I was once lucky enough to take a physics class taught by the head of the department, and I remember one of his policies on tests or homework was that if you got an answer that was off by 1/2 or 2*pi or anything like that, he'd nonetheless issue full credit because "you got all the physics right."
A trick to eliminate 2π (sometimes)
141–150 of 179 posts
Re: A trick to eliminate 2π (sometimes)
#142Earlier quoted context omitted.
That's actually exactly the question I asked my math teacher when I first learned about radians. I mean, I learnt degrees when I was very little, at an age when one tended not to question why, but I learned radians at an age old enough to question why. The answer I received was about making trigonometric identities cleaner: the derivative of sine becomes "just" cosine rather than a hypothetical turn-based sine (calle…
Due to the fact differential operators are linear and the nature of the accumulation of constants of integration it’s pretty easy to prove that the differential operator proposed is in fact equivalent to the standard derivative operator. Source: used to tutor calculus and differential equations in college. Generally speaking this is not a useful trick. There are _plenty_ of amazing ways to leverage Euler’s identity b…
I too fail to see a problem. There is a certain degree of trickiness in all walks of life. Sometimes there is a decent hack that simplifies things and sometimes there is a notion that is just as complex to deploy as the current state of the art and hence isn't worth pursuing.
Pi has a definition for a good reason. Sometimes a discipline has to put up with perceived oddities until the real, deep problem is surfaced, grappled with and kicked in the nuts until it gives up and allows a paper or two to emerge without ridicule. With luck it might really show some ankle and a Nobel heaves into view 8)
This isn't it for (n)Pi n Phy Sci.
Re: A trick to eliminate 2π (sometimes)
#143Earlier quoted context omitted.
If there is a bug in the screen reader then that's on the vendor. However the overwhelming number of cases where a website is unusable with a screen reader are due to lack of proper semantic tagging, like alt text and so on. The last thing a screenreader user wants to hear on a site is "button," "button," "button..." There's certainly room for tooling to help though. I definitely think it sucks that many accessibilit…
Did you even bother looking at the website source before writing this?
What part of if there’s a bug in the screen reader it’s on the vendor did you fail to understand?
Re: A trick to eliminate 2π (sometimes)
#144Earlier quoted context omitted.
That's actually exactly the question I asked my math teacher when I first learned about radians. I mean, I learnt degrees when I was very little, at an age when one tended not to question why, but I learned radians at an age old enough to question why. The answer I received was about making trigonometric identities cleaner: the derivative of sine becomes "just" cosine rather than a hypothetical turn-based sine (calle…
Due to the fact differential operators are linear and the nature of the accumulation of constants of integration it’s pretty easy to prove that the differential operator proposed is in fact equivalent to the standard derivative operator. Source: used to tutor calculus and differential equations in college. Generally speaking this is not a useful trick. There are _plenty_ of amazing ways to leverage Euler’s identity b…
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Re: A trick to eliminate 2π (sometimes)
#145To address the problem they discuss at the end with defining Θ = e^2πi, they could instead define Θ(x) = e^2πix, the circular analog to the exponential function exp (which is really more fundamental than exp(1) = e anyways).
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
> tau = 1
Who knew that if you go around in a circle you get back where you started?
Re: A trick to eliminate 2π (sometimes)
#146Earlier quoted context omitted.
The reason it is confusing is because an angle measure is a kind of logarithm of a rotation, and logarithms (sort of) have a unit: the base. The appropriate canonical representation of a rotation is a unit-magnitude complex number z = exp iθ = cos θ + i sin θ , which has a planar orientation (whatever plane i is taken to represent; if you want to represent a 3D rotation you can replace i with an arbitrary unit bivect…
I think I fundamentally disagree with you. Angles do not have a "base" any more than meters do (being embedded inside some metric space could be considered a base I suppose). But you seem to be drawing a distinction between meters and angles in your analogy where I assert none exists. The base of a number system only affects representations. This is not true for divisions of lengths. 1 meter divided by 2 meters is 0.…
The logarithmic base for an angle is something like "degrees", "radians" or "turns".
This is analogous to the way a scalar logarithm can have a base of "octaves" (doublings), "decibels", or "powers of the golden ratio" (as found in the Zometool construction toy). Or pick your favorite other logarithmic system.
Both are "units" in a certain sense, but neither one is quite the same kind of "unit" as light years or foot–pounds or amperes.
> 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space
Just as 1 meter straight ahead divided by 2 meters straight ahead is the unitless scalar number 0.5, we can likewise treat angles (i.e. rotations) as ratios: 1 meter straight ahead divided by 1 meter to the right has the unitless bivector-valued ratio i, oriented like the ground you are standing on. You can multiply this bivector by some other coplanar vector to rotate it a quarter turn. For example, you can multiply it by the vector «3 inches due North» to get the new vector «3 inches due West»; notice how the units do not change because our bivector i is unitless.
Re: A trick to eliminate 2π (sometimes)
#147The argument about why not to include the i in 2 pi i is incorrect. The problem is he says (e^x)^i2pi = e^i2pix does not work because ln(e^i2pi)=0. But he needs to use the complex logarithm. And for the complex logarithm ln(e^z) =z for z in C. If that wasn't the case calculation rules of logarithms and exponentials would depend on if arguments are complex or real, a lot of physics would become much more complicated s…
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.
Re: A trick to eliminate 2π (sometimes)
#148Earlier quoted context omitted.
I think I fundamentally disagree with you. Angles do not have a "base" any more than meters do (being embedded inside some metric space could be considered a base I suppose). But you seem to be drawing a distinction between meters and angles in your analogy where I assert none exists. The base of a number system only affects representations. This is not true for divisions of lengths. 1 meter divided by 2 meters is 0.…
The logarithmic base for an angle is something like "degrees", "radians" or "turns". This is analogous to the way a scalar logarithm can have a base of "octaves" (doublings), "decibels", or "powers of the golden ratio" (as found in the Zometool construction toy). Or pick your favorite other logarithmic system. Both are "units" in a certain sense, but neither one is quite the same kind of "unit" as light years or foot…
I understand your analogy, but I reject it's validity. Degrees/radians/turns map to meters/feet/angstrom. Decibels and octaves are true "number" multipliers. You could for instance talk about degrees in octaves or in dB if you like. It's just not particularly useful for the domain
Edit: another example difference. I can't measure an octave or dB. I can measure a degree
Edit2: we've reached reply limit but I concede you can measure a decibel. Point about dB degrees still stands though
Re: A trick to eliminate 2π (sometimes)
#149Earlier quoted context omitted.
The logarithmic base for an angle is something like "degrees", "radians" or "turns". This is analogous to the way a scalar logarithm can have a base of "octaves" (doublings), "decibels", or "powers of the golden ratio" (as found in the Zometool construction toy). Or pick your favorite other logarithmic system. Both are "units" in a certain sense, but neither one is quite the same kind of "unit" as light years or foot…
I understand your analogy, but I reject it's validity. Degrees/radians/turns map to meters/feet/angstrom. Decibels and octaves are true "number" multipliers. You could for instance talk about degrees in octaves or in dB if you like. It's just not particularly useful for the domain Edit: another example difference. I can't measure an octave or dB. I can measure a degree Edit2: we've reached reply limit but I concede y…
You absolutely can measure an octave or decibel. It's just a relative quantity, in just the same way any quantification of orientation is relative.
(For example, you can measure an octave by marking out a particular fret on your guitar; it will make an octave change whichever particular note you start with.)
You can feel free to "reject" whatever you want. You'll just be wrong/confused. ;-) (But you'll be in good company. Most working engineers, scientists, and mathematicians don't have or need a particularly clear philosophical understanding of angles.)
Re: A trick to eliminate 2π (sometimes)
#150It's a common idea in analysis / number theory in mathematics - see for example https://en.m.wikipedia.org/wiki/Exponential_sum