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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?

#31

Post by jcranberry »

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.

Not really. Different decimal systems will make divisibity by the primes of the base number more obvious (and thus polynomials whose solution set contain those primes will be more obvious), and fractions whose prime factorizations are contained in the primes of the base will be finite as opposed to "repeating", but all the structures beneath remain the same.

Re: How Did Anyone Do Math in Roman Numerals?

#32

Post by tpeo »

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)

It really isn't that ambiguous in cursive, though. When writing out "dx" as product in cursive, the "d" and "x" characters are written separately. When writing "dx" as an infinitesimal, the characters are connected. Never had any issue with higher degree derivatives, either.

But I see how it can be confusing with printed characters. I guess Leibniz just took ligatures for granted when he came up with his stuff.

Re: How Did Anyone Do Math in Roman Numerals?

#34

Post by nnq »

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

About this particular case: why hasn't Euler's notation (https://en.wikipedia.org/wiki/Notation_for_differentiation#E...) become the clear winner?

I mean, its the only one that is both not confusing for beginners (because it doesn't trick them into the "cool, let's `simplify` the dx at the denominator with the next one" mindset) and it also translates easily to code (or other 1D encoding), like you can write "second derivative of f with respect to x" as D[f, x, 2] and "integral of a with respect to t" as "D[a, t, -1]".

My point is mainly that there is nothing rational at all in how humans choose notations... without some obscure historic events, we'd probably still be using some derivative of roman numerals or maybe even sexagesimals! (https://en.wikipedia.org/wiki/Sexagesimal)

(I imagine the reason is because "highly performing" individuals have their own "internal" language to think in, so general language is just a for communication, so a political decision... unfortunately for education :()

Re: How Did Anyone Do Math in Roman Numerals?

#35

Post by jeffwass »

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 (without repeated decimals like 0.33333 for 1/3). Counting by fives (and twos) is very easy.

But choosing prime factors of 2 and 5 is a strange choice in itself. Why skip 3? Why is it more useful to easily represent fraction 1/5th as 0.2 instead of 1/3rd? How often do we use fifths?

Hexadecimal in one sense is easier, all prime factors are two. So we can represent 1/2, 1/4, 1/8, and 1/16 exactly.

Duodecimal (Base 12) is very convenient for having a high proportion of exact fractions. Eg - 1/12, 1/6, 1/4, 1/3, and 1/2 can all be represented exactly. I'd argue in everyday use we're more likely to consider 1/3rd of something than 1/5th. Counting by twos, threes, fours, and sixes is easy. Watch, let's count to 20 (24 in decimal) by 3's : 3, 6, 9, 10, 13, 16, 19, 20. In 4's : 4, 8, 10, 14, 18, 20. By 6 : 6, 10, 16, 20.

Base-12 offers four handy subdivisions (excluding 1) instead of two for decimal or three for hexadecimal. That beats hexadecimal using fewer unique digits. It beats decimal by two using only two extra unique digits.

And I think it's these reasons it was chosen for various historical subdivisional units (inches per foot, pence per shilling).

The other item to consider is the relative number of unique values per digit. I'm not sure of the utility of having 10, 12, or 16 here.

At one extreme, while binary is useful for discretising signals in digital logic, using only zeroes and ones becomes cumbersome for daily use at higher numbers.

Once we're at base 10 and higher, I'm not sure how much here extra digits help or hurt.

Re: How Did Anyone Do Math in Roman Numerals?

#36

Post by tgb »

nnq wrote: post #34
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

About this particular case: why hasn't Euler's notation ( https://en.wikipedia.org/wiki/Notation_for_differentiation#E... ) become the clear winner? I mean, its the only one that is both not confusing for beginners (because it doesn't trick them into the "cool, let's `simplify` the dx at the denominator with the next one" mindset) and it also translates easily to code (or other 1D encoding), like you can write "secon…

It's actually pretty common notation in a lot of fields, and mathematicians frequently switch between whichever of these notations is most convenient for a particular problem. Your point of course still stands for why isn't it the main one used in Calculus 1 courses, but it's common in differential equations courses in my experience (and PDE's often simplifies it further to just have D_x f as f_x). And many times where the "derivative" is not necessarily the "calc 1" derivative, you use notation similar to that, for example the covariant derivative [1] is sometimes written like that but with a 'nabla' instead of a 'D'. Some authors even use 'D' for covariant derivative along a path.

[1] https://en.wikipedia.org/wiki/Covariant_derivative

Re: How Did Anyone Do Math in Roman Numerals?

#37

Post by bonoboTP »

nnq wrote: post #34
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

About this particular case: why hasn't Euler's notation ( https://en.wikipedia.org/wiki/Notation_for_differentiation#E... ) become the clear winner? I mean, its the only one that is both not confusing for beginners (because it doesn't trick them into the "cool, let's `simplify` the dx at the denominator with the next one" mindset) and it also translates easily to code (or other 1D encoding), like you can write "secon…

I think a lot of my confusions when I first learned calculus would have been eliminated with a notation that clearly expressed that derivatives operate on whole functions, not on values. So the derivative of f(x) at x=0 is not some function of f(0), but it is derivative(f)(x). Also, even without derivatives, sometimes the expression f(x) refers to the whole function, sometimes just a particular value of it at a specific x.

I don't know if higher-order functions are really that tough to teach/understand but I think it would actually simplify and demistify many things.

Currently, of the widespread notations I like this one best:

f(x0) = d/dx (sin(x)*cos(x)+x^2) | x=x0 (with the last part in subscript)

I also miss variable scoping from math writing and it disturbs me that variable names often carry semantics, like p(x) and p(y) can be the probability density functions of random variables X and Y (so p is a different function depending on the name of the input variable that you substitute so it doesn't actually operate on real numbers, but (string, number) pairs). I'd prefer to explicitly mark the functions as p_X(x) and p_Y(y).

Similar things come up a lot with differential equations where you don't really know whether something (like y or u) is supposed to be a function (of x or t) or "just" a variable.

Despite the general opinion among laypeople that math notation is very precise and unambiguous, I find that it's often very sloppy and unless you already understand the context very well, it can easily be misleading. Math notation is somewhere between normal natural language and programming languages, and depending on the writer it may be closer to one or the other.

One can argue that this is necessary for compactness.

Re: How Did Anyone Do Math in Roman Numerals?

#38

Post by yorwba »

ewanm89 wrote: post #28
amelius wrote: post #13

Earlier quoted context omitted.

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 would point out it is d because latin typesetters back then often didn't have a greek typeface to print with and it is the closest to the greek letter delta δ. Once one understands it's δx and δy (or Δx and Δy) and today still today most people don't know how to get delta characters on their latin keyboards, then it is easy just to not use d in algebra and use for differential calculus only. Finally (Δx)^2 and Δ(x^…

If by (Δx)^2 you mean the finite difference (x₂-x₁)² = x₂² - 2x₂x₁ + x₁², then it isn't the same as Δ(x^2) = x₂² - x₁².

If you mean the infinitesimal difference, then Δ(x^2) = 2Δx still isn't (Δx)^2.

Re: How Did Anyone Do Math in Roman Numerals?

#39

Post by noir_lord »

bonoboTP wrote: post #37
nnq wrote: post #34

Earlier quoted context omitted.

About this particular case: why hasn't Euler's notation ( https://en.wikipedia.org/wiki/Notation_for_differentiation#E... ) become the clear winner? I mean, its the only one that is both not confusing for beginners (because it doesn't trick them into the "cool, let's `simplify` the dx at the denominator with the next one" mindset) and it also translates easily to code (or other 1D encoding), like you can write "secon…

I think a lot of my confusions when I first learned calculus would have been eliminated with a notation that clearly expressed that derivatives operate on whole functions , not on values. So the derivative of f(x) at x=0 is not some function of f(0), but it is derivative(f)(x). Also, even without derivatives, sometimes the expression f(x) refers to the whole function, sometimes just a particular value of it at a spec…

I've noticed that I understand what is going on much more when a function is written in code than in its mathematical form.

A lot of that is familiarity but I don't think all of it is.

Re: How Did Anyone Do Math in Roman Numerals?

#40

Post by jessriedel »

ewanm89 wrote: post #28
amelius wrote: post #13

Earlier quoted context omitted.

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 would point out it is d because latin typesetters back then often didn't have a greek typeface to print with and it is the closest to the greek letter delta δ. Once one understands it's δx and δy (or Δx and Δy) and today still today most people don't know how to get delta characters on their latin keyboards, then it is easy just to not use d in algebra and use for differential calculus only. Finally (Δx)^2 and Δ(x^…

> Finally (Δx)^2 and Δ(x^2) are the same thing in differential calculus.

Isn't Δ(x^2) = 2xΔx ≠ (Δx)^2 ? The object Δ(x^2) has one infinitesimals while (Δx)^2 has two, and the number of infinitesimals is conserved. (You can only get finite quantities by taking the ratio of equal numbers of infinitesimals.)

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