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How Did Anyone Do Math in Roman Numerals?

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Re: How Did Anyone Do Math in Roman Numerals?

#41

Post by jessriedel »

amelius wrote: post #13
WalterBright wrote: post #6

It's interesting how the notation used can encourage or retard progress. For example, Leibniz's calculus notation was vastly superior to Newton's, and calculus theory advanced much more quickly where Leibniz's notation was used. https://en.wikipedia.org/wiki/Leibniz%27s_notation

I prefer Mathematica's notation over Leibniz's, because it is not as ambiguous (dx can also mean d TIMES x, as a simple example, but also dy^2 can mean d(y^2) and (dy)^2)

I'd go stronger than this and say that Leibniz's notation is actively harmful. It is very useful for quickly doing certain kinds of computations, but at the expense of conceptual understanding for students. Obviously, it's fine to use whatever computational aids you want when you understanding things, but most students are taught nothing but this fragile notation.

Re: How Did Anyone Do Math in Roman Numerals?

#42

Post by shuntress »

WalterBright wrote: post #6

It's interesting how the notation used can encourage or retard progress. For example, Leibniz's calculus notation was vastly superior to Newton's, and calculus theory advanced much more quickly where Leibniz's notation was used. https://en.wikipedia.org/wiki/Leibniz%27s_notation

I think it's a bit of a stretch to say that "Better tools/process results in faster progress" is 'interesting'

I think the interesting part is perceiving language as a tool.

Re: How Did Anyone Do Math in Roman Numerals?

#43

Post by dmreedy »

CogitoCogito wrote: post #10
WalterBright wrote: post #6

It's interesting how the notation used can encourage or retard progress. For example, Leibniz's calculus notation was vastly superior to Newton's, and calculus theory advanced much more quickly where Leibniz's notation was used. https://en.wikipedia.org/wiki/Leibniz%27s_notation

Notation is extremely important. It's basically a way to organize how you abstract a problem. If your abstraction is "bad" it will be harder to solve certain problems. It's really not much different everywhere else in society. Different programming languages/frameworks/etc. are doing essentially the same thing (if you ignore the speed of execution). All the languages are Turing complete and can do more or less the sa…

This couldn't really be stated any more clearly[1]; well put. I'll only add that this is true for any variety of abstraction, natural language included. Abstractions encode the biases of their creators[2]. The 'power' of an abstraction comes from the set of things that are easily and concisely expressible; its primitives. However, this is balanced by the truths that are no longer easily expressible, because the encoding doesn't allow for it. There's a certain intuition that semantics and abstraction are tied tightly in this sense; you don't can't really convey what something means unless it's concisely expressible in the abstraction you're using. Slang, idioms, calculus, etc.

---

[1] and yet, I guess I'll just babble on adding more words anyway...

[2] be they mathematical concepts, programming paradigms, or cultural norms and quirks.

Re: How Did Anyone Do Math in Roman Numerals?

#44

Post by [deleted] »

sbuttgereit wrote: post #27
maxxxxx wrote: post #24

Since we have a lot of math experts here I thought I'd ask a question I was always wondering about: Is there an inherent advantage or disadvantage to using the decimal system as we do? Somehow I think octal or hexadecimal would be easier but I am not sure.

I apologize for not being one of the experts you speak of, but I do know of others that say there are superior number bases to use for everyday counting. http://www.dozenal.org/drupal/content/brief-introduction-doz... These people believe that base 12 would be superior to base 10. And they do make good points, expressing fractions like those in 3rds become easier. Is it superior in the grand scheme? I'll leave that d…

[deleted]

Re: How Did Anyone Do Math in Roman Numerals?

#45

Post by ocschwar »

WalterBright wrote: post #6

It's interesting how the notation used can encourage or retard progress. For example, Leibniz's calculus notation was vastly superior to Newton's, and calculus theory advanced much more quickly where Leibniz's notation was used. https://en.wikipedia.org/wiki/Leibniz%27s_notation

Mohammad al Khwarizmi's treatise on Algebra (the word comes from the title of the treatise) was written in plain Arabic prose. No notation at all.

It takes a good amount of intelligence to read it, and it took an amazing genius to write it. But put it in plain notation, and it's a collection of 7th grade algebra problems.

Notation is important.

Re: How Did Anyone Do Math in Roman Numerals?

#46

Post by gruturo »

sbuttgereit wrote: post #27
maxxxxx wrote: post #24

Since we have a lot of math experts here I thought I'd ask a question I was always wondering about: Is there an inherent advantage or disadvantage to using the decimal system as we do? Somehow I think octal or hexadecimal would be easier but I am not sure.

I apologize for not being one of the experts you speak of, but I do know of others that say there are superior number bases to use for everyday counting. http://www.dozenal.org/drupal/content/brief-introduction-doz... These people believe that base 12 would be superior to base 10. And they do make good points, expressing fractions like those in 3rds become easier. Is it superior in the grand scheme? I'll leave that d…

Maybe this is just me being very naive, but how does it help? It doesn't actually change the properties of the underlying numbers.

10 (decimal) is not divisible by 3, and rewriting it in base 12 as A won't change that. You simply get 29.4 instead of 33.3...

Re: How Did Anyone Do Math in Roman Numerals?

#47

Post by yomritoyj »

blattimwind wrote: post #12
WalterBright wrote: post #6

It's interesting how the notation used can encourage or retard progress. For example, Leibniz's calculus notation was vastly superior to Newton's, and calculus theory advanced much more quickly where Leibniz's notation was used. https://en.wikipedia.org/wiki/Leibniz%27s_notation

An important facet to mathematics in general, that most are unaware of before studying it, is that the majority of proofs, especially those done in bachelor university courses, are purely notation. Other problems often become trivial to solve by using a different notation (e.g. polar form vs. points on the complex plane) as well.

Could you please elaborate. Just to take two elementary examples: the intermediate value theorem and the theorem that any two bases of a finite-dimensional vector space have the same number of vectors. I would have thought that both require mathematical ideas and not just notation.

Re: How Did Anyone Do Math in Roman Numerals?

#48

Post by genmon »

flipp3r wrote: post #18

> the Romans greatly preferred the simpler IIII to IV, XXXX to XL, and so on. (The IIII-for-4 notation survives today on the faces of clocks.) looks at watch Well, damn, it's IIII. However my watch does use IX over VIIII, what's up with that?

There are a few theories detailed here:

http://mentalfloss.com/article/24578/why-do-some-clocks-use-...

The reason I always heard is about ease of casting: it requires a more complex mold if 4 is written as IV instead of IIII.

Re: How Did Anyone Do Math in Roman Numerals?

#49

Post by colechristensen »

jeffwass wrote: post #35
maxxxxx wrote: post #24

Since we have a lot of math experts here I thought I'd ask a question I was always wondering about: Is there an inherent advantage or disadvantage to using the decimal system as we do? Somehow I think octal or hexadecimal would be easier but I am not sure.

I was about to post that in my opinion base-12 is superior to base-10. But someone beat me to it. In a six-fi novel I'm writing, an advanced alien civilisation uses base-12. As to your question specifically regarding base-16 instead of base-12, it depends. Decimal itself is just a bizarre choice, most likely due to humans having literally ten digits. In decimal we can represent exact fractions of 1/2, 1/5, and 1/10 (…

>I'd argue in everyday use we're more likely to consider 1/3rd of something than 1/5th.

Because, I'd think, many things follow power laws which makes values closer to 1 occur more frequently.

Re: How Did Anyone Do Math in Roman Numerals?

#50

Post by sharpercoder »

ldjb wrote: post #23
flipp3r wrote: post #18

> the Romans greatly preferred the simpler IIII to IV, XXXX to XL, and so on. (The IIII-for-4 notation survives today on the faces of clocks.) looks at watch Well, damn, it's IIII. However my watch does use IX over VIIII, what's up with that?

There's a theory[0][1] that it's mostly down to aesthetics. It looks visually more pleasing that way when split into three groups of four numbers: I, II, III, IIII (consisting of I only) V, VI, VII, VIII (consisting of I and V) IX, X, XI, XII (consisting of I and X) [0] http://mathtourist.blogspot.co.uk/2010/08/iiii-versus-iv-on-... [1] https://www.hautehorlogerie.org/en/encyclopaedia/glossary-of...

Personally, I find distinguishing III and IIII quite hard in several fonts. However, the difference between [II and III] and [III and IV] is easier to read.
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