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

washingtoncitypaper.com

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

#31

Post by Sharlin »

hencq wrote: post #26
dwheeler wrote: post #8

In general, with an abacus. Roman numerals for generally used for recording information, not for calculating with them. The article emphasizes how easy it is to add and subtract with Roman numeral notation, but everything else I've read emphasizes the Abacus even for that. After all, most people today calculate with calculators as well, and we have a snazzy Hindu-Arabic system for numbers.

It's actually somewhat surprising to me that the Romans didn't invent the concept of zero, when they used the abacus for daily calculations. With the benefit of 20:20 hindsight you'd think that the concept of zero would follow quite naturally from the concept of an empty column on the abacus.

But that's the thing about Roman numerals: you don't need a placeholder number to represent empty columns. And for "what is XVI subtracted from XVI" they could just use a word meaning "nothing", such as nihil or nihilum. The need for the concept of zero as we understand it really only arises together with a place-value system.

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

#32

Post by Sharlin »

gowld wrote: post #10
monadic2 wrote: post #7

Earlier quoted context omitted.

...in terms of digits, which is a very odd quality to optimize for.

...in terms of only the worst case numeral. Common numerals like II III VII VIII have worse length, while I IV VI are same, and V X IX are better. And factoring in per-digit cost, Roman numerals up to 999 have 5 distinct digits, 46% more cost per digit than base 3, making it worse than base 3 in almost every case, information theoretically. (You could win some back with a huffman encoding, though)

Note that as mentioned in the article, Romans very rarely used the subtractive notation. So in general 4 was IIII and 9 was VIIII.

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

#34

Post by DNied »

k__ wrote: post #18
bhntr3 wrote: 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…

"Did the clumsiness of doing calculations in Roman numerals keep them from developing more complex systems of numerical calculation?" Probably? I mean, look what the world achieved after it left roman numerals behind.

> look what the world achieved after it left roman numerals behind.

Look what the world achieved after we started wearing button shirts.

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

#35

Post by AnimalMuppet »

goatinaboat wrote: post #22
k__ wrote: post #18

Earlier quoted context omitted.

"Did the clumsiness of doing calculations in Roman numerals keep them from developing more complex systems of numerical calculation?" Probably? I mean, look what the world achieved after it left roman numerals behind.

Probably? I mean, look what the world achieved after it left roman numerals behind. Yet the Romans were able to construct aqueducts that are still standing, and a road network spanning thousands of miles, and many other great feats of civil engineering.

Sure. But they did those things by experience and rules of thumb. They didn't do a real stress analysis on those aqueducts, for instance.

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

#36

Post by mdiesel »

dmurray wrote: post #25
forinti wrote: post #19

Earlier quoted context omitted.

Suppose there was a basketball game between Athens and Rome (go Athens!) and you have a 3 digit scoreboard. You would need 19 base 3 digits and only 15 roman numerals (plus N for zero) in order to represent every number. In base three, you sometimes have all three digits equal. With roman numerals, you can reuse the same digit in different positions.

I don't understand this. 999 is less than 3^7, so you can represent any number up to 999 with just seven base-3 digits. Where does the 19 come from?

I play village cricket, and the scoreboard has cards with numbers on, then hooks to hang them up depending on the score.

The problem is: Given the full range of possible (or at least plausible) scores, how many of the cards do we need for a full set?

So let's simplify it to just a run tally. You could be 111, so you'd need at least 3 of the 1 cards etc. Allow for scoring up to 999 (unlikely) and that's 29 cards to keep somewhere (only 2 zeros needed)

In base 3, you need 7 digits, but only 3 cards per, so we are doing better with 19 cards needed (21=3*7, but don't need all zeros, and that gets you to 1093 so for 999 you could save another)

In roman numerals, You'd need an M, a D, 3 Cs, 3 Ls, 3 Xs, 1 V and 3 Is. Total is 15 cards.

Can we do better? Good question.

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

#37

Post by crazygringo »

Sharlin wrote: post #31
hencq wrote: post #26

Earlier quoted context omitted.

It's actually somewhat surprising to me that the Romans didn't invent the concept of zero, when they used the abacus for daily calculations. With the benefit of 20:20 hindsight you'd think that the concept of zero would follow quite naturally from the concept of an empty column on the abacus.

But that's the thing about Roman numerals: you don't need a placeholder number to represent empty columns. And for "what is XVI subtracted from XVI" they could just use a word meaning "nothing", such as nihil or nihilum . The need for the concept of zero as we understand it really only arises together with a place-value system.

Thanks for explaining that clearly. I've always been so baffled by people who claim that some society didn't have a concept for zero, as if "inventing" zero marks some major advance in intelligence.

Every culture has a concept of "nothing" which works for zero. The ancient Greeks debated over whether nothing was a number or not, but that's just a semantic splitting of hairs.

At some point a symbol for nothing becomes useful so you invent a number-like notation for it. But that's just a matter of convenience. It's not some great conceptual leap.

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

#38

Post by kccqzy »

Grustaf wrote: post #14

Why would it be significantly harder than using some other system of numerals?

When I was learning programming, one of the beginner's exercises was to implement addition, subtraction, and multiplication of natural numbers "by hand" (division wasn't covered). There you learn to formalize the grade school algorithms you know by heart into concrete code, a necessary mental process for beginning programmers.

At the end of the session, the teacher asked us to try the same with Roman numerals. That was when I knew it's significantly harder.

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

#39

Post by mdiesel »

AnimalMuppet wrote: post #35
goatinaboat wrote: post #22

Earlier quoted context omitted.

Probably? I mean, look what the world achieved after it left roman numerals behind. Yet the Romans were able to construct aqueducts that are still standing, and a road network spanning thousands of miles, and many other great feats of civil engineering.

Sure. But they did those things by experience and rules of thumb. They didn't do a real stress analysis on those aqueducts, for instance.

For those interested in reading further, the ideal (unloaded) shape of an arch isn't a semicircle, but a catenary.

It sounds so simple: so hangs the chain, stands the arch. Took until Hooke in the 17th century before that was written down, though there are earlier (15th century) examples in architecture.

The Romans were still working on the Greek ideology that the circle was the perfect shape. Not to belittle what they did, but the key advances were really in concrete and having an authoritarian empire giving unprecedented resources to public works.

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

#40

Post by rozgo »

andrewl wrote: 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…

This is absolutely the case for programming languages. From Haskell to Rust. Programming languages shape the way we think and reason about the problem. Sometimes it does this so well, it leaves us starring into the abyss. Facing dead on the problem domain—after all sides quests have been removed. Once you can delegate away all the busy work, you can focus on bigger and better abstractions.
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