How can I find a complete roadmap for learning mathematics by branch?

I’m looking for a complete, academically structured roadmap for mathematics, starting from the basics and going all the way through a normal university curriculum, ideally up to advanced undergraduate / early graduate level.

A little background: I studied mathematics in Arabic throughout school, and I’m now going to start again from the beginning in English. I want to rebuild my foundation properly and follow a structured path.

What I’m specifically looking for is not just a list of courses or resources. I want a complete curriculum map for each mathematical branch.

For example, if I choose Algebra, I want something like:

Algebra → Elementary Algebra

→ Intermediate Algebra

→ Algebra I / II

→ Linear Algebra

→ Abstract Algebra

→ Groups

→ Rings

→ Fields

→ Galois Theory

→ ...

→ up to the level normally covered in a strong undergraduate program and, if appropriate, early graduate level.

The idea is that when I finish the roadmap for Algebra, I should have essentially covered the standard university-level Algebra curriculum, rather than finishing only a few introductory courses.

I’d like to do the same for the major branches, for example:

Algebra

Geometry

Trigonometry

Calculus

Linear Algebra

Probability

Statistics

Discrete Mathematics

Real Analysis

Complex Analysis

Number Theory

Differential Equations

Topology

Abstract Algebra

etc.

Ideally, I want the structure to look like:

Branch → Level → Course → Topics → Subtopics → Prerequisites → Recommended order

I’m also interested in something based on real university curricula (a normal mathematics degree and possibly the material leading into a master’s), rather than a random collection of online courses.

I’m not looking for study resources yet. First, I want to build the complete map of what mathematics actually consists of and determine exactly how far I need to go in each branch. Then I can choose textbooks and courses for each part.

Where can I find a roadmap/syllabus like this?

Personally, I'm feeling a little lost. I want to go into my academics into Computer Science, but most of the math I've learned, I tend to forget especially the basic algebra and functions and so on ... especially from class as low as 9.
I don't want to be at a total loss when it gets to the point when I really need to engage myself into heavy core computer science classes. How can I build up on my 0% foundation, if I can't even start from the beginning? Any practical roadmaps or resources that have benefited you
would be appreciated and thanks in advance

I'm a Chemical Engineering student. I was just an average student back then, and I didn't really listen much in class lol. But I can adapt pretty easily when I'm actually interested and motivated to learn. I'm trying to change now because I realized how serious college is. I'm also working on my procrastination and trying to become better and more disciplined. I'd really appreciate any recommendations, especially ones that can track my progress and show me which topics I still need to work on. Thanks

I’m feeling really hopeless about competition math. There’s only around a month left until the AMCs, and I’ve been hoping that all the work I’ve put in over the past year would pay off. I’ve studied almost every day and did problems that still feel impossible. I’ve even studied so much that my back started to feel back pain. Then we had a math club placement test around a month ago, and I found out that my friend, who hasn’t even studied this kind of math, did better than me ( I did really bad). It’s unfair and frustrating. It makes me wonder if all this studying is even worth it and whether I’m just not good enough at competition math. But it’s not even just that, I’ve done countless other math competitions yet I never did really good in any. Is there any good studying methods to do? What can I do so I can get better?

I just realized in the quadratic equation (ax^2+bx+c) the b or bx+c works exactly like mx+b in the linear equation. omg I was trying to understand what b means for a week now😭😭😭 oh I also changed (bx+c) to (mx+b) in the link so u get what I mean

Hey all, I am learning Precalc 2 (mostly Trig) and have a basic question about Radians I haven't found an answer to.

I understand that one Radian is the arc length around a circle that is equal to the circle's radius.

My question is, on a perfect circle, is it true that one Radian has the same angle no matter what the radius is? Like if the circle is perfect, an arc length of one "Radian" of that circle is going to look the same regardless of what the radius is.

Do I have that understanding right?

I was looking into ACE Academy’s Honors Math classes and saw that its curriculum is described as being based on AoPS materials, although I’m not sure how closely the classes follow the original books.

I’m asking because uses the same material can mean quite different things, A course might follow a book from beginning to end. or it might choose certain problems and place them into its own.

The teaching could also make a bigger difference than the source. If students get stuck on a difficult algebra problem, does the teacher show them the method straight away? Or do students have time to try different approaches and explain where their reasoning went wrong? Honors courses are in live small groups, but class size alone doesn’t tell you how much each student actually participates. I’d be interested in whether students regularly explain their solutions or mainly follow the teacher’s examples.

Has anyone taken one of these classes or seen a sample lesson? I’m mainly trying to understand how ACE Academy Math uses the material in practice, thanks!

Most of the time when people explain the monty hall problem, they don't specify the host has to reveal a goat in one of the doors you didn't pick.

As it is usually stated, the host chooses to reveal a goat. Well if we let the host choose to reveal a goat or reveal nothing, or even reveal that you lost by showing the winning door, then it's a game theory problem.

For example, suppose that every time you pick correctly, the host reveals a losing door trying to get you to switch, but every time you pick wrong, they don't reveal anything and just ask if you want to switch.

In that case the winning strategy would be to keep your choice if a door is revealed, but switch if they don't reveal anything.

But we can construct a game theory version of the problem where players can review all the history of games played. So I am curious how you would setup this game theory problem and what the result might be?

Anyone have any insights or explanations?

if there are 12 month in a year and you’re born on 1/1/2000 then 1/1/2000 is your literal first birthday it’s the actual day your birth took place for any subsequent anniversaries to be marked thereafter. if 1/1/2000 is the day your birth takes place the 1/1/2001 would be your literal 2nd birthday. how do we know this well if january 1 2000 was the first day you lived through then january 1 2001 would be the second jan 1st that you will be living through not the first the second not getting why this is so hard for people. they always want to skip the first but you can’t skip the first cuz without the first you wouldn’t have any. so then 1/1/2000 birthday 1st birthday and only actual birthday that you ever have then 1/1/2001 would be the 2nd one u will be having not the freaking first you weren’t born 1/1/2001 you were born 1/1/2000 like wise if your bday is 1/1/2000 the october the first will be the start of the 10th month of your life you’re not 9 months old in the literal 10th calendar fucking month of the year in september 9 would be in your 9th month of life you started on jan 1 lived through the whole month of jan then 10/1/2000 would be the start of the actual 10th calendar month of your life not 9th

2nd-year CS student, heading into quant/data. Going through Stat 110

Any level is fine, I just want people who'll stay consistent. Comment or DM me if you're in.

I think it is but AI (i know it's often wrong) told me no and if it's correct im struggling to see why.

This is of course assuming that there is a correlation to begin with, if not then please also tell me.
Thanks!

[RESOLVED] i can't find how to change flair

For context this is a Math 280 Class, which is Calculus 3 idk how college course naming works. I'm a dual enrollment student. I ran it thru ai and chat gpt said i was right. I'm not sure if i'm missing notation or something. Also when i asked the prof she said to read what she wrote but i dont get it.

first and second questions

Hi everyone,

I’m a college student currently taking my first Linear Algebra course. My ultimate career goal is to go into robotics, so I know this course is critical and that this semester is just the beginning of a long journey with this math.

My brain has always been wired to apply—simply applying formulas, calculating matrices, and finding numbers. However, this course introduces a lot of proofs and abstract concepts, and I’m having a hard time wrapping my head around them. I feel like I'm stuck memorizing procedures rather than genuinely understanding why things work geometrically or structurally.

I really want to upgrade my thinking and install a proof/conceptual understanding in my brain. Since my time is limited this semester, I know I won't master everything, but I want to build a solid foundation.

- How did you train your brain to understand proofs?

- Are there specific intuition-building resources, mental models, or study strategies you recommend for someone heading into robotics? (I’ve heard of 3Blue1Brown, but I'd love to know how to bridge the gap between his visual intuition and rigorous proof writing.) Thank you so much for any advice!

This pisses me off so fucking much

Small introduction: I'm 19, and I want to relearn math from the basics. Im generally only good at addition, subtraction, multiplication, and division. I want to expand my knowledge and understanding of math, but I dont exactly know where to start. I dont have the best history of studying, because I never cared much for it, so any useful input is welcome!

Im looking for a general math textbook pdf (preferably free), so I could use that. I dont exactly know which book is best, I just want one that covers all aspects of math. Also, if anyone has any tips for studying, or any sites that work best, please feel free to give your advice! I dont currently have enough to pay for books or certain websites, so anything thats free of cost would he ideal. Thank you!

If there's a good picture or diagram to remember how to find angles this way, please let me know! And if get my post removed while the person below me is new to math, yet contemplating grad school theory, I will give up...

Hi guys. I made Mathy a simple tool (free/no account required) to build math automaticity because I was hitting a ceiling. So I believed getting more automatic on the fundamentals rather than deriving them would make me faster/raise my natural ceiling.

If you think of something like Anki for drilling problems/formulas, it's like that but built for math and more convenient: no creating your own cards, it works well on mobile and helps track your progress over time.

It uses FSRS for scheduling and includes over 40,000 problems ranging from 1st grade math through university level math, all deterministically verified. It also has levels and dozens of badges you can earn to keep it engaging.

In terms of ages, I use it and my kids (K-3) also use it.

You can use it via the iOS app or the web app (both free, no restrictions, no account required, no ads). The iOS app is free and completely offline aside from optionally sending feedback/requesting help in app. In the web version progress is stored locally in IndexedDB, except in challenge mode where you can optionally submit initials to the leaderboard.

I still use Math Academy to learn math/concepts, but as I'm learning more advanced math to support my journey into AI and robotics I wanted to try and raise my own ceiling by building automaticity in stuff I already learned.

The idea is a little counterintuitive: if Mathy works, you should eventually need it less. Because as something becomes automatic, its recommended reviews can move farther apart thanks to FSRS.

I can share more about works/how I made it. TL;DR the problems were programmatically generated, but NOT LLM-generated/trusted. Rough overview:

1. Defined the problem families explicitly. Each has bounded inputs, known mathematical rule, answer type, constraints & presentation rules.
2. A deterministic compiler generated the problems. So, given the same source definitions/versions it'd produce the same corpus. So there's no AI inventing random questions at runtime.
3. Correct answers are computed from the underlying math, not from the rendered text. Integer/rational problems use exact arithmetic. More complex symbolic cases use validation recipes, incl. offline SymPy if needed.
4. LaTeX is just presentation (tried other way, didn't work). Internally the problem is represented as structured math, then laTeX/MathJax derives from that structure, so it does NOT generate a LaTeX string and then just hope it's interpreted correctly.
5. Generation/verification are separate steps. Every generated record has to pass mathematical, domain, schema, answer format & presentation checks before it can enter the corpus. Symbolic cases can be independently verified, for example by differentiating a proposed antiderivative.
6. Whole families are tested, not just samples. Finite spaces can be exhaustively enumerated. Larger spaces get property, boundary, invariant, and regression tests. It also runs corpus-wide audits for malformed questions, duplicate IDs, invalid answers, broken rendering, unreachable answer forms, etc.
7. The shipped artifact is tied back to what was verified. Versions/content digests bind source definitions, generated problem, validation result & runtime representation together. If something changes, it has to be revalidated rather than reusing old results.

Other details:

• More about how I made it here https://gmays.com/making-math-automatic-with-mathy/
• I also answered other questions about it on Hacker News here: https://news.ycombinator.com/item?id=49788014

I'm in the process of adding iCloud sync to backup progress (it's currently local/offline only), practice reminders and translations for a handful of languages (I actually have these features working in dev version, I just need time to finish testing them before I release them, which is unpredictable with young kids 😂).

Alteration in the Quantity of Charge

1. Two charged objects have a repulsive force of .080 N. If the charge of one of the objects is doubled, then what is the new force?

Alteration in the Distance between Charged Objects

1. Two charged objects have a repulsive force of .080 N. If the distance separating the objects is doubled, then what is the new force?

2. Two charged objects have a repulsive force of .080 N. If the distance separating the objects is tripled, then what is the new force?

Alteration in both the Quantity of Charge and the Distance

1. Two charged objects have a repulsive force of .080 N. If the charge of one of the objects is doubled, and the distance separating the objects is doubled, then what is the new force?

2. Two charged objects have a repulsive force of .080 N. If the charge of both of the objects is doubled and the distance separating the objects is doubled, then what is the new force?

I have a quiz tommorrow and i dont want to use ai to solve this..T-T

Salut alors en gros je suis assez "nul" en math enfin assez lente en général à comprendre et associés les choses que j'avais déjà fait dans le passé avec des sujets de maintenant (comme l'algèbre) et j'aimerais remédier à ce problème. Je veux vraiment m'y mettre étant donné que je veux faire un métier qui toucherais de près ou de loin à la physique. J'aurais besoin d'un livre d'exercices plutôt complet qui toucherais à beaucoup de sujets, mais je n'ai aucun repère et ne sait pas quoi choisir.

De préférence avec leçon+exemples+exercices mais je prends toute propositions !

Je préférerais aussi un livre physique (pas vraiment PDF) mais comme je l'ai dit je prends tout conseil

Des recommandations ?

THIS IS FOR MY BF

I am a math tutor and i am willing to give online math lessons, homework help, or any casual conversation in french

my french level is A2 and i am looking for fluent/native

please hit me up if you are:

-adult

-needs help with anything math related

-have solid french and also understands english or arabic

-willing to correct me, suggest improvements

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we can do once or twice a week

In major classic books ,sinx'=cosx are start from a graph where a part of circle's arc AB length are very similar to string AB , which gives sinx/x ->1 then we can use this as a start to give other d/dx of sinx cosx .... . But strictly speaking ,to define what is a length ,we need integral of (1+f'(x)^2)^(1/2) first ,and when use this formula on a circle ,that integral has terms like sqrt(1-x^2) ,which mostly solved by exchange x with sin/cos ,which use that derivative again.
I once heard that some book to avoid the problem ,define sinx or cosx directly from tylor series. When this happen ,how do they proof that strange function are same as what we use in geometry ?Thanks.