u/future_sponJ

Functions like Re(z), Im(z), |z|, arg(z), sgn(z), & z̄.

They're all very basic continuous but non-elementary non-differentiable functions ℂ->ℝ or ℂ->ℂ but are elementary & differentiable almost everywhere in ℝ. If you have any of them along with z, you can derive all of the others using elementary operations. They seem special but I haven't ever seen a term for them? Does one exist?

I'm 14 & have already learned linear algebra & calculus on my own. I sometimes read about particle physics & other things but I want to build a foundation & start studying HS-level physics online now. Where do I start?

A robot will only do something if it is commanded to do it & it is physically able to. So, assuming it wasn't specifically created for this purpose, it will only "take over the world" (or attempt to rise on humans) if the instructions were vague & it had the resources to do so. Otherwise, I don't see the problem.

Why would an electron bounded to an atom & a free electron have the same rest mass? Why does a down quark have the same fundamental rest mass whether it's in a proton, neutron, or any other hadron?

The same question could also be applied to the electromagnetic field too not just the Higgs Field.

I only know basic set theory so I apologise if this is stupid but I was thinking about the idea of sets of numbers that are rational multiples of each other.

Some examples (q ∈ ℚ , q ≠ 0):

• Set 1: q (all non-zero rational numbers)
• Set 2: q√2
• Set 3: qπ
• Set 4: q(π+1)
• Set 5: qe
• Set 6: qe²

Now how do I make a system for uniquely labeling them? It's not like I can choose the smallest or largest number in each set since they are unbounded & dense. What do I do then?

Most mathematical inventions/discoveries were "useless" at the time they were made/found. I can't imagine someone working their entire life on a single useless problem not knowing if they would solve it or not especially if there was no funding for it.

I know that a lot of modern mathematicians are either funded by companies they do applied math for or by universities but someone dealing with pure mathematics that had no real life applications (like Leibniz or Galois) wouldn't have that privilege.

I'm probably going to IGCSE next year (I'm yr9) & I am planning to study physics in college (with math, engineering, & CS as backup choices). I've never studied business before but bio has always been hell for me.

Will biology or Business matter for a physics degree? Which is better for me?

I'm currently in Gr8 Yr9 at a British curriculum school & my dad wants to make me go to a ministerial curriculum school. I'm going to sar IGCSE next year if I don't switch.

Which curriculum is better for both grades & college?

I'm also studying HS & uni-level subjects on my own to try to get into a college early (probably in another country though) if that matters.

I don't really understand the hyperbolic trig functions (e.g. sinh) from a geometric perspective but I think they represent the same quantities that normal trig functions (e.g. sin) represent but in hyperbolic space instead of flat space. So why aren't there (afaik) trig functions for spherical space too?

In 1D, you can move in 2 directions, & in 2D, you add 2 more, & in 3D you add 2 more, etc.

I was wondering what's so special about 2. Why not 1 direction? Or 3? Or 4? & so on. (time in a sense could be considered to be in 1 direction but that's a stretch)

It could be a consequence of the law of trichotomy of real numbers but that just begs another question, why is each physical dimension a real number line?

In 1D, you can move in 2 directions, & in 2D, you add 2 more, & in 3D you add 2 more, etc.

I was wondering what's so special about 2. Why not 1 direction? Or 3? Or 4? & so on. (time in a sense could be considered to be in 1 direction but that's a stretch)

It could be a consequence of the law of trichotomy of real numbers but that just begs another question, why is each physical dimension a real number line?

I understand why things like ratios (like angles or probabilities for example) are dimensionless but why are counts dimensionless too? I hear a lot of say that the mole is dimensionless because it's a discrete count but shouldn't the Coulomb be too?

I have been hearing about how abusive large-scale farms are in places like Europe & was wondering if that was the case with most halal farms here too or if I should buy from specific brands.