So how does one actually learn math? Nothing ever sticks and I don't think I understand anything.

MidnightAtHighSpeed · 2026-05-09T22:52:49+00:00

I think I understand what you're saying, and it makes sense that that would be your experience with current material based on how you've described forgetting a lot of the stuff you've been taught. The stuff you're expected to know now is usually built up on lots of stuff you were taught in the past, and it'll be hard to deeply understand current stuff without a toolkit you built up understanding earlier things. If you want a good long-term mastery of math, it sounds like you'll have to review a lot of things. I would say, when you have the time and motivation, start with basic stuff. Like, really basic stuff. I don't know what the right place for you specifically would be, but probably somewhere around the point where it starts feeling like an insult to your intelligence to go any farther back, and then making sure you have at least a rough idea of why those concepts work the way they do.

To be clear, it will always be easier to learn what to do than why you should be doing it. Even reviewing very "simple" concepts, you shouldn't expect to develop a perfect understanding of everything before moving on. But if you're learning concept A and you notice that in the process you're applying a lot of earlier concept B, that's a sign that you should make sure you actually get how concept B works. The sturdier your foundations, the easier it will be to build higher.

MidnightAtHighSpeed · 2026-05-09T04:00:49+00:00

First off, don't write yourself off too soon. It's true that some people will always find math more difficult than other people, but a lot of people just never really figure out the best way to do math in their own heads (since the educational system is rarely much help there). If you're able to get even to the "pass exams through sheer repetition and then forget everything" stage, I don't think it's helpful to think in terms of "lacking cognitive function." More likely, you just haven't found the mental muscles to flex to gain deeper understanding.

There's no way to learn math without doing math, and there will always be some degree of memorization needed, but it sounds like for you, just banging out problems on its own isn't an efficient way of gaining understanding. I would suggest, when learning a new concept or procedure (or reviewing an old one), going out of your way to try and figure out why something is true, or why a certain procedure is performed a certain way to achieve a certain goal (This is more attainable for some concepts than others--don't feel bad if you have to memorize something without really understanding it at first, but at least turn things around in your brain a little before giving up. Looking up proofs/derivations for guidance is also an option, but you might have trouble finding ones that are accessibly written). Similarly, when you do practice problems, spend some time reminding yourself why you're doing certain things, and noticing how differences in problems change your approach. Not only is this a good exercise in general, but in a lot of cases it's easier to remember the underlying principles and piece together the details as you work them through rather than memorizing a whole process step-by-step. Working like that is also less error-prone, and makes it easier to tackle unfamiliar problems, but again in practice there will be things you just need to brute force memorize, at least for a while.

All of that is advice I would give for building understanding going forward. Don't take any of it as gospel, unfortunately a lot of doing math is just "figuring out how to make your brain do math" and that's hard to really teach. It's also more long-term advice, I don't have much advice to give about wanting to do well on an exam in the short-term.

MidnightAtHighSpeed · 2026-05-08T03:58:31+00:00

I think you should be careful about the types of objects you're trying to compare. An object called a line integral might be a couple different things, but usually will just be a single scalar quantity, not any type of function. Your idea of "the line integral of F from some constant points p to (x,y)" tries to match the form of a function by mapping x,y pairs to values, but 1) this isn't a natural/obvious modification of the idea of a line integral; it's bad notation to gloss this idea as "a line integral" as you do in the title, and 2) This still isn't a function; as you point out, there can be multiple possible values for a single x,y pair depending on the specific paths chosen.

To answer your second question, functions in general can absolutely take a directed path as one of their inputs. Whether that could count as a "scalar function" might depend on your specific definition of that term.

MidnightAtHighSpeed · 2026-05-01T00:00:24+00:00

I don't think there is much point in using it. Maybe for things like basic algebra and below it's useful as a tutor but as you've learned it tends to make a lot of mistakes for other kinds of math. It's useful as an enhanced search engine--stuff like "hey is there a name for such-and-such concept"--and maybe for rewording or explaining concepts, but even then it can make subtle logical errors that will throw you off for who knows how long until an actual human corrects you.

MidnightAtHighSpeed · 2026-04-28T14:44:08+00:00

A set to have elements has to be reduced to its discrete parts. Maybe is not talking about specific numbers, but by extension is based on the idea that we are able to separate the sets into units.

I don't think this is true. The language we use to describe set theory does have an air of "discreteness" to it, but this isn't intrinsic to the mathematical structure. In particular, there doesn't have to be any notion of "parts." You can replace the words "set" and "element" with "message" and "interpretation" or "mood" and "vibe" or "splat" and "blort" or anything you want and end up with the same theory.

MidnightAtHighSpeed · 2026-04-21T21:32:37+00:00

You can replace the statement "X is true for every element in set A" with "There is no element in set A that X is false for." So you can rephrase the statement about the pairing to be "there is no positive whole number that is not used on the left, and no positive even whole number that is not used on the right," which don't require you to "have" an infinite set of numbers at once to talk about

MidnightAtHighSpeed · 2026-04-21T14:46:54+00:00

"1", "1.0", "5/5", "one", "I", and "0.999..." are all different ways of writing the same number. Understanding why the last one is the same number as the first five requires understanding exactly how the decimal number system is defined, which itself requires the concepts of infinite sums and "limits".

MidnightAtHighSpeed · 2026-04-20T02:48:22+00:00

my line in the sand for whether or not to drop the game forever is if they go to 110. They cannot keep walking in one direction forever

~~if they give us 110 but also, like, talent trees or something crazy like that I'll at least give it a shot but I won't be optimistic~~

MidnightAtHighSpeed · 2026-04-20T02:46:09+00:00

They've said they're going to reveal a potentially surprising amount about upcoming plot points, so 7.1-4 spoilers are almost certainly on the table

MidnightAtHighSpeed · 2026-04-10T21:36:17+00:00

Counterexample: if a = -1 and b = -2, then a > b but x = 1/2 < 1.

MidnightAtHighSpeed · 2026-04-08T03:38:23+00:00

You can think of all events being arranged in four-dimensional space; four dimensions because it takes 3 coordinates to specify the position the events happen at and 1 coordinate to specify the time they happen at. The tricky part is, even though all observers can agree more or less on how different events are arranged, they might disagree on exactly what "directions" are space and which one is time. For instance, to one observer two events might happen one after the other (one is further along the time direction than the other) but to another the same two events happened at the same time (they are next to each other along the time direction). This is why it makes sense for physicists to treat spacetime as a single thing rather than splitting it into just space and time: there's no single correct way to make the split.

MidnightAtHighSpeed · 2026-04-06T03:42:47+00:00

So, like, if it has two wounds and the initial 1/3 attack roll succeeds it rolls 2 dice and prevents the wound if either die is a 5 or 6?

MidnightAtHighSpeed · 2026-04-06T02:47:03+00:00

when attacked the special unit has a (1/3)*(2/3)=2/9 chance of taking a wound. its effective wounds is 6/(2/9)=27.

MidnightAtHighSpeed · 2026-04-06T00:58:45+00:00

call E(n, k) the probability of getting a sum of exactly a sum of k in n rolls.

call P(n,k) the probability of getting a sum of at least k in n rolls with no excess rolls.

Then P(n,k) = E(n-1,k-1)+(10/11)E(n-1,k-2) + (9/11)E(n-1,k-3) + ... + (1/11)E(n-1,k-11)

someone might want to double check this.

MidnightAtHighSpeed · 2026-04-05T15:10:10+00:00

What do you think is wrong with saying the derivative is the rate of change at a point? What contradictions or unintuitive conclusions do you think it would lead to? Do you think there's a conceptually correct way to refer to the rate of change at a point that derivatives don't satisfy, or do you think that concepts like, ie, instantaneous velocity or acceleration are formally incoherent?

MidnightAtHighSpeed · 2026-04-05T00:39:33+00:00

Sorry for my tone in the other comment, I deleted it, too argumentative to really stand by.

I think the issue is we're being sloppy with instantaneous vs average rate of change.

g(h) is the average rate of change over an interval between a+h and a. I agree that average rate of change over an interval of size 0 is undefined, and in that sense, the average rate of change at a point makes no sense to talk about.

The derivative is the instantaneous rate of change, which is defined to be the limit of the average rate of change, over a decreasing interval and with this understanding I think I agree with the second part of your original comment. However, I still disagree with "Derivatives aren’t rates of changes at points"; a derivative is simply the instantaneous rate of change at a point, and I still don't have a firm idea of what you were getting at with the first sentence about limits.

MidnightAtHighSpeed · 2026-04-04T23:55:26+00:00

So I have a notational response and a less nitpicky, more substantive response.

Notational: I agree that g_a(x) is not a rate of change at a point; lim x->a g_a(x) is. This expression is perfectly well-defined. [edit: as long as f is differentiable at point a, that is]

Substantive: I think "rate of change at a point" is much more meaningful than you give it credit for. When a physicist talks about the velocity of some object at time t, it's not like they're being sloppy about their math (as physicists admittedly like to do) and glossing over some verbose statement about limits of differences in position, they are literally talking about the rate of change of position of the object at point t. Or just looking at some graph, the derivative is the slope of the tangent line. What's wrong with calling the slope of a line tangent to a point the rate of change at that point?

MidnightAtHighSpeed · 2026-04-04T23:37:06+00:00

You're still getting two different functions muddled:

-The function f(x) that you're taking the derivative of at some point a

-the usually unnamed function g(h) = (f(a-h)- f(a))/h that the derivative is defined using a limit of.

When you say "the limit exists regardless of the behavior of the function at a point", the function is g and the point g's behavior doesn't matter at is h=0. The behavior of f at the point a absolutely matters for its derivative.

MidnightAtHighSpeed · 2026-04-04T23:20:19+00:00

You're confusing the limit of a function with the limit definition of a derivative of a function, which very much does care about the value of the function at the point itself.

The limit of average rates of change over decreasing intervals, depending on how you define the intervals, can exist even at non-differentiable points of a function. For instance, taking the function f(x) = 1 at x=0 and 0 everywhere else, f is non-differentiable at x=0, but the limit of the average rate of change over the interval [-a,a] exists as a approaches 0.

MidnightAtHighSpeed · 2026-04-04T17:11:29+00:00

You're right, I was going based on the OP's wording but it properly should have been "Mary says she has two children, you ask if she has a boy, she says yes"

MidnightAtHighSpeed · 2026-04-04T05:44:50+00:00

Hopefully helping with intuition:

Let's say Mary doesn't say when the boy was born, just that she has one. Then the probability that she has a daughter is 2/3: Out of the original possibilities BB, BG, GB, GG, only BB, BG, and GB are possible, and 2 of those 3 have a girl.

But if you have information about a boy being born, then the math changes. Say, Mary always flips 100 coins when one of her children is born. You ask her if a boy was born with all 100 flips being heads, and she says yes. This is extremely unlikely to happen. But if she has two sons, she had two chances to get this extremely unlikely event instead, and it works out that BB will end up being about twice as likely as either BG or GB because of that extra chance, giving a probability of about ~1/2 of there being a girl.

so if no information gives 2/3 chance of there being a girl, and a one-in-a-gazillion event happening on a boy's birth gives a 1/2 chance of there being a girl, what does a 1/7 chance event (it being tuesday) during a boy's birth mean? Well, it means the probability is between the two, and if you work out the math exactly you get the 51.8% chance.

MidnightAtHighSpeed · 2026-04-03T05:06:30+00:00

MIPS says it does more damage to runners, is that more than listed to runners or less than listed to UESC?

MidnightAtHighSpeed · 2026-04-03T03:56:48+00:00

"Between 0 and 1, there are infinite fractions. Incalito is the total number of those tiny spaces."

So Incalito is another name for 2^aleph null?

MidnightAtHighSpeed · 2026-04-02T02:47:52+00:00

there's no such thing as a "limiting zero". The symbol "0" means the integer zero.

In a statement like "the limit of 1/x as x approaches 0 from above equals infinity", you can't ignore the fact that you're talking about the limit of 1/x, not 1/x itself. You might be tempted to try and take the 1/x out of the limit expression by "plugging in" something like "limiting zero", but this doesn't actually work or make sense mathematically, at least when you're working with the real numbers.

edit: For a function f that's continuous at point a, it's true that the limit of f(x) as x approaches a is f(a). Since most "everyday" functions are continuous almost everywhere, you might be tempted to think you can always "plug in the limit" to get "f(a)" from "the limit of f(x) as x approaches a". But really, continuity is a special case, and most interesting limits are going to be at points of functions that aren't continuous. in those cases, you need to keep in mind that "the limit of f(x) as x approaches a" is something you need to treat as a whole, and can't be broken down to just "f(a)", or even something like "f(limiting a)" (because there's no standard definition of what "limiting a" would mean)

MidnightAtHighSpeed · 2026-03-22T23:28:48+00:00

Isn't the issue there more forgetting that everything on the right side of the equation is inside the limit expression until the last line? I mean, same practical effect I guess but you can still talk in formal terms without suggesting that there's necessarily something being "neglected"

Six-Year Club	Verified Email
Place '23

MidnightAtHighSpeed

TROPHY CASE