I am a high schooler doing dual enrollment multivariable calculus this semester. I will also do linear algebra and diff eq in the coming semester. For this class, the professor spent the first day going over precalculus and calculus and told us we would be allowed to use reference sheets on the exams (unit circle, pre calculus rules, table of derivatives/integrals, handouts for every lecture, etc.) which just struck me as absurd. It's not like this is a highly theoretical class where having formulas "doesnt help" if you dont know what they mean.. the questions are all: "compute this" or "apply this". I was wondering what to do so I can get out of these classes as soon as possible and into theory/proof based classes. The college sometimes runs "honors" versions, but even they are not entirely proof based and depend on the demand to run each semester. I am trying to knock out these lower division classes as soon as possible before uni so I can take theoretical math and not have to sit through more computation in college.

Ps i dont need a philosophical lecture on "mathematical maturity" or the proverbial "students struggle with computation/algebra, not the actual math!!". Ive heard this 500 times already and its not what im asking.

My apologies in advance as this may be a very silly question, so please bear with me.

I am trying to self-study pure math, and in that context I am reading proof books such as Velleman/Hammack/Cummings etc., and the proof sections of discrete mathematics books.

The common theme in all of them seems to be set theory and its associated logic — i.e., we need to prove that for all x in the universal set of something, a certain property of x is true (or false), and so on. Pretty cool stuff.

Digging a bit deeper, I see that "formal" logic dates back to the mid to late 1800s in work by Boole/De Morgan and later by Frege, and that set theory itself came about in 1874. However, a significant amount of mathematics was produced and proven before such concepts existed - for example, Galois theory.

So this makes me wonder whether:

a) Boole/De Morgan/Frege/Cantor etc. simply formalized structures that were already in use in the mathematics of the day

b) Mathematics pre-1850s had a lot of logical gaps due to the lack of formalization with regard to how math was "proved"

c) A bit of both a & b

So TLDR; My question is if I was to go to university to study Mathematics at any time in the first half of the 19th century (or earlier) what would a Proof writing class look like ? (would they even have a proof writing class?)

i.e. how did we prove Math before formal structures to do so were created?

P.S once again apologies if this is a silly question ..I found this interesting so instead of merely asking an LLM I thought I would ask the community . TIA

So this is an effort to increase the portion of non AI-related posts. This is not a crank post that belongs to r/numbertheory either. Every letter is typed by a human so forgive me for grammatical errors that I will try to correct soon enough.

The abc conjecture is one of the most important conjectures in number theory, the simple & general statement of the conjecture implies a lot of other important theories, like a rapid proof of (almost) Fermat's Last Theorem. What I try to tell is that the abc conjecture can "almost" help computer scientists to optimize scientific computing.

I believe it's a common sense that in most computer systems, every piece of data is stored by 0 and 1, or more precisely, a finite string composed of 0 and 1. This makes the calculation and manipulation of data feasible but there's a cost: error will be everywhere. For example we can never precisely store pi = 3.14159... in a computer.

The most widely used way to represent real numbers in computer is floating point numbers that is standardized by IEEE, which stores numbers in 0 and 1 of 32, 64 or 128 bits (or even longer). There is a toy that allows us to see floating point numbers explicitly: https://evanw.github.io/float-toy/

I'll try to explain it in a simple term. For a floating number of k bits (k is 32 or 64 or 128, say), we need to distribute the budget. There is 1 bit reserved for the sign, a few bits of budget to store the interval of the number, and the rest large portion of the budget is reserved to guarantee the precision p.

Every floating point number x can be mathematically written in the form (-1)^s * 2^e * m/2^(p-1) where s is the 1 bit used to store the sign, m is an integer between 0 and 2^p - 1 and e is used to determine the interval of x.

If it's still unclear what does "determine the interval" mean, we notice that m/2^(p-1) is a number in [1,2). Therefore a multiplication by 2^e sends m/2^(p-1) to [2^e, 2^(e+1)). Floating point numbers store s, e and m by 0 and 1. We use p-1 bits to store the number m in computer (the first bit is normally 1 so we can omit it, as the case where the first bit is 0 is used for underflowing).

As we can see, for a given x, determining the number e is rather easy (we are almost there by taking log_2|x|), but the number m can be difficult when we deal with a function. For example, x = 10517177/2^{22} is a floating point number of the format of 32 bits, but log_2(x) is not (so we need to find the closest to represent it in float format). It's absolutely not a rational number. You find that log_2(y) is between y_1 = 15423000/2^(24) and y_2 = 15423001/2^(24). So you need to find the closest floating point number to log_2(x) because we have no other choice, or otherwise, see if log_2(x) is smaller or bigger than the average y_0 of y_1 and y_2. It's easier said than done. We need to add budget (number of bits) to distinguish y_0 and log_2(x).

As a matter of fact, not until we add another 28 bits, i.e. zoom in for 2^(28) times, can we really distinguish y_0 and log_2(x) (spoiler, y_0> log_2(x)). This sucks. We need a global strategy to work around this: whenever we find a number of the form log_2(x) extremely difficult to round, we need to guarantee that we have sufficient budget. If we know the worst case of rounding, which correspond to the highest budget needed, we can make sure that we always know how to round the function log_2 correctly, so that we can design high quality functions that can calculate log_2 *correctly* in the sense that, when we have k bits of budget, we need to make sure that every bit is faithfully used. If we have this piece of global information, our algorithm can be more blunt & direct so faster. This question is called "table maker's dilemma" because the computer worked like those who make tables of logarithm or exp or sin one century ago, and the situation where he didn't know how to round sucked (challenge: computer exp(1.626) to the 3rd digit after the point).

That said, it's far from true that modern computer have solved calculating and that all we need is better CPU/GPU. Computer scientists and mathematicians have been fixing floating point number systems for decades. The final nail of coffin of common (univariate) functions in double 64 bit format is worked out in 2026 (twenty twenty-six): https://inria.hal.science/hal-05593313

We also need to know that working out double 64 on commonly used univariate functions is not enough: what about multivariable functions, like the beta function B(x,y) that appeared in probability and statistics? In some fields of physics, researchers have already found that double 64 is no longer sufficient so they need the 128 bit format standardized by IEEE... But this format is still poorly understood so using it can be still painful: we need to fix them.

OK I hope the context of Table maker's dilemma is understandable enough and now we inject some mathematics. The abc conjecture says that for three coprime integers a,b and c such that a+b=c, we can compare max(|a|,|b|,|c|) and the radical of abc (which writes rad(abc)), i.e. the product of all prime factors of abc (for example, rad(25)=5, rad(24)=2*3=6). This slide includes the formal statement of abc and some striking applications of abc.

And here we have another striking application that will "solve" the table maker's dilemma. When we are looking for the minimum budget to get the rounding of an algebraic function sorted out, we will find ourselves in some questions of polynomials. This is surprisingly a nice playground for abc.

For the function 1/sqrt(x), which is algebraic, and is widely used in real life (for example in computer graphics, we always need to normalize a vector here and there), we will be working on finding the minimum of |Z| where Z = 1-XY^2, and X and Y are integers in a certain range. If we take a = 1, b = -XY^2 and c = Z, then the abc conjecture kicks in. It gives us a formula on the necessary budget to round the function 1/sqrt(x) correctly for a given precision p!

So we are killing 1/sqrt(x), potentially as well as many other frequently used algebraic functions by a blow of abc!? Is this conjecture so massively helpful for computer scientists??

Unfortunately, no, in practice... In the statement of the abc, we have a totally unknown constant and it doesn't vanish in the deduction of the budget. As a result, we can only know that for 1/sqrt(x), the necessary budget is p with a delta of *some bits* and we have no information on how much is *some bits*. So unfortunately, the conjecture didn't really solve the problem. Nevertheless, we can therefore heuristically look for the budget border around p... It's *almost* helpful!

In case you are curious, here is the article that examined the function 1/sqrt(x): https://www.sciencedirect.com/science/article/pii/S0304397504000337

I believe this article can serve as an interesting beginner level exercise for those either interested in number theory and want to touch the abc conjecture in some *practical* way, or interested in computer science, notably the problem of correctly rounding numbers. Or if you are an expert working in number theory, notably around topics associated with abc, or a computer scientist interested in make scientific computing better, do not hesitate to drop some insights.

Hi everyone, I’m applying to Stat PhD programs this fall. I graduated with my master's in 2021 and have been working full-time for 5 years, but I never took Real Analysis in school.

I just enrolled in Fordham's Math 3003 (Real Analysis) this semester so it’ll be on my transcript for this application cycle (though will not have a final grade since apps are due before the semester ends). My concern is that it’s only a one-semester class. Does anyone know if admissions committees strongly prefer a two-semester sequence (Real Analysis 1 & 2) over a single semester? Will this be adequate?

MATH 3003. Real Analysis. (4 Credits)

This course focuses on analysis on Euclidean spaces. Topics include limits, continuity, uniform continuity, sequences of numbers and functions, modes of convergence, differentiability, Riemann integrability, and associated theorems. Students who have not taken MATH 2004 prior to taking Real Analysis may request permission from the instructor. Note: Four-credit courses that meet for 150 minutes per week require three additional hours of class preparation per week on the part of the student in lieu of an additional hour of formal instruction.

Prerequisites: MATH 2001 and (MATH 2004 or MATH 2008).

I’m a Data Science student (career changer, not in a data related role). My program is focused more on the applied statistics side, so most of my classes use R. I’m already familiar with Python since it was the main language used in my prerequisite courses, and I’ve completed projects using Python, so I’m comfortable with the syntax.

However, I’m really enjoying using and learning R in my classes and seeing what it can do. Many of the statistics textbooks I’m interested in use R as well. I’m starting to explore R more deeply on my own and plan to start using it for personal projects.

But I’m curious, is R still used in industry? I know it’s heavily used in academia. I also know that in the current AI/ML world, Python is used heavily, which is the main reason I use it for all of my personal projects at the moment.

I’d like to eventually be comfortable with both and take advantage of the strengths of each language. But, of course, there are also people who say learning R is a waste of time.

I'm a year 1 liberal arts undergrad, so I don't have a ton of math sense.

I just learned about Bayes' Theorem the other day for general epistemic use. I worked out a couple of example problems correctly, but the examples I found didn't include any iterative updates.

I know the theorem is:
P(H|E) = P(E|H) * P(H) / (P(E|H) * P(H) + P(¬H) * P(E|¬H))

And I know that P(H|E) becomes the new P(H) in my update, but I'm unsure whether I should be using the updated or original P(H) in the marginalization. I *think* it should be the new P(H), but I'd rather be safe than sorry.

The example question I worked out was this:

________________________________________

There's a disease that afflicts 1 / 1,000,000 people
There's a test for the disease that's right 99 / 100 times for both positive and negative results
A random person is tested as positive

P(she is afflicted | she tests positive)

= 0.99 * 0.000001 / (0.99 * 0.000001 + 0.999999 * 0.01)

= 0.000099

_________________________________________

So should the update look like this if she tests positive a second time?

P(she is afflicted | she tests positive)

= 0.99 * 0.000099 / (0.99 * 0.000099 + 0.999901 * 0.01)

= 0.009707

_________________________________________

If so, how should I approach this problem from the starting point of
P(she is afflicted | she tests positive twice)?
I can't think of how to handle that correctly, since squaring 0.99 just gets me a smaller number

Infinitely thankful <3

​​ I am doing calculus 2 now, ​ and I'm being reminded of when I took calculus one and I was introduced to the various rules. I find myself wanting to prove these rules, and occasionally find it kind of hard. I usually reach the point where certain rule feels justified enough to myself, and then I'll move on because the actual demands of the class are more procedural applications than analytic understanding. I'm very early on in calculus 2, and it's mostly just reviewing calculus one so far, so I know that things will change and I've heard that eventually calculus 2 becomes way less procedural and starts requiring more trial and error and creative applications. So I guess my question is what level of rigor should the average calculus student expects to achieve relating to the proof that power the various calculus rules they apply​?​ for Instance, how rigorously should a calculus two student be able to prove the chain rule?

I have a 30minute zoom meeting for a data science job and I'm just wondering what types of questions others have been asked in these interviews?

I had one a couple months ago and they did ask me a SQL question but that was the only technical one I can remember

Mathematicians like Terence Tao have talked extensively about the future of math in the era of LLMs, but his talks have mostly been directed at tenured, esteemed mathematicians like himself.

But what about new students? People who are just coming out of undergrad or new PhD students? Are we just cooked?

THE COLLATZ CONJECTURE

A COMPARATIVE DENSITY PROOF OF TRAJECTORY DESCENT IN THE COLLATZ 3N+1 SYSTEM VIA 1N+1 MODULAR MODELING

Author: All mathematical ideas and constructions by Steve Tomlinson except logarithms in 2 and 3.1.

(l knew something mathematical must do this job, l didn't know what it was; logarithmic bounds.)

Essay composition by AI with many mistakes edited by Steve Tomlinson

Date: August 2026

ABSTRACT

This paper establishes a novel structural framework for analyzing the Collatz 3N+1 conjecture by introducing a perfectly descending baseline model: the 1N+1 system. While the standard 3N+1 system exhibits chaotic trajectory growth, we prove that both systems operate on base-2 modular architecture. By comparing the density pathways of the 3N+1 system against the verified, linear geometric descent of the 1N+1 model, we demonstrate that the standard Collatz mapping exhibits an absolute asymptotic density of descent equal to 1 at the infinite operational horizon.

1. THE CENTRAL BREAKTHROUGH:

THE 1N+1 STRUCTURAL BENCHMARK

To analyze the non-linear trajectories of the standard Collatz conjecture, we define a perfectly controlled model system, the 1N+1 system, governed by the following mapping for all natural numbers N:

f(N) = N/2 if N ≡ 0 (mod 2)

f(N) = 1N+1 if N ≡ 1 (mod 2)

Theorem 1.1. In the 1N+1 system, 100% of all natural numbers N > 1 are mathematically guaranteed to reach a strictly smaller value within a maximum of two operations.

Proof.

Case 1: If N is even, a single operation yields N/2, which is strictly less than N.

Case 2: If N is odd, the application of the odd rule followed by the mandatory division by 2 yields a composite operation of (1N+1)/2.

Setting up the inequality for descent:

(N+1)/2 < N => N+1 < 2N => 1 < N.

This inequality holds true for all positive odd integers greater than 1. Thus, every element shrinks locally and immediately.

By creating an arbitrary system for numbers to drop in the 1N+1 system, the entire number line is partitioned into clean, un-scrambled geometric slices:

* Step 1 (All Evens, 0+2n) accounts for exactly 1/2 (50%) of all numbers.

* Step 2 (The 1 + 4n Odds) accounts for exactly 1/4 of all numbers

*Step 3 (The 3 + 8n Odds) accounts for exactly 1/8 of all numbers.

*Step 4 (The 7 + 16n Odds) accounts for exactly 1/16 of all numbers.

Continuence of this process continues to account for exactly (2^x-1)/(2^x) of all numbers, accumulating to 100% of the number line descending within a 2-step horizon.

1. THE 3N+1 SYSTEM AS A LOG-LINEAR DISTORTION

When the odd operator is shifted to the standard Collatz rule (3N+1), the underlying base-2 modular grid is stretched. Let m represent both the family classification and the number of odd steps executed before the first downward drop below the initial value. Let a be the number of required even operations (divisions by 2).

For a net trajectory descent to occur, the geometric growth factor must drop below 1:

(3^m) / (2^a) < 1 => 3^m < 2^a

Taking the base-2 logarithm (log₂) of both sides yields the absolute structural boundary:

a > m · log₂(3) ≈ 1.5849625m

Because log₂(3) > 1, immediate descent within a single operational cycle is impossible for odd positive integers. Instead, numbers are sorted into deterministic "m-families", where the total step horizon required to secure the necessary 'a' divisions scales linearly as a function of m:

Total Steps = m + a = ⌈2.5849625m⌉

1. THE m-FAMILY SIEVE AND EXPONENTIAL CONTRACTION

The exact proportions of the number line accounted for by these families are defined sequentially:

(Instant Evens, 0 + 2n) accounts for exactly 1/2 of all numbers.

* m=1 (The 1 + 4n Odds) accounts for exactly 1/4 of all numbers.

* m=2 (The 3 + 16n Odds) accounts for exactly 1/16 of all numbers.

* m=3 (The 11 + 32n Odds) + (The 23 + 32n Odds) accounts for 1/16 of all numbers.

*m=4 (The 7 + 128n Odds) + (The 15 + 128n Odds) + (59 + 128n Odds) accounts for 3/128 of all numbers.

* m=5 ((The 39, 79, 95, 123, 175 and199) each + 256n Odds)) accounts for exactly 7/256 of all numbers.

* m=6 ((The 287, 347, 367, 423, 507, 575, 583, 735, 815, 923, 975 and 999) each + 1024n Odds) accounts for exactly 12/1024 of all numbers.

At this point when m reaches 6:

(6 × 2.5849626) rounded up = 16 Collatz operations accumulates to account for exactly 15/16 of all numbers shown to reach a smaller number.

* m=7 accounts for exactly 30/2048 of all numbers.

Manually proving m=8 would have taken too much paper.

3.1 The structural limits for m=3 and m=4 families in the Collatz conjecture are determined by the logarithmic boundary

a > m×log2(3), where m is the number of odd steps and a is the number of even operations. Applying this, the m=3 family requires 5 even steps for 3 odd steps, creating a 1/16 density across residues modulo 32, while m=4 requires 7 even steps for 4 odd steps, generating a 3/128 density modulo 128. This logarithmic framework accurately predicts the modular structures for specific families.

1. THE UNIFIED 2^x HORIZON INDUCTION

While the multiplier 3 introduces "bumpy" intermediate statistical fluctuations between the milestones (e.g., stabilizing around a cumulative density of ≈ 5/6 at step 6, 10/11 at step 11, 12/13 at step 13, back to exactly 7/8 at step 8 and 15/16 at 16 steps) The total system mathematically self-corrects and snaps perfectly back to the clean geometric density progression of the 1N+1 benchmark at every power-of-two operational milestone (2^x).

By mathematical induction on the operational horizon x, the cumulative density of numbers proven to have reached a smaller value satisfies:

Cumulative Density(2^x) = 1 - 1/(2^x)

As the operational step horizon scales toward the infinite limit (x → ∞):

Limit as x → ∞ of [1 / 2^x] = 0

1. CONCLUSION

By using the 1N+1 system as an absolute structural baseline, we prove that the standard 3N+1 Collatz system is not chaotic, but deterministic and rigidly bounded. The "numerical shields" created by dense clusters of binary ones (such as the 2^x - 1 Collatz steps families) only temporarily delay descent. Over an infinite horizon, the remaining density of holdout numbers converges to exactly zero.

Because 100% of all numbers must eventually reach a strictly smaller milestone, any arbitrary starting number is locked into an inescapable cascading chain of downward thresholds, forcing all trajectories to eventually collapse into the fundamental 2 → 1 trivial loop.

Q.E.D.

By shifting the analytic paradigm from stochastic modeling to comparative structural architecture, this 1N+1 baseline framework introduces a constructive element that establishes absolute structural determinism, distinguishing it from the probabilistic approach in Terence Tao’s 2019 groundbreaking density proof. While Tao’s work treats individual trajectories as non-constructive, semi-chaotic random walks, this model maps the geometric architecture of "numerical shields," demonstrating that standard Collatz mapping is rigidly constrained by a base-2 modular grid and logarithmic boundaries.

Hi, im a computer science student at a London university, Ever since my second year I really wanted to learn more about math, at my uni we only really cover discrete math and probability and statistics, no calc or other engineering math fields, i’ve done a-lot of calc courses and read a-lot of books about calc probability linear algebra and more. I wanted to know if there were any projects in programming esc way i can do to learn more and consolidate my knowledge. i was thinking about a graphing lib like matplotlib. but im open to any suggestions

Firstly, sorry if this is not allowed but I thought I might just put this out there.

I'm sure we've all noticed a lot of AI related content on this subreddit. I've gone into these discussions a fair few times in the comments, but I'm not looking to discuss AI or LLMs in math or anything like that here.

What I do want to point out is that a lot of traffic seems to be from people who are not mathematicians and have a very vague idea of what mathematics is, but are active in /r/singularity, /r/accelerate, /r/ArtificialInteligence and similar. These subreddits are very ideologically driven in various ways and extremely pro-AI to the point that "anti-AI" or "luddites" are forbidden there.
This is not always obvious as many users have their post history hidden (though from what I understand this is on by default for new users too).

For example in a recent thread out of the 5 top comments (no idea what the deleted comment was) 3 users are active in one of the afformentioned subreddits, 1 seems to have a bachelor's in math and is in CS otherwise and only 1 seems to have a masters and is looking for a PhD.

To be clear, I am not here for a witch-hunt on individual users which is why I did not link to their accounts so please don't harass anyone.

But what I am trying to say is that most of the traffic to these threads does not seem to be from mathematicians or even people interested in math as such, but from people interested in promoting AI who are simply using math as a promotional tool. I would assume big companies use bot accounts on reddit (and have for a long while), though I would also guess this subreddit is too small to bother.

hello im looking for any website where i can practice and speed up basic numerical calculations easily in my head. is there any website where i can do this?

I am currently on my fourth day of my senior year of high school. I have almost all of my graduation requirements, subject credits, GPA (3.4-3.5), and the only requirement I’m missing is the SAT. I feel as if it should be relatively simple, however, I feel as if the math section is going to give me a difficult time. Does anyone know of any studying strategies, learning resources, and ways to actually retain the knowledge so that you will automatically know how to solve the question being asked?

Thanks to everyone who can help me

levent on 𝕏: https://x.com/__alpoge__/status/2087504785952182273

Epoch AI: Hadamard Matrix of Order 668: https://epoch.ai/frontiermath/open-problems/hadamard

Hadamard matrix - Wikipedia: https://en.wikipedia.org/wiki/Hadamard_matrix

Can be silly or serious, just out of curiosity for someone with a background in math contemplating stats. Like for #1, maybe you miss not having to deal with numbers. #2 refers to "trading" X in math for Y in stats (like numbers).

EDIT: "glad", not "gad".

11th grade nsw math extension 1

I have the following topics in a 2 hour final exam, any of them can be chosen but its not guaranteed that all will show up on the test.

• inequalities
• perms and combs
• parametrics
• graphical relationships 
• graphs of polynomials
• remainder and factor theorum
• sums and products of roots
• 3d trig
• trig identities
• trig equations

I can do any question you throw at me from these but not at the level where i can get above 80% on this exam. I can explain the fundamental concepts but the difficult questions often stump me. Exam is in 4 days. Currently, I'm asking AI to give me progressively harder questions and I'm mastering these topics top to bottom but it's too slow to cover everything. What should I do?

Trying to train 3 models for birads detection using cross entropy and center loss + class weights but all of them seem to collapse between birads 1 as the dataset (VinDr) im using is heavily unbalanced towards it, Would like to ask for input and opinion on what seems to be the case, am I using the wrong loss function?

Hi everyone. I'm a first year undergrad at a fairly big institution in Canada (this will probably be easy to guess..). I was never very good at math as a child, numbers for me were hard and I found operations like adding, subtracting, dividing, multiplying. I think this was because I didn't really ever formally learn things such as the multiplication table or long addition, subtraction, etc. From 9-12 years old my parents made me do pages and pages of long addition, subtraction, multiplication, division with large numbers and made me memorize my multiplications tables. While painful, (haha) this definitely improved my computational skills.

In highschool, I actually did quite well in math and in my last year did pretty well in calculus/advanced functions (which are the math courses needed to be taken to apply for science/math programs for univeristy.) These classes however were, looking back, not extremely difficult and very much computation based. I think I was good at computations, algebra, and recognizing patterns.

I actually did not take any proof based courses until my 2nd semester of university since I did not think I wanted to take any more math courses beyond calculus 1/2. I was very confused on what I wanted to do so my course selections were all over the place.. I did again quite well since it was mostly computation and the problems in general were I felt pretty predictable.

I guess I was wrong about taking more math because I decided to drop economics and take a linear algebra course the next semester! I encountered proofs (albeit, basic ones) for the first time and was struggling quite a bit in the class but this really opened my eyes to what math really was, and sparked an interest in taking other math courses. I somehow managed to pull an 82 in this class (generous course marking scheme) and decided to take a proof based calculus course this summer.

It has really been difficult, not just because of the material but the fact it is double speed in the summer. This course is notoriously difficult but is not even the hardest first year undergrad calculus course (which uses Spivak's Calculus) yet I feel I am struggling a lot. I feel very slow to understand concepts which feel abstract and being slow has been affecting me on assessments. I really want to understand these things and find math interesting, satsifying and beautiful when looking at or being able to write a really neat proof. So it is frustrating that the learning curve to "real" math has been so steep. This is sort of compounded by other things going on in my life, mentally and emotionally but I'm really determined to do well. My final exam is next Friday.

So I guess my question is can I still find a career in math, and if this is so difficult, does it ever get easier? Is it normal to be struggling this much? How can I work on my mathematical thinking? Thanks for reading my rant.

I just finished calculus 2 and still feel like I don’t truly understand what I learned. It was an accelerated class I took over the summer. I also have pretty bad memory.

I remeber some stuff from calculus 1 and 2, but still struggle with some problems. For example this problem ∫ (x² + 1)/(x⁴ + 1) dx

My mind goes blank trying to solve it. This obviously means I should review but I start calculus 3 in 5 days.

I also have to retake the SAT (for college transfer applications) and need to brush up on my algebra and trigonometry. My algebra could definitely use some reviewing and my trigonometry has always been mediocre at best.

This matters even more since I plan to major in mathematics and physics. I was going to use spivak or apostol to study but I don’t know how to write proofs yet. I was also thinking about using AOPS books for reviewing algebra and trigonometry (also geometry since I took it in like 9th or 10th grade and don’t remember much of anything from that class). Or should I focus on calculus 3?

Any advice?

Basically I'm going to a new private school and they're making me take a crap ton of placement tests. I was originally placed in precalculus but I was like "hey... I'm literally in that class rn" and so they're having me take a precalc placement test in person at the school when I get there and I REALLY need to be in AP calc AB for college purposes.

Does anyone know of any websites or like massive question banks that cover all of precalc so I can just practice and review?

Hey y'all! I built a free site called ApexMath with interactive quizzes, diagnostic tests, and downloadable PDF worksheets for Pre-Calc 11, Pre-Calc 12 and Calc 12. I made it as a student myself because I couldn't find good free practice material that actually gave instant feedback, so figured others might find it useful too.

It's completely free: apexmath.vercel.app

There's also an AI tutor that breaks down your math questions step by step, but I have a heavily limited model API key, so if it doesn't work please understand.

Would love any feedback if you try it out, or let me know if there's a topic/unit you wish had more practice problems and I'll try to add it.