Why do physicists care so much about dimensions but then pretend that their dimensionless quantities don't have units ?
A lot of people in the comments here don’t seem to understand the difference between the two.
A trick to eliminate 2π (sometimes)
161–170 of 179 posts
Re: A trick to eliminate 2π (sometimes)
#162I was with him until this point: >This notation could be abused even further by denoting đx = 1/(2π) dx , which can then simplify some integral formulae, But now you're screwing up all of your previous integral formulae!
It’s insane… I’m still shocked this isn’t trolling.
Re: A trick to eliminate 2π (sometimes)
#163Earlier quoted context omitted.
That's not his argument; he says defining Θ = e^2πi is not useful because e^2πi = 1, so Θ and Θ^x would also be 1. That's why he defined Θ = e^2π instead, so that Θ^x (or possibly Θ^ix) is a useful operation.
he writes: > where ln(e^2πi)=ln(1)=0, that is incorrect because the logarithm of a complex number is multi-valued. He even cites the correct source on wikipedia, but his argument is incorrect (and because the exponent is 2πi he actually would get a meaningful results I believe).
Hmm, this indeed seems to be the case! I think I was confused by ln(1), since 1 is a real number, but the multi-valued complex logarithm of 1 is indeed k 2pi (and now I see the appropriate notation should be "log" instead of "ln"). Perhaps the whole thing could indeed be reduced to "1^x" with an asterisk that 1^x means complex exponentiation. I'll have to update this section in the post.
Re: A trick to eliminate 2π (sometimes)
#164Earlier quoted context omitted.
All mathematical formulas can be inter-converted based on the identity: e^(x + i*y) = 2^(x/ln2) * 1^(y/2Pi) Most formulae from textbooks are written in such a way to be simpler with e^x and its inverse, but it is almost always possible to move the constants ln2 and 2Pi between various equations so that in the end they will disappear from most relations, with the exception of the derivation or integration formulae. In…
so you express x (the simple polynomial) as ln(2^(x/ln2)? or as an infinite series expansion?
You have misunderstood my point. I have not said anything about polynomials.
I have said that the so called "natural" exponential, normally written as e^x or exp(x) of either real or complex argument and its inverse, the hyperbolic a.k.a. natural logarithm, and any other functions derived from it are neither needed nor useful when computations are done by computers, as opposed to computations done with pen and paper.
In all traditional formulae where the "natural" exponential function or functions derived from it occur, all occurrences can be replaced using a pair of functions of real argument, the function 2^x with real value and the function 1^x with complex value, either directly or with functions derived from this pair, e.g. the binary logarithm.
In computer programs this substitution results in both higher accuracy and higher speed and it has as a side effect that the units radian and neper are never needed.
It should be noted that even in the 19th century, when the "natural" exponential and logarithm and the trigonometric functions with argument in radians were useful for symbolic computations done by hand, they were never used for practical numeric computations.
All practical numeric computations were done using the function 10^x and the trigonometric functions with argument in degrees and their inverses, by using mathematical tables where the values of these functions were tabulated (or equivalently, by using slide rules).
The use of the "natural" exponential and logarithm and of the trigonometric functions with argument in radians for practical computations has become widespread only after the development of the electronic computers, after programming languages like Fortran have included them as standard functions.
I consider that this has been a mistake, similar to the use of decimal numbers in some computers. Both the use of decimal numbers and the use of the "natural" exponential and logarithm and of the trigonometric functions with argument in radians are sub-optimal in all their possible applications.
Re: A trick to eliminate 2π (sometimes)
#165Earlier quoted context omitted.
The simpler derivation formula was important when such symbolic computation was done by hand. Now, except perhaps for school exercises, anything complicated is done with a computer and this advantage is much less important. The increased accuracy and simpler formulas in other places when measuring angles in cycles a.k.a. turns vastly outweigh the advantage of radians for differentiation. In real applications you almo…
If you’re using a computer for symbolic algebra or whatever, none of this matters anyway. The whole post is about simpler notation for the sake of making things less error-prone when working by hand.
For a computer it matters because it is possible to compute with higher accuracy and speed the trigonometric functions with the argument in cycles instead of radians (similarly for the binary exponential and logarithm vs. the "natural" exponential and logarithm).
The main reason is that the reduction of the argument to the range where a polynomial approximation is valid becomes much simpler.
Also, the primary inputs or the final outputs of any really complete computation are never in radians, because in physical devices it is not possible to realize radians with high accuracy, but only the angles that are in a rational relationship with the cycle. This is true both for geometric angles and for the phase angles of oscillations and waves.
Re: A trick to eliminate 2π (sometimes)
#166`Θ^i = 1` does make this really slick, imho. I often wonder if someday when we meet alien intelligences, they'll have a completely different set of constants, derivable from our own but different. Θ=535.491... may be such an example.
I don't think Θ makes for a good fundamental constant, though, because it's composed of pi and e, which must persist as distinct concepts in the end.
While pi and e must persist as two distinct concepts, they may be substituted by a constant pair that is much more useful for practical purposes is 2*pi and ln 2 ("natural" logarithm of two).
When using this alternative pair of constants (which are the ratios between two pairs of units, cycle vs. radian and octave vs. neper), there is no longer any need for pi or e.
Re: A trick to eliminate 2π (sometimes)
#167Earlier quoted context omitted.
> You can't discuss math while getting rid of the math symbols. That's not a reasonable proposal. If we assert that some topic can’t be discussed with plain prose, then the only logical conclusion is that you can’t discuss the topic at all. > Math on the web is broken, the the affected people should be up complaining about that. There is really two different topic there. One is, how can screen readers deal appropriat…
> If we assert that some topic can’t be discussed with plain prose, then the only logical conclusion is that you can’t discuss the topic at all. Please bear in mind that mathematical notation is a language , and that mathematical formulas are perfectly valid plain prose in that language . I imagine that some screen readers will fail gracelessly when faced with Chinese script or Hindu as well. Especially if they're no…
>Please bear in mind that mathematical notation is a language, and that mathematical formulas are perfectly valid plain prose in that language.
In that sense its more a DSL than a generalist language. And anything you can express in a DSL, you can also express it with a more generalist language. Most likely the DSL will provide a far more compact way to communicate what it allows to express, but that compactness doesn’t come for free: it takes time to compress and decompress and you have to also communicate the DSL specification in some preexisting medium.
On my side, I suppose "discuss" to mean we can talk on the topic face to face with spontaneous expressions means like oral expression and listening comprehension, or sign language. I don’t mean that there is no other mean to discuss, but when we can’t transpose the topic in such a medium, we are probably beyond the realm of discussion. For example when we make love, there is more happening than what can either hope to realize through mere talk.
Once again, I’m not again DSLs and so on. I just mean that there are not the proper tools for maximizing accessibility.
Re: A trick to eliminate 2π (sometimes)
#168Earlier quoted context omitted.
Note there's an existing notation which I've mostly seen in lower-class settings like high-school textbooks: r theta, for the complex number r e^(i theta). Optionally leave out the r. So you have that "most beautiful formula in all of mathematics": tau = 1
And here it is in Unicode glory: ∠τ = 1
Thanks, I should've figured there'd be a character.
Re: A trick to eliminate 2π (sometimes)
#169Earlier quoted context omitted.
so you express x (the simple polynomial) as ln(2^(x/ln2)? or as an infinite series expansion?
You have misunderstood my point. I have not said anything about polynomials. I have said that the so called "natural" exponential, normally written as e^x or exp(x) of either real or complex argument and its inverse, the hyperbolic a.k.a. natural logarithm, and any other functions derived from it are neither needed nor useful when computations are done by computers, as opposed to computations done with pen and paper.…
Ah interesting! I thought you were saying we can express all the other analytical functions with 1,2^x.
Re: A trick to eliminate 2π (sometimes)
#170Earlier quoted context omitted.
Thing is, angles don't really have units (the technical term is they are dimensionless). They are a length (the subtended arc of a circle) divided by a length (the radius of the circle). When you want to do something like get a sine wave of period T, you inevitably have to include a 2π somewhere. Speaking as someone who had to write down many 2π's in university (especially as I find angular quantities like angular fr…
This is not true. Angles very much have units and it's why you can express the same concept with different numbers. Pi equals 180 degrees equals 0.5 turns. 1 radian has different units than 1 steradian and if they didn't there wouldn't be a need for two different words to denote them. The quantity is a ratio of two lengths, and the length measure does "drop out". But it's not just any ratio, it's a very particular ra…
Jeez, I've really kicked off quite the heated discussion in the replies... I don't want to get bogged down in lengthy arguments, but I feel I should explain my reasoning in more detail.
Essentially, I think that whatever angles are, they are not like other dimensionful physical quantities. I have two arguments.
The first: Someone mentioned symmetries in a reply. I wanted to mention them too but didn't have time to structure my thoughts into a coherent argument. But the gist of it is that dimensionality is just a kind of scale invariance, and the scale invariance of angles is fundamentally different from that of linear quantities due to their periodicity — to apply a unit transformation, you have to scale the quantity _and the period_.
The second: Consider units from a "type theory" perspective instead. If you are considering exclusively linear trigonometry (no arcs), it's trivial to assign a dimensional type structure to expressions (e.g. cos takes angle type and maps it to dimensionless type). But as soon as you allow arc lengths, it becomes cumbersome to type common expressions.
I think these distinctions form the crux of the disagreement. Ultimately, it depends on your intuitive notion of what "dimensionality" actually means, and how it ought generalise to other kinds of quantities.
Here is an example to highlight my point. Let there be a circle C of centre O and radius r. Let A be a point on the circle. Let there be a point M outside the circle such that (AM) is tangent to C. Let B be the intersection of C and [OM]. Let s be the arc length along C from A to B. Then we want to write AM = r tan(s/r).
How does one get s/r to resolve to an angular dimension? Ought we instead ascribe s dimensions of length-angle? Imagine, then, that the circle is in fact a pulley, and we wish to measure a change x in length of rope as the pulley rotates through the angle of the arc from A to B. We would want to write x = s. But this is now dimensionally inconsistent.
It's certainly possible to make all these expressions correctly typed by introducting appropriate conversion constants. But this seems to me to be cumbersome. Since in physics, arc and linear lengths can convert freely into one another, it seems more economical to just let angles be dimensionless.