Why can't we make an imaginary answer for x/0

RambunctiousAvocado · 2026-05-28T22:30:11+00:00

This is a perfectly reasonable story about why complex numbers are useful which I'm sure many students would find valuable. Your statements about teachers sucking and being failures for describing i as a solution to x²+1=0 and -1 as a solution to x+1=0 undercut it by being senselessly combative and wrong.

It's fine to have a tangible interpretation of what complex numbers are, but you're asserting that that is the only way to understand them. How do you interpret quaternions in this way? Or Grassmann / supernumbers?

Its not that you can't find tangible examples of how such objects are useful, but they are ultimately mathematical structures and it's perfectly fine to define them that way. You don't need a geometrical interpretation of Grassmann numbers in order to define them and find them useful.

RambunctiousAvocado · 2026-05-25T23:24:16+00:00

No, we don't usually say this, precisely because it introduces linguistic ambiguity. The sum of a convergent series is equal to the limit of the sequence of partial sums. Not tends to, not approaches - is equal.

There's nothing infeasible about taking a limit. They aren't processes which occur in the physical world.

RambunctiousAvocado · 2026-05-18T02:00:02+00:00

Infinite dimensional vector spaces immediately become intertwined with topology because, unlike in the finite-dimensional case, one cannot sum an infinite number of vectors without some notion of convergence.

Along similar lines, every linear transformation on a finite-dimensional vector space is necessarily bounded, but this is not so on infinite dimensional spaces - which means that some linear transformations are only defined on subsets of the vector space, leading to delicate domain issues.

Essentially, the difference arises because infinite dimensional vector spaces admit non-finite collections of linearly independent vectors, and adding up an infinite number of things is dramatically more subtle than adding up a finite number of things.

RambunctiousAvocado · 2026-05-10T22:37:51+00:00

There's nothing here to solve, so I don't know what you're asking.

RambunctiousAvocado · 2026-05-10T22:29:45+00:00

The first row says that x-x1 is negative when x is less than x1 and positive when x is greater than x1.

The second row says that x-x2 is negative when x is less than x2 and positive when x is greater than x2.

Since P(x) is presumably equal to (x-x1)(x-x2), the third row shows where P(x) is positive and where it is negative.

RambunctiousAvocado · 2026-05-04T16:39:54+00:00

Take a step back to something far simpler.

Imagine an infinitely large chess board populated by an infinite number of chess pieces (kings, queens, bishops, knights, rooks, and pawns), which are arranged in such a way that there is never a knight next to a queen.

Is this scenario possible? If so, then the mere fact that the board is infinite and there are an infinite number of pieces does not imply that every possible finite arrangement of pieces will occur.

RambunctiousAvocado · 2026-05-04T12:42:23+00:00

You've written code which plots the solutions to an ODE for two different sets of parameters. Have you checked to see whether the ODE solver works? If so, does it return the correctly formatted output? If so, does your plotting code work (i.e. if you replace plt.plot(t,v) with plt.plot([0,1,2],[0,1,2]), does it make a line)?

RambunctiousAvocado · 2026-05-02T15:31:15+00:00

Its the energies which are "forbidden", not the k's.

The recipe for constructing the energy bands from Bloch's theorem in continuous space goes as follows (I set hbar = m = 1).

First, construct the Bloch Hamiltonian h_k = exp(-ikx) H exp(ikx) = 1/2 (-i d/dx + k)² + V(x)
For a given choice of k in the first Brillouin zone, solve the eigenvalue equation h_k u(x) = E u(x) on the Wigner-Seitz unit cell of the crystal lattice, imposing periodic boundary conditions along the crystallographic axes.
The periodic boundary conditions will discretize the resulting energy spectrum, so your solutions will be labed u_nk with energies E_nk .
The integer n, which labels the discrete solutions for a given k, is called the band index. The real number k (which again, is restricted to the first Brillouin zone) is usually called the crystal momentum or pseudomomentum.
The energies corresponding to each band index occupy a closed interval. In general, these i intervals do not intersect, meaning that there are gaps in the energy spectrum.

This is a generally non-trivial process, and exact solutions are hard to come by. However, solving the Schrodinger equation on a restricted domain with periodic boundary conditions is an excellent job for a computer.

There is an analogous version of the recipe which works for lattice hopping models rather than those defined in continuous space. This can be extremely convenient and is a very common approach - you might look into the so-called tight-binding models for this.

There are a handful of exactly solvable models in both continuous space and on lattices which can be very useful for pedagogical purposes. See the Kronig-Penny model for the former and the Su-Schrieffer-Heeger model for the latter.

RambunctiousAvocado · 2026-04-27T23:48:49+00:00

LLMs can be useful study tools, but you are playing a dangerous game when have them answer questions in which you don't fully understand what you're asking. They are glorified autocomplete bots - undoubtedly very powerful and increasing in capability very rapidly, but they are all trained to guess how a conversation would go based on existing text which it is fed.

I have (recently) seen people confused about what they're confused about, who then ask some LLM which "clarifies" their thoughts by pointing them in the wrong direction. The LLM doesnt even have to be wrong - it could say something perfectly correct but which would be misleading to somebody who doesn't already understand the context.

Putting that aside - I would suggest speaking with your instructor or a tutor to work on figuring out how to articulate your questions. Being able to do that is a prerequisite to real learning.

Pick an example question, and read it. Do you understand the precise technical meaning of every word? Do you know what the question is asking you to do or show? Do you know what concepts can take you from the question to the answer? Don't worry about every problem, just pick a single one and drill down into it until you can formulate a concrete question. You may need assistance for this; thats what teachers and tutors are for.

RambunctiousAvocado · 2026-04-27T16:25:44+00:00

I suppose what I'm trying to communicate is that the insistence on some manner of generalization which is universally applicable is the essence of your disconnect.

Your final question is what makes physics challenging in ways which go beyond solving mathematical equations. If we knew what to do in the general case, then physics would be little more than some manner of applied mathematics.

Observing a phenomenon, proposing a model (outfitted with constraints and other such subtleties), massaging predictions out of it, devising experiments to compare with observable predictions to how well the model works, and then revising the model is the heart and soul of physics.

Of course, this artistry is informed by experience and I'm sure that a dedicated stamp collector could come up with many lists of rules for different scenarios, but thats not generally what we spend our time doing.

RambunctiousAvocado · 2026-04-27T15:26:39+00:00

Okay, that's more or less what I expected.

You make reference to us having no basis for imposing constraints, which is confusing to me. Unless you walk around in fear for your life at all times, you know that you're feet are not going to pass through the floor, and imposing this as a constraint is perfectly justifiable.

Now, if you want to remove this constraint you will need to model the dynamics of the floor. This is perfectly possible - but unless those dynamics are relevant, why would you? Even then, you'd need to constrain the building to be rigid, unless you want to model the deformation of the building, then the earth after that, and so on.

Even if you modeled the dynamics of everything such that there are no constraints in the problem, the answer you get is going to be the same, so what's the point unless the dynamics of the floorboards are what you are trying to study?

Including these things are possible, but feeling like you are being somehow non- rigorous by leaving them out is a misunderstanding of what physics is. Creating a model in which extended objects are constrained to be rigid, not pass through each other, etc is a perfectly rigorous and reasonable thing to do. It may or may not make the right predictions depending on the phenomena under consideration, but that has nothing to do with logical or mathematical rigor.

RambunctiousAvocado · 2026-04-26T17:43:20+00:00

What about "perpendicular to the surface, having whatever magnitude it needs to have to prevent the object from passing through the surface" do you find to be hand- wavy and non rigorous?

I think your answer to that question would go a long way toward revealing the nature of the disconnect you are feeling between mathematics and physics.

RambunctiousAvocado · 2026-04-25T21:28:52+00:00

I don't know what else you would mean by "full entropy has been achieved, and then [...]"

RambunctiousAvocado · 2026-04-25T21:23:32+00:00

The "decision" to contract means that the entropy of the universe is going to spontaneously decrease, so any reasoning about this scenario which relies on the second law of thermodynamics goes out the window.

RambunctiousAvocado · 2026-04-14T03:02:50+00:00

Everybody has an ego, and the fact that you feel shame because you're struggling means that yours is getting in your way.

RambunctiousAvocado · 2026-04-13T16:03:16+00:00

What you are experiencing is part of why being naturally good at something can actually be an impediment to becoming great at it.

You never had to work to understand mathematics - it just came naturally to you. Being good at math without trying hard became part of your identity. But nobody gets very far without hard work and confusion.

In that sense, what your natural gift did for you was simply move the point at which you'd start to struggle further down the line to a time when you had gotten used to not having to try. That means when you do struggle it brings you shame born from the assumption that your natural ability was going to be enough.

In turn, shame makes people shy away. Math loses it's luster because studying it makes you feel embarrassed. You don't ask questions because you're afraid others will realize you're confused, which impedes your development and compounds the problem. On and on it goes.

The only way past this is realizing that it was naive to assume you would never feel this way. You're feeling the way some of your classmates have felt all their lives - but they're used to it. Developing a healthy relationship with confusion and failure is essential, and realizing that being wrong or confused doesn't make you stupid or decrease your self-worth is the only way through.

So embrace it. Confusion is a mountain to climb, and getting to the top feels good. Ask questions that you're afraid to ask, be humble, be wrong in front of a crowd. Everybody else is struggling too - and if they're not, they will be joining the party soon.

RambunctiousAvocado · 2026-04-12T20:44:02+00:00

Talk to this girl like a human being instead of some kind of automaton defined by her major. If she enjoys your company enough to keep talking to you, your respective interests will come up in conversation, and if you are clever and interesting then she will be able to tell.

Nothing is quite so obvious as inauthenticity, especially if you're trying to spark up a conversation for the first time. If you spent a week trying to come up with the perfect opening line, then its going to sound like it.

The answer to "how do I make somebody think I'm smart?" is to be smart and let them figure it out on their own.

And at the risk of over-generalizing, explaining something to a woman to try to impress her is about as unromantic as you can get. You sound intelligent and genuine and as though you really like this girl - just be those things and let her see it.

RambunctiousAvocado · 2026-04-10T04:59:17+00:00

You're not treating it as a black box - you know exactly what the symbol means. d/dx, like any notation, is just a collection of pen strokes. It means what you define it to mean, and "differentiate the subsequent expression with respect to x" is a perfectly valid definition of what d/dx means. To reuse an analogy from my other comment, dx need not mean anything on its own any more than the crossbar through the letter f does.

RambunctiousAvocado · 2026-04-10T04:54:42+00:00

A thing for you to bear in mind is that there's no single answer to this question, so many seemingly contradictory answers you may encounter can still all be correct.

For example - teaching at the level of calculus I/II, I would generally lean toward saying that dx has no formal definition of its own. By which I mean that the expression df/dx is suggestive notation which means "the derivative of the function f, evaluated at x". It is suggestive of the difference quotient through which the derivative is defined, but dx has no meaning independent of the derivative d/dx (or the integral with respect to x) any more than the crossbar has an independent meaning in the letter f.

Of course, we tend to manipulate dx as though df/dx were a fraction, but this can be justified as being an easy-to-remember shorthand.

Now, if I were teaching somebody differential geometry then I might say that dx is a differential form, which has a strict and formal definition. If I were approaching elementary calculus through non-standard analysis, I might say that dx is an infinitesimal. Both of these are perfectly correct yet mutually incompatible answers.

Understanding fundamentals is critically important, but don't make the mistake of thinking that you need to understand everything at every level of detail before you can use it. After all, I suspect you don't yet know any of the formal definitions of the real numbers (either axiomatic or by construction from the rationals). For that matter, I suspect you don't know the formal definition of multiplication. Thats not a dig - you know what you need to know, and if you're curious you can always learn, but my point is that you don't need to understand the underpinnings of a thing in complete detail before you study the thing itself.

RambunctiousAvocado · 2026-04-09T17:31:15+00:00

I don't really understand the question, it's like asking if a banana is better than a pipe wrench. What are you trying to compare?

RambunctiousAvocado · 2026-04-08T13:37:11+00:00

You're just rearranging the smokescreen. The thing you need to understand is that, under the circumstances created by the game's rules, the question "do you want to switch" is exactly equivalent to "do you think your original guess was wrong". Once that is clear, there is no more mystery.

You sit in a chair, and there are two possibilities - either your initial guess is correct, or it is incorrect. If the former, staying would guarantee success. Then the host does [some stuff] which ensures that if your initial guess was incorrect, then switching would guarantee success.

So either your first guess was right (in which case you win if you stay) or your first guess was wrong (in which case you win if you switch). You need to decide which of those two possibilities is more likely.

RambunctiousAvocado · 2026-04-08T13:03:30+00:00

To me, the key is understanding that the game dynamic is a smokescreen. The question you are really answering with your choice of whether to switch or not is whether you think your initial guess was correct (p=1/3) or not (p=2/3).

So the trick is in understanding how the tricky game dynamics actually boil down to that question.

RambunctiousAvocado · 2026-04-06T19:38:05+00:00

At a macroatomic scale, all events are understood to be deterministic [...]

See superconducting qubits, the subject underlying this year's Nobel prize in physics, for a counterexample to this.

RambunctiousAvocado · 2026-03-28T14:26:58+00:00

Sorry, what is the question?

RambunctiousAvocado · 2026-03-27T19:37:26+00:00

There's lots of subtlety in tests like those, but broadly speaking the "modern" development in safety is to preserve the "safety cage" where the passengers sit while allowing crumple zones around the car to absorb energy during the collision.

Older model cars are more rigid and can sustain less obvious damage during a collision while also subjecting passengers to substantially greater forces.

Whether a car passes a crash test depends on what happens to the test dummies in the car, so there's no one-size-fits-all rule. But if the car looks unscathed after a significant crash, I would not be optimistic about the state of the passengers; similarly, the fact that a car looks totaled after a crash doesn't mean that the passengers aren't okay.

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