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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)

#41

Post by ciceryadam »

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.

Plus Eratosthenes and Cleomedes got pretty close on calculating the Earth's circumference.

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

#42

Post by dwohnitmok »

crazygringo wrote: post #37
Sharlin wrote: post #31

Earlier quoted context omitted.

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…

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 whereas the other has a more confusing and less well-known notation.

Therefore I think ancient arguments over whether 0 is a number (and acceptance thereof) are representative of a greater paradigmatic shift, similar in essence to the arguments over whether the square root of -1 is truly a "number."

Viewed that way 0 is the first step in a journey of an understanding of numbers from purely counting discrete entities, to abstract parts of computation.

So basically I would posit that it is in fact a great conceptual leap (just as the negative numbers are) that only seems like an obvious fluke of notation when every schoolchild has learned it.

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

#43

Post by Koshkin »

Grustaf wrote: post #14

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

Well, you can try and then you'll appreciate the difference, I am sure. Even if implemented in computer hardware, operating with Roman numerals would either be slow or take many more transistors (or both). As to why in a philosophical sense, it is because the positional system was invented specifically as a computational device, which only happened many years after people learned how write numbers down. Optimization effort takes time, and a random solution is not guaranteed to be optimally suited for a particular application (such as performing calculations).

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

#44

Post by boogies »

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…

> Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation.

Great quote. I wonder if this process will continue with adoption of duodecimal numbers.

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

#46

Post by DNied »

shagie wrote: post #23
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 ans…

Well, thanks for repeating what Wikipedia says, down to the same example.

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

#47

Post by Someone »

mdiesel wrote: post #36
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?

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 ke…

Of course we can do better. Every card that is an identical copy of another in the set loses you some flexibility in picking different sequences.

6 different cards give you

  - 1 zero-card sequence
  - 6 different 1-card sequences
  - 30 different 2-card sequences
  - 120 different 3-card sequences
  - 360 different 4-card sequences
  - 720 different 5-card sequences
  - 720 different 6-card sequences
That’s enough to almost get you up to 2000. And I don’t think the resulting encoding is objectively weirder than Roman digits.

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

#48

Post by crazygringo »

dwohnitmok wrote: post #42
crazygringo wrote: post #37

Earlier quoted context omitted.

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…

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 or annihilates a number, and so doesn't need to be treated like other numbers. AFAIK, zero as a number arises first in figuring out how to "get to" negative numbers, e.g. what is two minus four (one, zero, negative one, negative two), where zero is required as a numeric concept.

Negative numbers were a big step forwards. Zero, I still don't see it -- either it was just convenient notation for "nothing", or part and parcel of the shift to negative numbers. Unless I'm missing something in the historical record?

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

#49

Post by mdiesel »

mdiesel wrote: post #36
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?

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 ke…

The answer is: It reduces to a question of permutations. 7! > 999, so we could do it with different arrangements of just 7 cards.

Can we do better? Good question...

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

#50

Post by mdiesel »

mdiesel wrote: post #49
mdiesel wrote: post #36

Earlier quoted context omitted.

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 ke…

The answer is: It reduces to a question of permutations. 7! > 999, so we could do it with different arrangements of just 7 cards. Can we do better? Good question...

Now if we arranged them on a 2d grid, could use 4 cards, then the different shapes (even discounting the similar looking shapes) would get you to >1000. You'd have to allow more than just the standard tetrominoes.
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