notation similar to that, for example the covariant derivative [1] is sometimes written like that but with a 'nabla' instead of a 'D'. Some authors even use 'D' for covariant derivative along a path.
It's not much more useful than Newton's. The greatest point of Leibniz' notation is that you can do those things such as simplifying dx, or moving dy to the other side of the equality.
It is confusing to beginners, but it's very useful once you understand it.
What I can't understand is why anybody still used Newton's notation. Also, at least for me Leibniz notation tends to make pen-and-paper symbol manipulation simpler.
Notation is extremely important. It's basically a way to organize how you abstract a problem. If your abstraction is "bad" it will be harder to solve certain problems.
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.
This couldn't really be stated any more clearly[1]; well put. I'll only add that this is true for any variety of abstraction, natural language included. Abstractions encode the biases of their creators[2]. The 'power' of an abstraction comes from the set of things that are easily and concisely expressible; its primitives. However, this is balanced by the truths that are no longer easily expressible, because the encoding doesn't allow for it. There's a certain intuition that semantics and abstraction are tied tightly in this sense; you don't can't really convey what something means unless it's concisely expressible in the abstraction you're using. Slang, idioms, calculus, etc.
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[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.
I learned this in designing the D programming language. The original syntax for lambdas seemed natural from a compiler point of view, but from a user point of view it was awkward and pretty much unusable. Changing the syntax was a complete transformation in its usage.
Another excellent example of the power of good notation to aid understanding and increase efficiency is Dirac, or Bra-Ket notation[1] used in Quantum Mechanics.
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.
Mohammad al Khwarizmi's treatise on Algebra (the word comes from the title of the treatise) was written in plain Arabic prose. No notation at all.
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.
I prefer Mathematica's notation over Leibniz's, because it is not as ambiguous (dx can also mean d TIMES x, as a simple example, but also dy^2 can mean d(y^2) and (dy)^2)
I'd go stronger than this and say that Leibniz's notation is actively harmful. It is very useful for quickly doing certain kinds of computations, but at the expense of conceptual understanding for students. Obviously, it's fine to use whatever computational aids you want when you understanding things, but most students are taught nothing but this fragile notation.
I would point out it is d because latin typesetters back then often didn't have a greek typeface to print with and it is the closest to the greek letter delta δ. Once one understands it's δx and δy (or Δx and Δy) and today still today most people don't know how to get delta characters on their latin keyboards, then it is easy just to not use d in algebra and use for differential calculus only. Finally (Δx)^2 and Δ(x^2) are the same thing in differential calculus.
It really isn't that ambiguous in cursive, though. When writing out "dx" as product in cursive, the "d" and "x" characters are written separately. When writing "dx" as an infinitesimal, the characters are connected. Never had any issue with higher degree derivatives, either.
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.
A similar notation is Lagrange notation, which is really useful for studying derivatives as a member of a larger class of operators.
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.
An important facet to mathematics in general, that most are unaware of before studying it, is that the majority of proofs, especially those done in bachelor university courses, are purely notation. Other problems often become trivial to solve by using a different notation (e.g. polar form vs. points on the complex plane) as well.
An important facet to mathematics in general ... is that the majority of proofs ... are purely notation.
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?
Could you please elaborate. Just to take two elementary examples: the intermediate value theorem and the theorem that any two bases of a finite-dimensional vector space have the same number of vectors. I would have thought that both require mathematical ideas and not just notation.
"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race."
I always thought that roman numerals would be a simpler way to do basic arithmetic and might lend itself more to simple commerce. For example: III represents 3 things, so III + II = IIIII For simple commerce application that is simpler, I just have to then remember that IIIII = V, and VV = X and XXXXX = L, LL = C. Armed with just those simple rules I could probably get by in the market place in Rome.
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.
>1 is one thing, 2 represents 2 things, 3 represents 3 things and so on
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.
You’re speaking from a position of bias and ignorance (we all do this sometimes, but it’s important to be aware of), with a lifetime of familiarity with Hindu–Arabic number notation and almost no experience reading/writing Roman notation or translating back and forth between written numbers and pebbles or other tokens on a counting board, which is how most calculations were done in Roman times (persisting to this day in our word calculate, from the Latin word for pebble).
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 remember learning Roman numerals in 1st or 2nd grade and finding them to be a lot easier for basic arithmetic.
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.
I'm not 100% sure, but through anecdotal experience and from what I've read (although haven't vetted), we lose track of counters at ~8 of them. That is, III is easily discernable as 3 'I's, but take IIIIIIIII, and try to recognize how many 'I's are there immediately.
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.
Curious, why do you have to remember that 2 + 2 = 4 & 3 + 2 = 5? Once you know the values the symbols represent, at that point isn't it similar in simplicity to roman numerals?
II + II = IIII
2 + 2 = 4
Don't see how the latter problem lends itself to any more memorization beyond symbols
Because if you don't memorize your 'primitive' algebraic rules you'll end up just doing a pullback with roman numerals in the middle. Integers don't have any values, they're mathematical objects with certain properties. Asking about 'the value of 5' doesn't make sense unless you're trying to convert to another, already known, number system.
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.
You'd think you could just collect all the numbers and then translate, i.e. if I add some things and get XXVVVVVIIIIIIIIIII, I know that's XXVVVVVVVI = XXXXXVI = LVI.
> For example: III represents 3 things, so III + II = IIIII For simple commerce application that is simpler, I just have to then remember that IIIII = V, and VV = X and XXXXX = L, LL = C. Armed with just those simple rules I could probably get by in the market place in Rome.
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.
Just thinking about how they work here, it looks like Roman numerals are effectively a base 5 notation, with a special behavior for 5n - 1.
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.
In the general case maybe not, but to get by in a marketplace you'd need just those one he listed... like.. 3 or 4 things. 5, 10, 100, 1000. Most people can remember 3 or 4 things. Children can do it, let alone market vendors who are used to considering that 5 chickens = 1 goat. (Inflation these days!)
According to the article, that notation didn't come into use until the middle ages. So using IIII instead of IV was almost certainly acceptable in the context of transactions. Even if it wasn't, doing the conversion before addition/subtraction would be trivial (and the first step).
I believe some Roman inscriptions were found to use additive, i.e. III + I = IIII, VI + III = VIIII. Certainly I feel that nobody would look at IIII and be confused as to what it represents.
Lets say I am a farmer and I take eleven pigs to market. I start with XI pigs and sell three III. I know X = VV and I know V = IIIII so I expand XI = VVI = V IIIII I, then take way III sticks and am left with VIII. It is easier for simple primitive thing because the stick I represents a thing and the V represents 5 I's and X represents 2 V's. It is perhaps easier for someone like a farmer in Roman times to use than remembering arithmetic rules. It is more directly related to the physical world. At least that was my hypothesis.
Okay so the article is wrong about some points. We have evidence for both IIII and IV notation in classic roman archeological finds. The obvious example is the entrance fee doors around the colliseum in Rome, they're are engraved with numbers, in both forms of notation.
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.
It seems the subtractive notation was used for engravings, where keeping it short was important and nobody was going to do calculations with the numbers anyway.
> 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.
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.
Personally, I find distinguishing III and IIII quite hard in several fonts. However, the difference between [II and III] and [III and IV] is easier to read.
Arabic numeral: Cardinal number name | Ordinal number name, with conjugation information
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
From Cassell's Latin English Dictionary, second edition.
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.
...which is funny (the Asterix thing), because spoken French isn't exactly brilliant with numbers either: 99 for example is expressed as "eighty nineteen", 70 as "sixty ten".
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.
> 99 for example is expressed as "eighty nineteen"
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.
Same thing with Belgian French, much to my dismay.
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.
73 is soixante treize and is just this : a word (ok, two) one learns by heart. No French think that this is 60+13, but just 73.
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.
Readit News
https://en.wikipedia.org/wiki/Leibniz%27s_notation
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.
[1] https://en.wikipedia.org/wiki/Covariant_derivative
It is confusing to beginners, but it's very useful once you understand it.
Deleted Comment
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.
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[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
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.
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 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.
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The problem is IV isn't 6, it's 4.
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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.
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.
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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?
[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??
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.
http://www.youtube.com/watch?v=xnuetbfDLpE&t=1m14s
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.
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.
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