51–60 of 179 posts
You should call it d/dx bar. Anyway, I love the choice of theta because a while ago I came up with a nice notation for sin and cos and this fits it really well. When I first learned trig, it was by way of skipping into physics early. I only understood cos as the magic button for getting x components from angles, and y as the button for y components. So my notation is based on this very literal brute understanding. Al…
Also, π is the wrong constant. The very definition is awkward: the ratio of two radiuses to the circumference. How about one radius? It is much more natural to work with 2π. Some people use the letter τ (Tau) to denote 2π, and it simplifies almost all naturally occurring expressions. For example, what is more elegant? e^(π*i) = -1 e^(τ*i) = 1
basically the point of the Tau Manifesto: https://tauday.com/tau-manifesto
And would ya look at the date today... 6/28...
Earlier quoted context omitted.
Angles are absolutely more dimensionless than lengths are. For an easy check you can't add quantities where the dimension differs, which means it doesn't make sense to add a length to its cube. On the other hand it does make sense to add an angle to its cube - this is a necessary component of computing sin(angle) by the power series sin(angle) = angle - (angle^3)/6 + ...
How can we compute angle - (angle^3)/6? 360 - (360^3)/6 = -7M degrees or is it this? 2*pi - (2 * pi)^3 / 6 = -35 radians = -2k degrees Or maybe this? 1 - (1^3)/6 = 0.8 turns = 300 degrees They're wildly inconsistent because I'm not taking the units into account and we have to take the units into account.
As far as you three examples go, which is "correct" depends on what you are trying to calculate - if you want this to approximate the power series for sin close to 0 you should use radians. Otherwise you use something else.
Earlier quoted context omitted.
I'm curious how blind people normally engage with math. For me, engaging with math almost always means conjuring up a visual representation in my mind. Failing that, an equation. Since visualization is so fundamental to doing math, and since mathematical symbols and equations are a written language for which there is no spoken analog, I really can't imagine engaging with math without my eyes. Even reading equations a…
The way I've heard those distinguished in spoken math is x + b^2 is said "x plus b squared" and (x + b)^2 is said "x plus b all squared. There's a similar approach for divide "x plus b over 8" vs "x plus b all over 8". That was often enough but if it wasn't you'd be reduced to pronouncing brackets.
Unrelated to the content but complaining about the website is a popular thing to do here so I'd like to share my experience, as a blind user. Here is what my screen reader sees for this page: Recently, I came up with a trick that can get rid of in many cases. It’s pretty simple, but it has some interesting implications. This is the trick: I just define a new derivative operator, like so: That’s all. You just take the…
I just define a new derivative operator, like so: dxđ f(x)≡2π1 ⋅dxd f(x). That’s all.
... structure possible. I have basically zero web dev experience so I don't know how to fix this, maybe I need to tweak the KaTeX settings.
I think it's quite sad that math is so difficult on the web. While setting up the blog, I looked around and it seemed like FF is the only browser with proper MathML support, but I think that was also being phased out because it's apparently buggy and hard to maintain. IMO, the screen reader version should just basically be the LaTeX source, which is probably kind of awful when read out loud, but at least it would be unambiguous.
On a similar note, when doing quantum physics, I like to introduce h dot, which is i × h bar. There are tons of formulas where you either get i × h bar or -i / h bar, but these are just h dot and 1 / h dot, so this removes a little sign confusion and saves a little handwriting.
People will argue that real constants are more natural, but maybe they're not. Maybe radians are naturally imaginary, so if h bar is meant to have dimensions of energy time per radian, then it's better to use the imaginary h dot.
Also, π is the wrong constant. The very definition is awkward: the ratio of two radiuses to the circumference. How about one radius? It is much more natural to work with 2π. Some people use the letter τ (Tau) to denote 2π, and it simplifies almost all naturally occurring expressions. For example, what is more elegant? e^(π*i) = -1 e^(τ*i) = 1
e^pi + 1 = 0, of course
Also, π is the wrong constant. The very definition is awkward: the ratio of two radiuses to the circumference. How about one radius? It is much more natural to work with 2π. Some people use the letter τ (Tau) to denote 2π, and it simplifies almost all naturally occurring expressions. For example, what is more elegant? e^(π*i) = -1 e^(τ*i) = 1
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 suddenly.