4 ms·

That really surprises me. I've never found mathematical notation to be anything than very useful and terse. Can you make it any more clear what bothers you? It

by kmm 11y ago

That really surprises me. I've never found mathematical notation to be anything than very useful and terse. Can you make it any more clear what bothers you? It has always been very clear for me, and I'm not sure what could be improved. I'd think any attempt to make it less ambiguous or more expressive would eventually devolve in something very similar to what we have now, because of our compulsion to simplify repeated tasks (eg Einstein notation, bra-ket etc..)

mrob 11y ago
>Can you make it any more clear what bothers you? A few things that spring to mind: dx/dy notation, and especially manipulating it like it's a real fraction. Implicit multiplication, which forces use of single character variable names, which forces use of weird fonts and greek letters. Writing sin^2(x) for (sin(x))^2 Writing sin^-1(x) for arcsin(x) (especially bad because it conflicts with sin^2(x)) The base for log(x) is ambiguous. Writing |x| for abs(x) (abs() is obviously a function, it shouldn't have special notation) Calling abs "modulus" And the names "imaginary number" and "complex number" seem almost deliberately designed to intimidate outsiders. IMO they should be called "oscillating numbers" and "rotating numbers".
- Maken 11y ago
  >Writing |x| for abs(x) (abs() is obviously a function, it shouldn't have special notation) >Calling abs "modulus" Both of those are because the absolute value of a number is the norm of the one-dimensional space.
  - mrob 11y ago
    I've been successfully using abs() for a very long time, and I have no idea what "the norm of the one-dimensional space" is. If the notation requires understanding something of so little practical relevance to make sense then it's bad notation.
    - schoen 11y ago
      It's a generalization, and it makes sense as one: https://en.wikipedia.org/wiki/Norm_%28mathematics%29 https://en.wikipedia.org/wiki/Norm_%28mathematics%29 It allows you to think about |x| when x is a vector (or even other kinds of objects!) rather than just a real number. This kind of generalization can be a really powerful thing about mathematics, making existing insights more broadly applicable.
      - mrob 11y ago
        So it's a special notation that I can overload to use with multiple types. I can already do that with normal function call notation. How does the added complexity benefit me?
        Maken 11y ago
        Well, using "abs()" for it would be a bad idea, because "|x|" (or "||x||") means "whatever is the distance I've defined between two elements of this particular space". For all that I know, the only reason to write "|x|" instead of "norm(x)" or "length(x)" is because brevity and because it is a well accepted notation. Edit: wrong post.
        kaitai 11y ago
        I force my students to use this overloaded notation because I want to make very clear to them the commonality of ideas. They need to know that determinant of a matrix, length of a vector, and absolute value are all ways of measuring "size" in different contexts that fit together into a harmonious whole. abs(Matrix) just doesn't... do that. Doesn't make sense.
      - deleted 11y ago
        [deleted]
- jdminhbg 11y ago
  > Writing |x| for abs(x) (abs() is obviously a function, it shouldn't have special notation) And like nearly every piece of mathematics notation, this one is overloaded: https://en.wikipedia.org/wiki/Determinant https://en.wikipedia.org/wiki/Determinant
- serge2k 11y ago
  > And the names "imaginary number" and "complex number" seem almost deliberately designed to intimidate outsiders. IMO they should be called "oscillating numbers" and "rotating numbers are you serious?
  - mrob 11y ago
    Those names are related to the common uses. Anybody doing something uncommon is probably advanced enough that the names shouldn't bother them.
    - zodiac 11y ago
      What "common use" are you thinking about?
      - mrob 11y ago
        AC circuit analysis mostly. And it's not a bad fit for Mandelbrot sets and similar either, if you think about how the points in the set behave.
        zodiac 11y ago
        I see. While it is probably true that the number of people who use it for AC circuit analysis and other applications for which the rotation analogy is appropriate is rather large, complex numbers are not "fundamentally about" rotation. In fact, the reason they're called "imaginary numbers" makes perfect sense if you were a mathematician 500 years ago - one method of solving cubic equation manipulated the square root of negative numbers in intermediate steps, which of course doesn't make sense, but by the final step these square roots canceled out and the final answer was correct. So they named them "imaginary numbers" to kind of tell people "don't worry whether this makes sense, it's only used in an intermediate step, they're not real numbers". So, naming a concept after its most common use is exactly what got us into this situation! In addition to the modelling-rotation and solving-cubics aspect of complex numbers, last year I used them in a factoring-things context in a number theory class, for example to prove https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squares https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_o... (read the "Dedekind's Proof" section).
- iheartmemcache 11y ago
  I'm with you 100% on all of those ambiguous terms. Conventionally, arcsin is now used. Log(x) edit: was presumed to be base 10 in most domains. Historically a carry over from when logarithmic tables were used very often by those who operated in base 10 (*edit: read down for a historical exposition). Mathematicians were concurrently publishing with log(x) with e as the base. Calculators ultimately (legend has it, at least) formalized it to log = base 10 because engineers designed it not mathematicians, though it might just be angry mathematicians who perpetuate that lore ;). Some subsets of mathematics might use e as the implicit base, but it'd be obvious from the context of the paper. Ln is e. Lg is 10. I can honestly say though every single one of those ambiguities I had an issue with in the past. This post reignited latent of ire of mine which I haven't experienced since 16. Denotational semantics are important!
  - heinrich5991 11y ago
    Most (all?) logs I encountered (in mathematics) were base e, simply due to the fact that base 10 isn't special in mathematics, but for base e, you have log' x = 1/x. Also, it seems very common to denote the inverse of a function (even sine or cosine) with f^-1. See also Wikipedia on that subject (https://en.wikipedia.org/wiki/Logarithm https://en.wikipedia.org/wiki/Logarithm): >The logarithm to base 10 (that is b = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number e (≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler derivative. The binary logarithm uses base 2 (that is b = 2) and is commonly used in computer science.
    - iheartmemcache 11y ago
      At least in the northeast US, ln(x) was the preferred nomenclature as to eliminate ambiguity. Again, it depends on what course you were taking. Lg and Ln. Log was explicitly avoided. In higher mathematics, f^-1 does not mean the inverse of a function, unless that function is injective. Most people are aware of that at that point, but it's still a mental hurdle you have to unlearn and can lead to dangerous composition errors. (Edit: since I can't respond to you, poster below me, I mean the latter of course) Historically, in yore days of sextants, slide rules and log tables, engineers could "lift" multiplication into addition with said log tables which were of convention base 10. For anyone who wasn't a peer in the Royal Society or a professor, that's how complicated math calculations were performed. Banks balanced books and calculated interest payments via tables like this, and the precision of your log tables had a mantissa (the original floating point error, heh).
nilkn 11y ago
As an example, I dislike how overloaded symbols like 'dx' are, which are sometimes purely syntactic sugar (specifying the variable of integration), sometimes are just a part of another indivisible symbol (dy/dx), sometimes are treated as things that can be manipulated on their own but don't actually have a value or any rigorous meaning (physicists...), sometimes represent entire entities such as differential forms, and sometimes the 'd' is actually a fully qualified, completely rigorously defined operator (the exterior derivative, or the boundary map in a chain complex, especially for de Rham cohomology). I wouldn't be surprised at all if I'm missing other ways in which this exact same symbol occurs with different meanings. To the experienced eye, all these usages overlap in subtle ways, so that it all feels reasonable and even justified. But to newcomers, I think the notation just feels like a complete mess.
- OscarCunningham 11y ago
  People sometimes use it to denote a measure, and of course sometimes it means d lots of x.

by kmm 11y ago

mrob 11y ago

>Can you make it any more clear what bothers you? A few things that spring to mind: dx/dy notation, and especially manipulating it like it's a real fraction. Implicit multiplication, which forces use of single character variable names, which forces use of weird fonts and greek letters. Writing sin^2(x) for (sin(x))^2 Writing sin^-1(x) for arcsin(x) (especially bad because it conflicts with sin^2(x)) The base for log(x) is ambiguous. Writing |x| for abs(x) (abs() is obviously a function, it shouldn't have special notation) Calling abs "modulus" And the names "imaginary number" and "complex number" seem almost deliberately designed to intimidate outsiders. IMO they should be called "oscillating numbers" and "rotating numbers".

Maken 11y ago
>Writing |x| for abs(x) (abs() is obviously a function, it shouldn't have special notation) >Calling abs "modulus" Both of those are because the absolute value of a number is the norm of the one-dimensional space.
- mrob 11y ago
  I've been successfully using abs() for a very long time, and I have no idea what "the norm of the one-dimensional space" is. If the notation requires understanding something of so little practical relevance to make sense then it's bad notation.
  - schoen 11y ago
    It's a generalization, and it makes sense as one: https://en.wikipedia.org/wiki/Norm_%28mathematics%29 https://en.wikipedia.org/wiki/Norm_%28mathematics%29 It allows you to think about |x| when x is a vector (or even other kinds of objects!) rather than just a real number. This kind of generalization can be a really powerful thing about mathematics, making existing insights more broadly applicable.
    - mrob 11y ago
      So it's a special notation that I can overload to use with multiple types. I can already do that with normal function call notation. How does the added complexity benefit me?
      - Maken 11y ago
        Well, using "abs()" for it would be a bad idea, because "|x|" (or "||x||") means "whatever is the distance I've defined between two elements of this particular space". For all that I know, the only reason to write "|x|" instead of "norm(x)" or "length(x)" is because brevity and because it is a well accepted notation. Edit: wrong post.
      - kaitai 11y ago
        I force my students to use this overloaded notation because I want to make very clear to them the commonality of ideas. They need to know that determinant of a matrix, length of a vector, and absolute value are all ways of measuring "size" in different contexts that fit together into a harmonious whole. abs(Matrix) just doesn't... do that. Doesn't make sense.
    - deleted 11y ago
      [deleted]
jdminhbg 11y ago
> Writing |x| for abs(x) (abs() is obviously a function, it shouldn't have special notation) And like nearly every piece of mathematics notation, this one is overloaded: https://en.wikipedia.org/wiki/Determinant https://en.wikipedia.org/wiki/Determinant
serge2k 11y ago
> And the names "imaginary number" and "complex number" seem almost deliberately designed to intimidate outsiders. IMO they should be called "oscillating numbers" and "rotating numbers are you serious?
- mrob 11y ago
  Those names are related to the common uses. Anybody doing something uncommon is probably advanced enough that the names shouldn't bother them.
  - zodiac 11y ago
    What "common use" are you thinking about?
    - mrob 11y ago
      AC circuit analysis mostly. And it's not a bad fit for Mandelbrot sets and similar either, if you think about how the points in the set behave.
      - zodiac 11y ago
        I see. While it is probably true that the number of people who use it for AC circuit analysis and other applications for which the rotation analogy is appropriate is rather large, complex numbers are not "fundamentally about" rotation. In fact, the reason they're called "imaginary numbers" makes perfect sense if you were a mathematician 500 years ago - one method of solving cubic equation manipulated the square root of negative numbers in intermediate steps, which of course doesn't make sense, but by the final step these square roots canceled out and the final answer was correct. So they named them "imaginary numbers" to kind of tell people "don't worry whether this makes sense, it's only used in an intermediate step, they're not real numbers". So, naming a concept after its most common use is exactly what got us into this situation! In addition to the modelling-rotation and solving-cubics aspect of complex numbers, last year I used them in a factoring-things context in a number theory class, for example to prove https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squares https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_o... (read the "Dedekind's Proof" section).
iheartmemcache 11y ago
I'm with you 100% on all of those ambiguous terms. Conventionally, arcsin is now used. Log(x) edit: was presumed to be base 10 in most domains. Historically a carry over from when logarithmic tables were used very often by those who operated in base 10 (*edit: read down for a historical exposition). Mathematicians were concurrently publishing with log(x) with e as the base. Calculators ultimately (legend has it, at least) formalized it to log = base 10 because engineers designed it not mathematicians, though it might just be angry mathematicians who perpetuate that lore ;). Some subsets of mathematics might use e as the implicit base, but it'd be obvious from the context of the paper. Ln is e. Lg is 10. I can honestly say though every single one of those ambiguities I had an issue with in the past. This post reignited latent of ire of mine which I haven't experienced since 16. Denotational semantics are important!
- heinrich5991 11y ago
  Most (all?) logs I encountered (in mathematics) were base e, simply due to the fact that base 10 isn't special in mathematics, but for base e, you have log' x = 1/x. Also, it seems very common to denote the inverse of a function (even sine or cosine) with f^-1. See also Wikipedia on that subject (https://en.wikipedia.org/wiki/Logarithm https://en.wikipedia.org/wiki/Logarithm): >The logarithm to base 10 (that is b = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number e (≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler derivative. The binary logarithm uses base 2 (that is b = 2) and is commonly used in computer science.
  - iheartmemcache 11y ago
    At least in the northeast US, ln(x) was the preferred nomenclature as to eliminate ambiguity. Again, it depends on what course you were taking. Lg and Ln. Log was explicitly avoided. In higher mathematics, f^-1 does not mean the inverse of a function, unless that function is injective. Most people are aware of that at that point, but it's still a mental hurdle you have to unlearn and can lead to dangerous composition errors. (Edit: since I can't respond to you, poster below me, I mean the latter of course) Historically, in yore days of sextants, slide rules and log tables, engineers could "lift" multiplication into addition with said log tables which were of convention base 10. For anyone who wasn't a peer in the Royal Society or a professor, that's how complicated math calculations were performed. Banks balanced books and calculated interest payments via tables like this, and the precision of your log tables had a mantissa (the original floating point error, heh).

nilkn 11y ago

As an example, I dislike how overloaded symbols like 'dx' are, which are sometimes purely syntactic sugar (specifying the variable of integration), sometimes are just a part of another indivisible symbol (dy/dx), sometimes are treated as things that can be manipulated on their own but don't actually have a value or any rigorous meaning (physicists...), sometimes represent entire entities such as differential forms, and sometimes the 'd' is actually a fully qualified, completely rigorously defined operator (the exterior derivative, or the boundary map in a chain complex, especially for de Rham cohomology). I wouldn't be surprised at all if I'm missing other ways in which this exact same symbol occurs with different meanings. To the experienced eye, all these usages overlap in subtle ways, so that it all feels reasonable and even justified. But to newcomers, I think the notation just feels like a complete mess.

OscarCunningham 11y ago
People sometimes use it to denote a measure, and of course sometimes it means d lots of x.