Oh, I like this one

Alright, so, we'll revolve around two parts, first the advantages of the Hindu-Arabic numerals, then the history of mathematics and how they came to standardize Hindu-Arabic numerals

It's rather hard to say that a certain language is superior compared to others. Research has shown that when you compound the density of information with the speed of speech of languages, all languages tend to have the same rate of transmission of informations, which is 39bits per second. The only defining way of defining what would be an optimal language would depend on the tradeoff you're trying to strike between ease of learning (Esperanto for example) and precision, but since it's a tradeoff there is no absolute better language. There is no revolutionizing feature a single language has and that makes it superior to the other. Moreover, the fact that everyone learns language from their environment also limits widespread efforts of language uniformization - as seen, for example, under the attempts of the French 3rd Republic of eradicating local french dialects.

On the other hand, there are ways of counting that are superior to others - especially the ones that have what we call positional notation. What does it mean? It means when we use numbers, the contribution of a digit to the value is the value of the digit multiplied by a factor determined by the position of the digit. This factor is called the base of the system. In arabic numerals we use base 10 but there are many bases in positionnal number systems - base 2 (binary), base 8 (octal), base 10(decimal) , base 12(duodecimal) , base 16 (hexadecimal), base 60(babylonian)...

What it means is, for example, in base 10, 42 is 4x10^1 + 2x10^0. In base 2, 10 is 1x2^1 + 0x2^0=2.

So why are arabic numerals superior? There are several advantages.

-Positional notation allows for easy representations of arbitrarily large numbers

-The idea of positional fractions come rather easily, as, say, 4.2 = 4x10^0 + 2x10^(-1) - some traces of it were found as far back as the Xth century. So, easier to express non interger values

-It's much easier to use to calculate stuff. For addition, substraction and multiplication, you just align the numbers, do your individual operations, put them in the right position and you're done. Think about how hard it would be to calculate, for example, 48x11 (equals 48 times 10 plus 48, equals 528)... But it's actually XXXXVIII times XI. (quick parentheses - there is no evidence to suggest the current interpretation of roman numerals that "a symbol before a higher value symbol is a substraction" - as in "IV = 5-1=4" was ever used historically - the roman numeric system is already complicated enough as is to not further complicate it with perfectly avoidable subtractions. So XXXXVIII rather than XLVIII)

Even knowing that X times X = C, X times V = L, that makes CCCC + L + XXX + XXXXVIII = CCCC + L + L + XXVIII = CCCCC + XXVIII = DXXVIII. It's much lenghtier because one you have multiplied the numbers you need to squash them together to recover the expression using the least amount of symbols. What about divisions? The answer is... We don't know. The most comprehensive source we have on the topic is Friedlein (1869) and he hypothesizes that romans used abaci (plural of abacus) to compute division. Unfortunately, unlike the Greeks, Romans were not really interested in writing down theoretical mathematics, so we have no idea of how they calculated really - the aforementioned example is just me using the most efficient possible way, but they may have just used repeated additions, or the much more efficient powers of 10 methods - which the Greeks knew. We simply don't know.

-Arabic numerals have this small dot that is called sifr which is the root for the french "chiffre" (digit) and the english "cipher" : the zero. The influence of the idea of zero on the entire field of mathematics cannot be overstated and would deserve its own post. Let's just say that you cannot build an advanced theory of mathematics without the 0 (it has been independently invented no less than 3 times), and roman numerals do not have a zero.

-As a freebie : it's much easier to check calculations made with arabic numerals rather than roman numerals. In roman numerals in most cases the only way to check a calculation is to redo it entirely. In arabic numerals it's not necessary. That is a huge advantage.

Thanks to the use of figures and fractions, Arabic mathematics allowed to handle rational numbers with an efficiency that could not be matched by the previous Roman system. The fundamental financial instrument of the period – the bill of exchange – could not have been developed in the absence of Hindu-Arabic numerals, as it was based on a proportion between fractions. (source below)

So in a few words, roman numerals were unwieldy and generally not terribly great at doing what numbers do. Gauss once lamented on the fact that Archimedes had imagined a positional numeric system but failed to realize the importance of his discovery. Gauss said that if Archimedes had pushed with his system, mathematics would have earned centuries worth of progress.

As for why the positional babylonian numerals weren't adopted, base 60 (59 symbols) made them hard to use for non specialist. It is today generally agreed upon that the superior base is 12 as it minimizes the amount of non simplifiable fractions while retaining easy readability. In any case, we're stuck with base 10.

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This may be a question that needs input from linguists or mathematicians, because both of these fields are at play here.

I think you may be getting at this in your post, but it's worth noting that there actually are many different and diverse numeral systems used in modern language, not all in base 10. Many languages use mixtures of bases - for instance, 79 in French is soixante dix-neuf (60 + 10 + 9) and in Welsh is pedwar ar bymtheg a thrigain (4 + (5 + 10) + (3 × 20)) or (somewhat archaic) pedwar ugain namyn un (4 × 20 − 1). In English, one could say "four score less one", as in the famous Gettysburg Address representing "87 years ago" as "four score and seven years ago".

On the mathematical side of things, mathematicians like the authors of this textbook make the point that what makes the hindu-arabic system unique among numeral systems is its efficiency and its utility as a tool for arithmetic. This functionality is ascribed to the combination of its positional system and use of zero:

Positional System: It is also known as place-value notation....working in base-1-, ten different digits 0,1,...,9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in 304=3100 + 010 + 4*1..Note that zero, which is not needed in the other [numeral] systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu-Arabic numeral system, which originated in India and is now used throughout the world, is a positional base-10 system. Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 1-; a positional sytem needs only ten different symbols (assuming that it uses base 10).

The authors are basically saying that the everyday counting done in spoken languages is simple enough that you could go about it in many ways, which we do. However, doing lots of counting, with large numbers and complex operations, requires efficiency to be practical. If it took as long to write "1,245,345.84" as it does to say it verbally, ("One million two-hundred and forty-five thousand three-hundred and forty five and eighty-four hundredths"), then it would be really challenging to, say, keep a complete ledger of foreign currency and commodity exchange data, or record the measured results of complex scientific experiments. You CAN do these things with other systems, but it's just not as efficient as using a positional system with zeroes as placeholders. In this sense you could say that people adopted the system 'organically' as a matter of practicality, because it was simply the best system to accomplish increasingly complex quantitative tasks of science, engineering, finance, and administration that most modern societies have gone through.

This is where the historical factors come in - yes, a positional numeral system with zeroes as palcehodlers is the most efficient, but why was it spread the way it was? Because, it was first adopted in Europe as a practical system to manage trade with the Islamic / Indo-Iranian world, and European dominance and reach during the colonial period meant that this system was not only spread directly through colonization but also copied by societies that were genuinely shocked by how far they had fallen behind the European powers and their technological, economic, and administrative accomplishments in the modern era. If you are a late-1800's Japan or China or Vietnam, and the British and French and Dutch and American and German navies are running armored steam ships with naval guns into your harbors while your navy is still using wood-hulled sailing ships, then your best bet for survival is to take whatever systems or knowledge allowed the European powers to do that, and copy it - put it in your universities, send your brightest students abroad, create a national curriculum, and catch up to them as quickly as you can. This is why the arabic numerals became so ubiquitous in education and technology, even though everyday speach in societies across the world use traditional numeration. This dynamic is also why most modern sciences are biased towards western conventions, despite being practiced on an international scale. Students and academics all over the world routinely use European nomenclature as standard practice in fields like medicine, chemistry, taxonomy, etc. so it's not just an "arabic numerals" thing, it's a "European colonialism and scientific advancement" thing, too.

To me, the question was like comparing apples and sausages, because languages are very different things than written notation for math. To me, it would be better to compare numeric systems to alphabets.

Languages contain information which may be particular and unique to the locations and cultural history. As such, there can be a cultural interest in not making it too accessible to outsiders (shibboleths, slang, inside terms, jargon, allusions to events and stories.) One language never maps on another exactly. We describe it as what we speak, gesture (in the case of sign languages), and write. The grammar and rules of languages do vary, even in the same language over time.

Mathematics is formulated on logic, and is more about quantities. Crudely, "One is one is one" no matter what the language, at least as far as math is concerned, as long as you understand the numerical system being used. So it is easy to translate from native numerical systems to the international number systems used.

Alphabets are a way to write down languages by their sounds (syllabaries-- Sanskrit is one-- and pictograms predated alphabets and still are in use.) Languages have different sounds. Even English sounds different from dialect to dialect. So even the Roman alphabet diversified over time, and is not completely phonetic. The International Phonetic Alphabet is rarely used except by linguists; by 2005 it had 122 distinct symbols, many which represent sounds that are not found in many languages. Even when you can write down the sounds, you can't always write down the tone of voice, the nonverbal aspects of language. By their nature, alphabets are a simplified representation of spoken language. They are also a late invention.

Writing math is also a culturally late invention. Some cultures never actually had any math words to speak of before contact in the 20th century.

Even when they did math, for even literate societies, it wasn't just speaking or talking math.

They used their fingers and tallies on sticks, abaci, clay balls, and checkers, sticks and string. These were fairly diverse. The Venerable Bede recorded one finger counting system for math. There were many others.

I even own a book, "Fingertalk" which purports to make it easy to add, subtract, multiply and divide on your fingers, and large numbers too. It's a lot easier than the awkward ASL number system, IMO.

For a fuller answer to your question on the history of numbers and counting, I recommend "Pi in the Sky: Counting, Thinking and Being" by John D. Barrow. He covers finger (and body) counts, tally systems for livestock, linguistic-- word based counting systems, and other ways of counting things.Instead of comparing to language, think of math also as simplified representation of abstract reality and relationships. By focusing on quantities and other concepts, you can boil down a complex activity such as cooking into symbols and a formula.

As such, math too is always evolving with our understanding, and is applied to various problems, often culture-bound concepts. But while it can be expressed symbolically, with symbols we can manipulate, it's not language. We don't use it the same way. The Greeks used to do geometry with sticks and string to do basic computations. We have computers do math-- which they do very well-- in binary (on and off.)-- well beyond our capacities.

Even across countries, 1,00 may mean 1.00 (number 1 to two places) in India, and look like an error here in the US for 1,000 (meaning one thousand.) With a set formula like scientific notation, it's easy to correct for such local quirks. Math mistakes can cost money, time, and lives. Reading math needs not to be too hard, especially across international commerce. Many local number systems are used traditionally/culturally only, to stand for the number words themselves. Chinese and Japanese certainly have their own traditional symbols for numbers, but it's easy to be fluent in Arabic numerals too, and more useful as it allows for using zero.

But beyond basic math, things get complex. A gander at calculus, topology, algebra, geometry, trignometry, differential equations, statistics, exponential notation, and symbolic logic will show you different notations (greek, etc.) and operations being added to-- or completely replacing-- the basic Arabic numerals.

Computers run on binary (on/off) which is often notated as 0101011... But they can create various symbols as programmed, to express math notation. Unfortunately a lot of advanced math notation is quite hard to type on computers without special software, which is why you see a lot of borrowing of pre-existing symbols. I look forward to the day I see math notations done in emoticons! I know that day is coming.

There is also hexadecimal, often used in codes and computers. These are symbolized by the letters 0-10 + ABCDEF. If you have ever had to input a code key with letters mixed in numbers (but never anything higher than F), that key is likely hexadecimal.

In addition, cryptography often uses substitution codes, and prime numbers also feature in secure communications on computers since they are hard to deduce. For more information on this kind of math, "Things to Do and Make in the 4th Dimension" by Matt Parker has lots on the usefulness of prime numbers in cryptography and banking.

In addition, there's a field of number theory. This math analyzes the very concept of number itself. A wonderful book to try is Eugenia Cheng's "How to bake Pi"

An internationally understandable numerical system is very useful for science, overall flow of knowledge, and global commerce, and yes, there's a colonial aspect to all that, as mentioned by others.

Math books written in native languages can be hard to obtain, and when it comes down to the symbols, why not just teach commonly used math symbols, so they can study and use advanced math resources from aboard and use computers to do math? Many mathematicians in the past learned math from books written aboard in foreign languages-- precisely because they could understand (and deduce) the math notation itself.

But to me, if you can do math with stones (mandela game in Africa), abaci, computers, on your fingers, or with checkers, and written symbols, then there is no need to change your language to fit the latest math-- in fact, the only changes needed are to teach it better.