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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)

#21

Post by aswanson »

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…

Amazing, the nth level effects a superior notation can have in problem solving. The same consequences of a higher-order programming language.

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

#22

Post by goatinaboat »

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.

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.

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

#23

Post by shagie »

dhosek wrote: post #3

When I got my math teaching credential, there were bunches of interesting historical things we learned along the way including Egyptian Fractions https://en.wikipedia.org/wiki/Egyptian_fraction Never actually used any of it so most of it has evaporated from my memory along with calculating square roots by hand, but it's nice to know at least enough to be able to look up the information if I want it.

Egyptian fractions have an interesting property that may have been useful back then.

Consider the problem "How do you divide five things for eight people?"

Simple - cut everything into 1/8 and give each person five.

But Egyptian fractions give an even easier way. 5/8ths is 1/2 + 1/8. Divide four wholes into halves - give each person a one half. Take the unit and divide it into eights and give each person one. The answer to this problem is the way you write the number itself. This makes division of the items easier and simpler.

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

#24

Post by setr »

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?

The book Mathematics for the Million[0] suggests a number of limitations were created from the roman numeral system, largely due to the difficulty of division, making infinite series and even large (and extremely small) numbers difficult to work with, intuit and even see outright. As a specific example, it suggests the Achilles and the tortoise paradox[1] is trivially intuited and resolved in the decimal system, whereas no relationship between each division is made clear in the roman numerals

[0] https://archive.org/details/HogbenMathematicsForTheMillion/p...

[1] https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_an...

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

#25

Post by dmurray »

forinti wrote: post #19
gowld wrote: post #10

Earlier quoted context omitted.

...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)

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?

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

#26

Post by hencq »

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.

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

#27

Post by shultays »

Grustaf wrote: post #14

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

They probably had algorithms for it, but even then it sounds challenging.

Addition sounds easy and works mostly like how we do base 10 addition. I imagine they would first go for sub 5 part which is a bit exceptional and had to be manually. And then start grouping letters together like we do and create carries if they reach the next letter.

Subtraction sounds harder. It sounds close enough to our base 10 system but borrowing from next digit is much more complicated. Like subtracting D-I (500-1). The answer is CDXCIX but I am not sure how I can go there. And now imagine this with more complex numbers

But multiplication and division? in that weird base format? I won't even try

Keep in the mind that they don't convert or even think numbers in base 10 system like we do.

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

#28

Post by forinti »

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?

999 is 1101000. So, for MSB, you only need 1 (when it's zero, you just don't put anything there). All the other positions can be 0, 1, or 2. So you would need 1 + 6*3 digits for your scoreboard. You can't not have 6 2s, for example, because your score might be 222222.

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

#29

Post by dan-robertson »

I am very wary of just about anything published on the history of science or mathematics. Much of it tends to be very low quality full of just-so stories, or at best whiggishness. Unfortunately much of the work by academics on the subject is not much better as the field is full of former mathematicians or scientists with an interest in history rather than lots of historians with an interest in mathematics or science.

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

#30

Post by dmurray »

forinti wrote: post #28
dmurray wrote: post #25

Earlier quoted context omitted.

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?

999 is 1101000. So, for MSB, you only need 1 (when it's zero, you just don't put anything there). All the other positions can be 0, 1, or 2. So you would need 1 + 6*3 digits for your scoreboard. You can't not have 6 2s, for example, because your score might be 222222.

I see what you mean now. That's 19 with additive Roman numerals only, right? And for good measure, you don't even have to rearrange the digits, you can just order them as DCCCCLXXXXVIIII and cover the unused ones with a cloth.

With subtractive Roman numerals, you can get away with just 17 digits, but you lose the feature of not having to rearrange them.

Edit: actually, how did we get 19 here? I agreed at first, but my number above is just 15 digits.

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