r/learnmath

My daughter seriously over thinks math. She has real issues with saying the wrong answer. What I mean is if she even thinks the answer might be wrong, she won't say it. I'm asking here because this seems to be specifically related to math. She doesn't overthink in other classes the way she does for math. (That's why I'm asking here)

She knows almost all of her times tables. She only struggles with 7's and 8' once you get past 5. The rest she knows, and can do quickly unless specially promoted by an assignment. But she's so scared of getting a wrong answer she won't try.

Anyone have any tips to get past this road block? Or should I be focused more somewhere else? Again, she only seems to have this issue specifically with math which is why I asked here.

This post contains content not supported on old Reddit. Click here to view the full post

Hi everyone, I've just started with proof-based mathematics (I'm self-taught) and I made the mistake of starting with linear algebra done right.

The book is really good, but I can't do almost any of the end-of-chapter exercises (actually, the same thing happens to me with real analysis too). So, since I'd like to understand it 100%, and since the author himself says to use it as a second course, I need an intermediate book to use. Now, I hate non-proof-based books (I don't like recipe books), so I'd like one like this.

I'm undecided between linear algebra done wrong and linear algebra by Friedberg, Insel, and Spence. What are your opinions on these two? Is Friedberg's book practically a duplicate of Axler's book in terms of difficulty, or does it really make sense in my situation? I repeat, I'm really bad at non-mechanical exercises on proofs.

(One advantage of Friedberg's Linear Algebra is that it comes in paperback, which is a huge plus for me as I prefer physical books. By the way, if the answer is Friedberg, what are your thoughts on the Pearson International Edition of the book? I mean, the Indian one. Is it any good, or should I go for the classic fourth edition?)

How to improve at maths? I'm going to 9th grade when school starts again in September. And I wanted to learn like 10-11th grade math so I could be a mathematician when I finish school.

So is there a YouTuber or videos or even books I could watch and read?

I have been trying to figure this question out and would appreciate any help. I have looked this up but would appreciate it being explain simple simple.

Question is: 7km^2 to miles^2

Hey everyone, I have a bit of a confession: I absolutely suck at math.

I’m 27 years old, and honestly, I only know the absolute basics—addition, subtraction, multiplication, and division.

Deep down, I've always known how useful math is, and a part of me always found it interesting.

Unfortunately, bad experiences with teachers completely killed my interest when I was younger.

On top of that, dealing with my parents' divorce and a chaotic household growing up meant I never had the stability to try and self-study.

Now that things have settled, I’ve finally regained my interest and genuinely want to learn math from the ground up.

Can anyone recommend some good online resources for a complete beginner?

Thanks in advance!

How do express fractions in set theory?

Theorem. 

There is no continuous function f, defined on ℝ, such that every value f attains is attained exactly twice.

PROOF

Setup

 f(0) is some value. By hypothesis it is attained exactly twice; call these points a, b with a < b, so f(a) = f(b).

Step 1 — Dichotomy on (a,b). For x ∈ (a,b), f(x) cannot equal f(a): that would be a third occurrence of f(a), contradiction. So f(x) is strictly above or below f(a) for each such x. Suppose both occurred — some x₁ ∈ (a,b) with f(x₁) > f(a), some x₂ ∈ (a,b), x₂ ≠ x₁, with f(x₂) < f(a). By IVT applied between x₁ and x₂, f equals f(a) at some point strictly between them, hence in (a,b) — a third occurrence, contradiction. So f is entirely above f(a) or entirely below it on (a,b).

WLOG f(x) > f(a) for all x ∈ (a,b) (otherwise replace f by −f, which is continuous and has the same "exactly twice" property).

Step 2 — Interior maximum. By EVT on [a,b], f attains a maximum M at some c ∈ (a,b). Since f > f(a) on the interior and f(a) = f(b) at the endpoints, M > f(a), and c is interior (not an endpoint).

Step 3 — Outside is forced below f(a). Suppose some x₄ ∉ [a,b] has f(x₄) ≥ f(a). It can't equal f(a) — third occurrence. So f(x₄) > f(a). Pick v with f(a) < v < min(f(x₄), M). By IVT on [a,c], f attains v at some q₁ ∈ (a,c); by IVT on [c,b], f attains v at some q₂ ∈ (c,b); q₁ ≠ q₂ since the intervals are disjoint. By IVT between x₄ and the nearer endpoint of [a,b] (where f = f(a) < v, while f(x₄) > v), f attains v at some third point outside [a,b]. Three occurrences of v — contradiction. Hence f(x) < f(a) for all x ∉ [a,b].

Step 4 — M is the global maximum. For x ∈ [a,b], f(x) ≤ M by definition of M. For x ∉ [a,b], f(x) < f(a) < M by Step 3. So f(x) ≤ M for all x ∈ ℝ.

Step 5 — M's second occurrence lies in (a,b). M is attained (at c), so by hypothesis it's attained exactly twice; let c′ ≠ c be the other point. By Step 3, c′ cannot lie outside [a,b] (there f < f(a) < M), and it can't be a or b (there f = f(a) < M). So c′ ∈ (a,b). WLOG c < c′.

Step 6 — Dichotomy at (c,c′). For x ∈ (c,c′), f(x) > M is impossible since M is the global maximum (Step 4); f(x) = M is impossible since c, c′ are M's only two occurrences. So f(x) < M for all x ∈ (c,c′); in particular, pick any x₅ ∈ (c,c′), with f(x₅) < M.

Step 7 — Triple occurrence, contradiction. Take v with max(f(x₅), f(a)) < v < M. By IVT:

• on [a,c]: f goes from f(a) up to M, so f = v at some p₁ ∈ (a,c);
• on [c,x₅]: f goes from M down to f(x₅), so f = v at some p₂ ∈ (c,x₅);
• on [x₅,c′]: f goes from f(x₅) back up to M, so f = v at some p₃ ∈ (x₅,c′).

These three intervals are pairwise disjoint, so p₁, p₂, p₃ are distinct. Thus v is attained at least three times. But v is attained, so by hypothesis it must be attained exactly twice. Contradiction.

Therefore no such f exists. ∎

I dont know how I got in. I was very surprised! I got some books by that For Dummies group, but my ADD wont let me read them. Please tell me the tips and tricks! I am so nervous because I have so many other classes, which will fill up my schedule, so math being easy for me will help a lot! I haven't done geometry in awhile, but I was good at it. Algebra comes easy to me, but that doesnt mean that I know everything! My teacher said that she thinks I can do it, which I guess I should listen to her, but I am soooo nervous! Anything helps! Thank you!!!

I love math, I find it as a way to decompress myself sometimes and earlier this year I wanted to master math all the way to what Ive currently learned in high school and get working on the harder topics subsequently and i tried making a whole notebook with math ranging from elementary to calculus which i never did since i didnt know how to format or write math (i was only used to solving it) which kinda brought a new problem i wanted to fix and it was understand math conceptually rather than just write numbers like a robot! I want to expand my mental math skills, use an abacus, work visually, but not sure where to start or how to approach it efficiently! and this is just like elementary and intermediate maths... I dont know how i will tackle algebra in the way i want to at this rate! While i cant strive for perfection on the dot, i want to learn!!!

This might be a very stupid question, but, I have taken a few more adv math courses and haven't found an answer yet. Usually we treat multiplication as repeated addition during schooling and the separation came a bit suddenly to me in more adv math courses.

EDIT:

some people gave examples when repeated addition isn't 100% applicable, for example multiplying a number by pi.

I always thought of it as repeated addition going to infinity, at least thats how I remember it being taught. i.e Σ where the number is added for each index. So, 4x3(i.e 3+3+3+3) + 4x0.1(i.e 4+1 ) etc etc. note, I am aware that division is the opposite of multiplication, so it is quite weird to include it in the definition of multiplication, however if we restrict ourselves to only defining tenths as defined as needing 4 to be added 10 times to be equal to 4 we have again defined multiplication as repeated addition

I wonder and when a similar argument can be expended on

I need to present a math concept of my choosing, so im wondering what is a complex and interesting, but not totally out of depth concept or theory for a bunch of highschool students.

The problem was (x+1/x)^2

I solved it: = (x+x^-1)^2

using udv +vdu
=2(x+x^-1) (1-x^-2)
=(2)(x+1/x)(1-1/x^2)
This is the correct answer per my book.

Here’s my question: If I multiply it out and take the derivative I get a different answer:
(x+x^-1)^2 = (x+x^-1) (x+x^-1) = X^2 + 2x^0 + x^-2

= (2x) (1-x^-2)

I can’t seem to get it in to the same form as the given correct answer. Is it my algebra, or is it not a valid approach to multiply it out first and take the derivative? Thanks

I've been working through Calculus by Spivak to learn calculus, and I really enjoy doing the problems. I have finished Part II, and am eager to get on to Integrals and Derivatives, however currently I am working through the problems of chap 7 & 8. Henceforth, should I continue doing all of the problems? I enjoy them, but the starred ones can take me several days of work and iterative correction to solve. I try to contribute 4 hours of my day to these problems, but progress feels slow.

This post contains content not supported on old Reddit. Click here to view the full post

Hi, I'm a high school student with a strong interest in number theory, and I'd like to start learning mathematics from a more research-oriented perspective. Most of the books I've come across are aimed at olympiad training and focus primarily on problem-solving techniques. What books would you recommend for someone at my level who wants to make the transition from olympiad mathematics to research mathematics?

This is a quite long read, but I hope you will take your time and share your critique and wisdom. I also know it's quite a common topic these days and might be annoying by now, but still. I want discussion, your actual thoughts. If this interests you at all...

I am currently a first-year pure mathematics student. How did you change your approach to doing math when AI came, now that it can "understand" the vast majority of problems? No matter what math problem I throw at it, it solves it. I know this is because it's in its training data; we can assume it has no creative, original insights, and we can argue and debate about how these things operate. But the fact is: it solves the problem, it outputs the proof, it gives the answer. And it's just a matter of time before hallucinations are reduced to zero.

I understand that I am naive and may have a limited understanding of the world and of "real world" problems, not just textbook ones or olympiad ones. And I honestly started to hate it when people say that math is a human endeavor and that it's just so fun solving and exploring problems together. Like, yeah, it's fun, but that immediately reduces the significance of math to something like chess. What I mean by that is this: before AI, when you solved some kind of problem, it felt magical, meaningful, something that would let you solve harder problems and contribute to the world. Thinking about the problem, that was the fun part. But most importantly, not only that: it signified some kind of intellectual superiority, bonded tightly to one's identity; for some, the whole world could be looked at through a mathematical prism. Then, when you didn't understand what to do, maybe you asked your peers, or professors, or experts, and you always knew there would be someone smarter than you. And that's okay, totally okay: you would learn from them, you would get inspired, and you wouldn't really care whether you'd manage to get a Fields Medal or whatever. But you knew you would be needed. Just a small fraction of the population is actually good at math, and an even tinier fraction actually puts in the work to become even greater. Now it just feels... meh, to be honest.

Then some will say: if calculators were invented, why do we still teach basic arithmetic? Well, I agree with that. It is, of course, to develop thinking skills, basic skills, something all of us globally should know. But when it comes to dedicating your prime years to learning advanced mathematics... well, you know what I mean.

Still, the reason I chose a pure math degree is that I am good at math (well, based on what grades and standardized test results tell me) and, most importantly, I do like math and actually view it as the universal language and something beyond... I believe no one has yet invented a better way to train the mind than mathematics, and I have yet to meet anyone complaining of having too much general aptitude. Learning mathematics helps develop the ability to think logically, analytically, and critically, to structure and organize, to process information, and it also trains the problem-solving skills that help us explore and understand the world. "Mathematics should be learned if only because it sets the mind in order," as the Russian polymath Mikhail Lomonosov said.

I mean, it's quite paradoxical at this point. I remember when I was in middle school, I really liked word problems. I would solve dozens of them, searching for ever harder and harder problems; that's how I really ingrained my first glimpses of learning in my mind, of how applicable math can be. When solving problems, I would do most algebraic manipulations with a CAS calculator, because I already knew how to do those, and I often thought: what if there were a system that could actually understand word problems? And here, bang! We have LLMs that can now solve the vast majority of them. I would have loved to have that at that age. How much more I could have learned and accelerated! But now I find it very hard to find the same motivation. Maybe because it's the real world now: how will I make money?!!!!!! Maybe I just should have gone into medicine, the field least affected by AI in the long term. But I don't like medicine. Never did.

So yeah, math is sublime, powerful, a universal language, and applicable. In the same way, AI is just math and nothing else: a huge, complex mathematical structure, a function. And then: we are all going to die, aren't we? So the last day on Earth won't really matter that much. Then I think: why learn anything at all? If AI can do anything, why should we learn? Why should we exist at all, maybe just end our existence? What's the point of it all? So, of course: we learn, we get better at what we do, and we simply know that we can't avoid taking risks, and so on.

So, my question after this monologue. I am not asking you whether it's worth learning mathematics; I do have a plan: pure math as the equivalent of a hardcore brain gym, then maybe a master's in machine learning/AI, and then work on AI, improving it. But the thing is, I often find it hard to believe what I want to believe. Do I believe it, or what? Also, I will most certainly not take the fully academic path, like a math PhD; I am more interested in application, and the only reason I am taking a pure math degree is that it is purely abstraction-loaded, believing it will train me, and I enjoy it. But how do I keep learning? How do I stop just thinking and thinking and posting questions on forums, and actually just do things? Maybe I am going insane, or maybe I am just so f**king stupid.

So please, share your wisdom.

I am aware that by the Abel-Ruffini Theorem, there is no general quintic formula that could be expressed with an algebraic expression. However, I was wondering if this could be possible with tetration instead.

Tetration, as far as I know, is one of those things in math that doesn't really have any applications. But I was thinking that since tetration's inverse, the super-root, might not be algebraic, maybe they could be used to create a general formula for quintic and higher-order polynomials. Could this theoretically be used, and if not, why not?

EDIT - Tetration is algebraic since it's just repeated exponentiation. I still don't think the super-root is algebraic though.

Where n how can I start?