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How did anyone do math in Roman numerals? (2017)

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Re: How did anyone do math in Roman numerals? (2017)

#61
post #9

In An Introduction to Mathematics (1911) Alfred North Whitehead wrote: 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. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably…

I'm reading a book called Mathematics from the Birth of Numbers by Jan Gullberg and it covers these systems from various cultures in depth. Great book and easy to read even for the non mathematician. It reads like a cross between a history book and a technical reference. This is literally the first topic in the book.

Re: How did anyone do math in Roman numerals? (2017)

#62
post #9

In An Introduction to Mathematics (1911) Alfred North Whitehead wrote: 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. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably…

Adding Roman numerals, at least, isn’t hard. If it was I doubt Roman numerals would have ever lasted. Arabic is still, in my opinion, easier to add—from the perspective of a lifetime spent exclusively doing arithmetic in Arabic numerals—but it’s not much easier until you’ve memorized all 50 unique sums of one-digit numbers. Multiplication, though, that’s the real difference maker.

Re: How did anyone do math in Roman numerals? (2017)

#63
I have read that the uptake of Arabic numerals was actually fairly slow and fraught in Europe, but of course can't put my hands on any references.

There's a reddit thread [0] that might be some of what I saw, and my wife does paleography work where she runs across books of accounts that are rendered in Roman numerals, because that's how formal accounts were prepared, even if the actual accounting was done by other means.

That same reddit thread has link to an "algorists vs abacists " article [1] which purports to back this up, but I can't confirm because the article is paywalled for me.

Edit: Moved/fixed links.

[0] https://www.reddit.com/r/AskHistorians/comments/12m0vp/how_a...

[1] https://www.jstor.org/stable/pdf/2686479.pdf?seq=1

Re: How did anyone do math in Roman numerals? (2017)

#64
post #9

In An Introduction to Mathematics (1911) Alfred North Whitehead wrote: 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. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably…

I wonder if arabic numerals really are the best system.

Too much inertia is invested in them now but I wonder if a yet easier representation exists. For example, I know that there was a brief push to use quaternions in physics.

Re: How did anyone do math in Roman numerals? (2017)

#65
Contrast Roman numerals with the rod calculus[0] invented in Ancient China. Wikipedia has a list of algorithms for calculating with rods, from the usual arithmetic operations to fractions, division, square and cube roots, Gaussian elimination, and solving polynomials.

It would seem that such tasks would be extremely difficult for someone working with the Roman numerals.

[0] https://en.wikipedia.org/wiki/Rod_calculus

Re: How did anyone do math in Roman numerals? (2017)

#66
post #64
post #9

In An Introduction to Mathematics (1911) Alfred North Whitehead wrote: 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. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably…

I wonder if arabic numerals really are the best system. Too much inertia is invested in them now but I wonder if a yet easier representation exists. For example, I know that there was a brief push to use quaternions in physics.

Base 12 is likely better than base 10. Twelve is the number of phalanges on your hand that you can touch with your thumb, so if that counting system had caught on and stuck we'd likely have a better base for divisibility.

Re: How did anyone do math in Roman numerals? (2017)

#67
post #9

In An Introduction to Mathematics (1911) Alfred North Whitehead wrote: 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. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably…

Adding Roman numerals, at least, isn’t hard. If it was I doubt Roman numerals would have ever lasted. Arabic is still, in my opinion, easier to add—from the perspective of a lifetime spent exclusively doing arithmetic in Arabic numerals—but it’s not much easier until you’ve memorized all 50 unique sums of one-digit numbers. Multiplication, though, that’s the real difference maker.

> Adding Roman numerals, at least, isn’t hard.

I find it insanely difficult - but as you note, a lifetime of arabic numerals, and a lack of skill in appropriate tools (eg. an abacus) will skew that comparison.

> ... but it’s not much easier until you’ve memorized all 50 unique sums of one-digit numbers.

This doesn't feel right.

I don't think I have memorised the sum of all pairs of 1-digit numbers - but contemplating this now, it's impossible for me to be sure. I'm not sure where multiplication kicks in for breaking down larger numbers into quotient and divider for me, let alone 'most people'.

Given the maximum value (sum) of two x 1-digit numbers is 18, a naive assumption is the permutations don't really number 50 (I get 45) - and given that roman numerals didn't have a zero, a fairer comparison would be [1-9][1-9] (36 unique combinations)

Either way, in any counting system there's presumably a similar 'memorisation' gate you have to pass for the fundamental set. With roman numerals there was historically 7 I think - I, V, X, L, C, D, M - and summing those wouldn't be anywhere near as straightforward as summing sets of single-digits, so I don't think the comparison of this requirement is as skewed against arabic numerals as you suggest.

Re: How did anyone do math in Roman numerals? (2017)

#68

Earlier quoted context omitted.

I think that for the majority of people throughout history, numbers were inseparable from numerals, i.e. notation for numbers. This would explain why people are far more comfortable with the notion of real numbers (which despite their name are very very strange in a lot of ways) than imaginary and complex numbers. Even their names betray the difference. However, one has a common notation that everyone has learned whe…

I would clarify that to say numbers were inseparable from words for them. Writing has only existed a short time of our history, so numerals are pretty recent. And the "meaning" of zero as a number like others along a number line, rather than as mere notation for "nothing", I assume only ever became necessary with the invention of negative numbers. With addition, multiplication and division, zero simply does nothing o…

From my understanding (and I well could be wrong here), Europeans inherited from Greek a mathematics system that favored geometry and tended to abhor algebra. Concepts like integers, rationals, and irrational numbers are all pretty easy to explore and explain with geometry. By contrast, zero, negative numbers, and imaginary numbers create absurdities in geometry (how can a line have length 0? -2? 3 + 4i?). Moreover, even as algebra is introduced to Europeans via the Arabs, I can see people resisting algebra in part because it introduces these absurdities and paradoxes that need explanation.

As far as I can tell from the historical record (and it doesn't help that modern histories tends to describe historical mathematical discoveries in modern terms, meaning it's difficult to work out as a lay person in what terms the historical discoverer understood their own work), it looks like the acceptance of zero, negative numbers, and complex numbers are more or less concurrent, and this also seems to coincide with the shift in mathematics from being predominantly geometric to algebraic.

Re: How did anyone do math in Roman numerals? (2017)

#69
post #12

I feel like the article misses the most interesting question about Roman numerals and Roman (Greek really) math. How did the numerical system influence the math that they developed and used? The Greeks were really into geometry using the compass and straight edge so they actually did a lot of math without really needing numbers at all. They viewed calculation as less worthy of mathematicians and my understanding is t…

The Greeks had several competing numeral systems (https://en.wikipedia.org/wiki/Greek_numerals) including a decimal based system. What I do not get is how come the Romans adopted (if they adopted from the Greeks) the most unwieldy one.

Re: How did anyone do math in Roman numerals? (2017)

#70
Not really related to the article per se but I always find it interesting how one may become tempted to say "this alternative to a thing I already know makes so much sense, why don't we always use it?"

I felt the same way when encountering Chinese numbers via Japanese. If 二 is two, 十 is ten, 四 is four, and twenty-four is 二十四, that's so clear! Two tens and four!

I quickly decided that this number system, though something I'd obviously need to learn and become acquainted with if my Japanese learning were ever to progress, wasn't necessarily as easy as I initially imagined. Yes, there are no places, but numbers in this system are grouped at different boundaries — not every thousand but every ten-thousand.

So 六十七億八千三百一万五千四百二十一 breaks up as sixty-seven hundred-thousands eight-thousand-three-hundred-and-one ten-thousands five-thousand-four-hundred and two-tens-and-one — and obviously, that's not quite how we would represent six-(billion/thousand-million) seven-hundred-and-eighty-three-million fifteen-thousand-four-hundred-and-twenty-one, or rather 6,783,015,421.

I post this not to discuss the positives or negatives on the Chinese number system compared to the Arabic one or vice-versa. Rather, just how one's imagination can be so easily captured by the apparent simplicity of an alternative to that with which one is familiar, almost to the point of wanting to adopt it altogether. The realisation of where things get tricky for oneself, often not coming until quite a bit later, sometimes doesn't come until later.

For myself, I tried using roman numerals for my own math for a long time but stopped when I found division too brain-breaking!

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