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So one element which kinda wraps all of these concepts together is what some have called the "Golden Age of Islam".[1] The [literal] Empire which emerged from Muhammad's legacy became a source of great intellectual thinking (while Europe had just entered its dark age). This makes sense, being the place of some of the oldest civilizations on Earth, and surrounded by several other great civilizations.

During this time (a few hundred years after the fall of Rome) the texts of the ancient Greek philosophers and scientists became quite popular among Muslim thinkers. One could argue that this Golden Age included a natural continuation of the works of Aristotle, Plato, etc. Furthermore, they adopted a new system of representing numbers from India which turned out to be a convenient language for formalizing the concepts of Algebra (done by Muhammad ibn Musa al-Khwarizmi in the early 800's [2]).

Sadly I can't recommend any books -- this was conjured up from my memories of this subject from College some time ago (with the guiding help of Wikipedia to fill in some gaps).

[1] https://en.wikipedia.org/wiki/Islamic_Golden_Age

[2] https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi

For anyone interested in this era, I can recommend the four-part series hosted by Radio War Nerd: https://www.podchaser.com/podcasts/war-nerd-radio-subscriber....

The podcast in general is quite interesting, even if you're not as left-leaning as the hosts are. But this series in particular was fun to listen to.

I'm slowly reading through a book called the "History of Mathematical Notations" by Florian Cajori and it sounds like just what you're looking for.

It goes into far more detail than probably anybody would wish for on the histories of various symbols and ways of writing addition, roots, ratios etc. Many of these notations were used only by a few medieval scholars for a few decades but they usually try to optimize for efficiency or clarity in some creative way. Really fascinating stuff.

> All thanks to the folks that laid down the foundation to Arabic numerals.

As an Indian, it is particularly weird for me to see the achievements on Indian pioneer mathematicians attributed to a totally different group of people. Brahmagupta, Pingala, Aryabhatta etc. should be correctly attributed and celebrated for their towering contributions to laying the foundations of Maths as we see today.

Right, yes, sorry. I'm just a casual to the history of math. My bad.

As an Indian, it is particularly weird for me to see the achievements on Indian pioneer mathematicians attributed to a totally different group of people.

Unfortunately, the internet is full of this sort of thing. Wikipedia is especially rife with revisions that change inventors and originators in jingoistic ways.

See also: The eight thousand tiny museums in villages scattered across the U.K. all celebrating the local "inventors" of different things that were already invented in other countries long before.

The Internet did not trick Leonardo "Fibonacci" of Pisa into calling the number system he was advocating "Arabic" in the thirteenth century.

Fibonacci called them "Indian figures" and arithmetic the "Indian method". So you are technically correct that the internet did not trick him into calling them "Arabic": he was not tricked at all on this point!

> As my father was a public official away from our homeland in the Bugia customshouse established for the Pisan merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me to be in the study of mathematics and to be taught for some days. There from a marvelous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learnt from them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learnt from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method. Therefore strictly embracing the Indian method, and attentive to the study of it, from mine own sense adding some, and some more still from the subtle Euclidean geometric art, applying the sum that I was able to perceive to this book, I worked to put it together in xv distinct chapters, showing certain proof for almost everything that I put in, so that further, this method perfected above the rest, this science is instructed to the eager, and to the Italian people above all others, who up to now are found without a minimum. If, by chance, something less or more proper or necessary I omitted, your indulgence for me is entreated, as there is no one who is without fault, and in all things is altogether circumspect.

I just wanted to have a casual, nerdy discussion of math god damn it. That's all. How did this turn to a politically correct issue already?

> Maybe the actual number representation with our modern numerals is the worst way to track count, but the way to manipulate those numbers is the best that ever was from what I can tell. All thanks to the folks that laid down the foundation to Arabic-Hindu numerals.

Hmm? The manipulation and the written form are fully independent. Before the introduction of Arabic numerals to Europe, this manipulation was already common; that's what an abacus was for. It's just that you wrote your result down in Roman numerals rather than as a picture of an arrangement of beads.

Yes and no. The format of numbers is important to how you end up manipulating them. The way we do long division with 3 or more digit numbers cannot translate to roman numerals. A different method is required. This is part of my point. No one goes into how to do complex maths with Roman numerals, Cunieform or anything else. They're just "look at how you can display a count". Once you learn to add, substract, multiply or divide 2 digit numbers on paper with Arabic-Hindu numerals, 30 digit numbers are trivial, only time consuming. Having a system that uses entirely different characters to represent hundreds, five-hundreds, thousands and more, that changes the math system.

There was a British show I watched as a kid that played in the US which made a light hearted joke regarding this. I think it was called How 2 or something like that. But it was a Welsh woman that showed how 12/2 can equal 7. XII, when you erase the bottom half, it gives you VII. While a joke, it always bothered me that no one ever talked about how the Roman ACTUALLY did math with their number system. How in the hell do you XII/II on paper by showing your work? Because our 12/2 long division method can translate easily to 1212/2. How do you do the same with MCCXII/II? I find it hard to believe there isn't one. Just, no one talks about it.

> The format of numbers is important to how you end up manipulating them.

I just pointed out:

>> Before the introduction of Arabic numerals to Europe, this manipulation was already common; that's what an abacus was for.

The abacus doesn't represent numbers in Roman numerals. It represents them positionally. This poses no problems because conversion between representations is trivial.

> How in the hell do you XII/II on paper by showing your work?

That's as easy as it gets; half of X is V and half of II is I. This is like asking "how in the hell do you do 7 - 2 on paper by showing your work?". Dividing by 2 is a primitive that is always easy to do in Roman numerals, and is relied on heavily in the method you'd use to multiply them, which you can in fact read about. I'll do 47x9:

  XXXXVII  *  VIIII
    XXIII  *  XVIII
       XI  *  XXXVI
        V  *  LXXII
       II  _  CXXXXIIII
        I  *  CCLXXXVIII
  
  CCLXXXVIII + LXXII + XXXVI + XVIII + VIIII =
  = CC LL XXXXXXXXX VVVV IIIIIIIIIIIII =
  = CC C LXXXX XX XIII =
  = CCCCXXIII = 423

I thought 9 was IX

The prefix = subtraction convention came late in the history of Roman numerals. I think it might even postdate the introduction of Arabic numerals, but I can't say for sure since I'm going from memory here. If I were less lazy, I'd look at wikipedia which doubtless can give a more definitive answer.

Wikipedia has plenty of citations to classical Rome.

Colosseum gate 44: XLIIII (not, for whatever reason, XLIV, or XXXXIIII)

18th legion: XIIX

22nd legion: IIXX

There was evidently not a strict standard.

(Note also that the symbols we use for high values, D and M, weren't really what the Romans used. You can see Roman abaci with columns labeled with place values: https://en.wikipedia.org/wiki/File:RomanAbacusRecon.jpg https://thatsmaths.com/2019/07/04/what-did-the-romans-ever-d...

Those look the same; I don't know whether the one is a replica of the other or whether the form was just that standardized. But the columns are plainly labeled as, in ascending order, ϴ, I, X, C, ∞, ((|)), (((|))), and |X|.

I speculate that ∞ is a fancy way of writing (|); I believe our M descends from Roman (|) and our D from Roman |). )

You're free to write it that way if you want, but that just adds an unnecessary step to doing math with it.

It's why arithmetical calculation was a role one had. They used an abacus for the calculation of complex sums.

https://thatsmaths.com/2019/07/04/what-did-the-romans-ever-d...

So wait... math was never really done long hand in ancient times, but calculators (the abacus or any other form) were used instead?

In some respects, that makes sense. Paper like materials were never cheap and long-hand math uses a whole lot of paper. Bead calculators makes sense. I'd be curious to see if there's a correlation to "cheap" paper being plentiful right before the invention of complex math like Algebra.

With a quick 5 minute peek, "cheap" paper came from China to the middle east in the ~8th century. Algebra was recorded in the 9th in the middle east. So... eh, maybe. Requires more reading, but it's interesting. Sounds like an interesting infrastructure like correlation to technology.

> math was never really done long hand in ancient times, but calculators (the abacus or any other form) were used instead?

No. I provided an example of peasant multiplication in my response sidethread. (Well, "yes" if you mean using paper. Paper is hard to get. But you can multiply without needing a special tool; the algorithm is commonly executed by using piles of rocks and pits in the ground.) The abacus is a more advanced technology that took time to develop. (It does appear early in some form; see https://en.wikipedia.org/wiki/Abacus#History )

Yes this is largely correct. Calculation was not done with Roman (or Sumerian, Egyptian, Greek, ...) numerals per se, but mentally, with fingers, with piles of tokens, or with a counting board. But we are talking about moving tokens around on some markers drawn on the ground (or on a piece of cloth or animal skin or wooden board or whatever), not beads on rods like a Suanpan/Soroban used in East Asia today. My speculation is that many ancient game boards (e.g. backgammon, mancala) were also (or originally) counting boards.

(The Roman phrase 'calculos ponere' literally means 'place pebbles', as on a counting board; this is where our word 'calculate' comes from. To settle accounts with someone was to 'vocare aliquem ad calculos' or 'call them to the pebbles')

You should think of written numbers as a historical serialization and storage format. Something like the XML or JSON of the ancient world. And just like a modern computer, ancient computers first deserialized data to a more convenient internal representation for intermediate processing.

Arithmetic really took off in Europe with the advent of printed books (easier to teach written arithmetic tricks via books, whereas counting-board tricks were largely passed down orally and by demonstration) and of course with cheaper paper and widespread literacy.

The big advantages written arithmetic has vs. counting boards (despite being slower and having a steeper learning curve) is that all of the scratch work remains visible, which makes it easier to check for mistakes.

Roman numerals are a more direct representation of the state of a counting board which takes less practice to learn, especially for illiterates.

That's interesting that the root meaning of calculate is pebbles. I'm sure it's unintentional, but pebbles as a mnemonic for calculus is pretty good. Take the counting stones and pulverize them into dust and you get calculus.

The word is derived from calx [limestone/chalk]. Calculus is then a "little rock" -- the -ul- or -cul- infix is a diminutive form. (Analogous to "doggy" vs "dog" in English.)

> pulverize them into dust

While we're on the subject, the Latin for "dust" is pulvis. ;D

> The manipulation and the written form are fully independent.

No, they are related. The optimal written form for a base 10 manipulation system is a base 10 numeral written system.

> Before the introduction of Arabic numerals to Europe, this manipulation was already common; that's what an abacus was for.

And the abacus was base 10, wasn't it? So the natural (and optimal) way to write down that result is in a base 10 notation.

This Cistercian numerals are base 10000. Try doing a pen and paper multiplication or division using this system. No, seriously, try it. You'll understand why historically we settled on our decimal system. It's basically "divide and conquer" applied to mental math.

The good news is that 10000 has 10 as a factor, so base 10000 numerals come sort of natural to us, the learning curve isn't that high.

> This Cistercian numerals are base 10000. Try doing a pen and paper multiplication or division using this system.

Sure, in the same way that Babylonian cuneiform numerals are base 60, which is "they're in base 10, but you combine multiple symbols into the space of one glyph". The Cistercian numerals don't have 10,000 separate symbols. They have 10 symbols, and one glyph includes four positional slots in which a symbol appears. They're exactly equivalent to the Arabic numeral system we use now, and the pencil-and-paper multiplication system we use now transfers directly without problems.

It's a bit like hexadecimal and octal and binary like that, in that it's really trivial to convert between those and just see at a glance what the conversion is. Except with Cistercian it's even easier.

It would be exactly like hexadecimal and binary if the hex glyph for 11, instead of being "A", looked like

  1 0
  1 1

pedant alert: you mean B

Right you are.

Tangentially related: in modern math we do have alternative number systems to use for problems that are cumbersome with Arabic-Hindu numerals. Check out p-adic numbers on Wikipedia, for instance.

Yes, if I remember correctly, any n-ary number system is equivalent and you can translate any proof between them. It's just easier to work in one or another system.

p-adic is quite different though. It's not really that relevant to this conversation though as far as I can tell as it's not just about representing the integers or rationals in a different way.

https://en.wikipedia.org/wiki/P-adic_number

Cuneiform math isn't the most terrible. It's 'column' based, like the modern arabic version we have. However, at some times they lacked the 0 representation.

Roman numerals, however, are quite difficult. In addition to the problems with basic manipulations (27-9, or XXVI - IX), they also lacked any higher symbols than 1000 (M in Roman). So for 15,000 you have to write MMMMMMMMMMMMMMM. I'll spare you numbers like 1,000,000.

Here's a good video on how to do math in Cuneiform, and a brief discussion on Roman numerals:

https://www.youtube.com/watch?v=RR3zzQP3bII

Strangely, Numberphile has yet to do a video on Roman/Chinese number systems. Maybe because it's really hard to do math with them, and that's a bit boring to just show that it's hard.

EDIT: I have been misinformed, Chinese numbers are in fact easy to work with.

The Roman system is indeed difficult? What's difficult about the Chinese one? It seems basically identical to Arabic numerals except using different symbols.

> What was known on how accounting was done, the mathematic usage of that system was a complete pain in the ass.

Actually it wasn't that difficult and was even rather convenient given how they viewed the rest of the world (among other things they only had rationals, and a limited number)

Decimal systems didn't take over the circle, or timekeeping (and for the circle sometimes radians are easier). Decimal stock trading in the US is only a couple of decades old and within my memory countries switched to decimal currencies replacing systems that had survived centuries.

In fact the US uses a non-decimal measurement system which has numerous advantages (as well as disadvantages) for daily use.