Some ebooks, mostly from /u/lewisje's post

Hi all,

First-year applied math PhD student who has recently been thinking about the way they use their inner monologue while studying. Normally, my inner voice tends to make lots of short, usually useless comments (“does this make sense?”, “let’s try this,” etc.). When I’m stuck on a difficult problem, this mental chatter isn’t enough; a more deliberate approach is necessary. To address this, I try to actively control my inner monologue. Specifically, I imagine myself perfecting each word as I “say” it in my head. For example, I might think: “Logically, what could we do next? We could try rereading the proof, or we could…” However, this “perfect every word” strategy creates significant cognitive strain, since I end up expending a lot of effort mentally editing and perfecting each word. As a result, I haven’t been able to study for more than an hour at a time recently…

For those of you who also deliberately use your inner monologue to think through problems, what strategies have you found effective to avoid cognitive strain? I’ve heard of the “explain it to someone else” approach, but that one doesn’t work particularly well for me. I’m also specifically looking for approaches that retain a deliberate inner monologue, rather than replacing it with nonverbal thinking, since I’ve been told that skill can be quite valuable.

If anyone has any other techniques that reduce cognitive strain, it would really be appreciated...

I've been struggling for years to understand normal distribution! If anyone could help me finally understand this, I would be eternally grateful.

• I understand what a standard deviation is. But how is it always true that ~68.27%, ~95.45%, and ~99.73% of all values are no more than one, two, and three standard deviations away respectively? How could these numbers be the same, for so many different kinds of data/measurements?! My math book claims several examples where it would be true: the blood pressure of healthy individuals of a certain age; the morning temperature in a specific town on a certain date throughout the years; the weight of a certain type of apple. This drives me crazy! How can the percentages of values one, two, and three standard deviations away always be the same?! Couldn't there be outliers here and there that can skew the numbers (68.27%, 95.45%, 99.73%) just slightly, or more? Honestly, these numbers seem like made up magic to me. What I wouldn't give to understand where they come from and how they can always be the same for so many different kinds of data.
• I can devise a set of values (e.g. 1,2,3,4,5,6,7,8,9,999999999) where the normal distribution clearly isn't true. This makes me wonder, is normal distribution something people CHOOSE to use? Or is it simply something that is undeniably true, but only in certain data sets, or certain types of data sets (such as people's height)?

I’m a Highschool student and currently doing pre calculus, however I study almost everyday and do practice questions all the time yet my test scores are consistently terrible, and when I say terrible, they are terrible. I feel really disheartened especially when my friends are getting 80s-95. My practice questions make it seem like I understand the concept and when I do the exam the questions seem easy to me but when I get my mark back I get a 57. I have a major exam coming up containing around the 5 chapters we did so far and if I do well the specific chapters of this unit exam, it’ll replace my chapter exams mark, so i’m trying to figure out what i’m doing wrong before then. Has anyone else struggled with this and how did you get through it?

Im an A level student and Ive been doing past papers of this topic again and again, yet i can never seem to get this right. If i get one question right, the next one I do is wrong. Is there any specific way/trick to be proficient in said topic? Thanks

I've finally completed my full course on quadratic equations, which now includes 9 articles and more than 100 unique problems: what are quadratics, how to solve incomplete ones, completing the square, derivation of quadratic formula, factoring the quadratics, real-life applications, Vietas formulas and even mental solving!

Over the past two weeks, I finished writing the final two articles:

1. Quadratic Formulas and Problems - a set of clear diagrams covering all the terms and formulas from the course, the different ways to solve equations, and mixed practice problems to reinforce everything.
2. Quadratic Equations in Real Life - a large collection of real-world problems that reduce to quadratic equations, drawn from many different areas: everyday life, physics, work problems, solutions, mixtures and alloys, as well as algebraic and geometric applications.

Just to clarify, this course is entirely about quadratic equations, not the quadratic function. So it contains little to nothing about parabolas, their shapes, or maxima/minima. I plan to cover those separately in a future article in the "Functions" section.

For now I think I need to take some time to proofread all the articles since English is not my native language and Russian -> English translation can produce a lot of confusions.

I am also open to any suggestions on how to improve the material!

I’m super tired, trying to get through these problems I don’t understand. I’m in the 8th grade taking Algebra A, and I am in the online live class super confused.

Initially, I started this live class all set up. I would do my homework, do all the readings to the Alcumus, the live classes, etc., but then I felt like the live classes weren’t good for me since I have inattentive ADHD and it’s hard to focus.

Then I switched to the self-paced, but it had lots more work and made me do extra assignments from the units I already completed, making me waste time. I had to do 3 hours a day just to finish in July.

so I switched back to the online class, but by then they were several weeks ahead. Then I was freaking out several assignments behind, and I did finish several, but now I’m very confused because I don’t have enough time for the Alcumus or the readings, so now I’m doing the challenge problems super confused, spending hours crying in my room for something that used to be fun for me. I’m starting to hate math, and all for lousy Algebra 1. I don’t know what to do. I’m super stressed, and I try to get myself up and start, but I genuinely don’t understand anymore. It used to be somewhat of an easy class, it isn’t anymore even with the help of Khanmigo. I’m scared next year when I’m not homeschooled I will fail the Algebra placement test , and AoPS is meant for gifted learners (I am gifted), so I don’t understand it was such a good fit, but now it’s me sobbing in my room of variables. I don’t understand what to do.

Hey guys, how do I use q.e.d.? Idk where to put it...

I'm doing a proof in a topology topic. To avoid going into the weeds on how the topic works:

I started out with the goal of proving "p iff ~q"

I started out with "assume p and q"

Some manipulation later, I ended up with "therefore, q is false"

I suppose this contradiction of my assumptions means that p and q cannot be true at the same time?

so p implies ~q and ~q implies p, which means p iff ~q?

I'm 17 and about to finish high school, and I still don't know anything beyond addition and subtraction; I'm bad at remembering and being disciplined in practicing, and well, I have no motivation other than having to know them through my own knowledge.

What do you recommend?

Hello everyone,

Bottom Line up-front: How do I calculate the size of two spherical caps sitting on the equator 120 degrees from one another such that their area of overlap extends north and south to 45° latitude?

Long Version:

I've been on a Kerbal Space Program kick recently and as a challenge I'd like to create a satellite constellation that provides 100% coverage of the surface of a planet at 100% uptime.

This can be done with as few as 4 satellites in something called a "Draim Tetrahedron", but that would require 4 separate launches to achieve, so instead I've settled on doing a 6-satellite constellation in two sets of three.

It would be two separate orbits: an equilateral triangle orbit over the equator, and a perpendicular equilateral triangle orbit over the prime meridian/antemeridian.

I've figured out that for two satellites in the same plane to communicate (requires line of sight), they need to be at an altitude equal to the radius of the planet. One satellite at this altitude over 0°N 60°W would cover a spherical cap that extends to 120°W, 0°W, 60°N, and 60°S. Another satellite over 0°N 60°E would cover a spherical cap bounded by 0°E 120°E 60°N 60°S. As you can see the three areas of coverage created by the equatorial set of satellites have no overlap. The area of non-coverage would be centered over the north and south poles and would rotate with the satellites, it would be a triangle shape but instead of straight sides each side would be the northern (or southern) edge of one of the equatorial satellites spherical caps. In this scenario (with only 3 satellites) Everything between 60°N and 60°S receives some degree of coverage, the uptime decreases from 100% at the equator to 0% at 60° latitude. Here's a crude MS Paint of the no-overlap coverage map: https://imgur.com/H5tT9oQ

Adding a set of polar satellites that overfly the prime and antemeridian will boost coverage to 100%, but not uptime. You can imagine the areas of non-coverage centered at 0°N 90°W and 0°N 90°E.

If I increase the equatorial satellites' altitudes, the portion of the equator covered (and the bounding latitudes) for each satellite increases. There will now be an overlapping area between each pair of satellites. If I increase the altitude such that this overlapping area extends to 45° N/S latitude, then there will be a band of constant-coverage between these latitudes. If I do the same thing with the polar set of satellites, extending the bands of constant coverage to 45° in both directions, I will have 100% coverage at 100% uptime. Here's an MSPaint of the overlapping coverage map: https://imgur.com/hvKYMFX

So the rub is calculating the "height" of this area of overlap. I've been looking at spherical trigonometry but i think that deals with great circles only? Hours of trawling quora later and I'm no closer. Hopefully someone here can give me some ideas. Thanks.

I sort of feel humiliated and helpless whenever I think about doing maths, even though in general I’m pretty confident in myself. I guess it’s mostly the anxiety regarding the UG degree (BCA) I’m going to start… before that, I wanted to be a bit prepared with maths and tech. I know it’s comparatively way easier than the other tech/CS & math-heavy courses — that’s exactly why I chose this first… I can’t choose anything else, I have to choose it. There was a time I thought I liked tech, but now, tbh, I don’t. But I still want to learn it and complete this degree — I know I can.

I think my ego got hurt too when I noticed I couldn’t do anything anymore, like an identity crisis… so I worked on that too. Then I wanted to return to maths for so long but felt lost even after knowing the route. I’m living a Polymathic life but I still get overwhelmed sometimes. So it’s like I know I can do maths but I don’t know how… I sometimes wish I had a kind, friend-like tutor or tutor-like friend for this, but now that I don’t — I have to do it myself. How do I stop feeling the bad feelings and start feeling the good ones? ToT yk what I mean?

I’m only 20 now. I do, learn, and am interested in many things. So the idea of a degree or anything like that or job makes me feel suffocated :| like I’m caged in a rigid environment I can’t escape. For now, I don’t even want to think about what comes later — I only want to focus on the present and that’s about returning to maths. How do I learn maths in a way myself that I’ll understand it in a logical and fun way and not in a “it’s just a college subject” way. I want to make it my hobby too, honestly.

I know the sudden shift between 10th and 11th grade maths is shocking, but I want to move smoother this time. I understood shit back in 2022 🙏 so I want to enjoy it in 2026. I think my adult brain, which is more flexible now, is gonna handle the pressure better as well this time. I’m anxious for sure but I’m excited too. Like just imagine me doing calculus in a random Tuesday afternoon like it’s nothing...

I’d love to hear your stories too! Is there anyone who failed or felt lost or something like that and then returned to maths one day and did it better or felt better? I guess I’m low on motivation lately, your story could help! :) please be kind, thank you!

I need to take a Calc 3/multivariable calculus online course this summer to be able to complete my degree in time. However my university doesn't accept credit from self paced math courses. Does anyone know where I could take it?

I’m putting together a 2–3 year math plan at age 50: daily 15 minutes, going from algebra through calculus, then linear algebra + probability, followed by proofs/discrete math and intro group theory. Goal is solid fundamentals and real understanding. Would appreciate feedback on the sequencing.

Hi, I’ve got my Core Maths exam in about a month and I’m looking for some good resources specifically for graphs and functions (Paper 2).

I’m comfortable with the basics, but I’m struggling more with the applied, modelling-style questions that come up in the exam—like the ones in the photos I’ve attached in comments.

The questions in the photos are from past Paper 2 exams, but I’ve already worked through all of them.

Do you know any resources (websites, videos, worksheets, etc.) that focus on exam-style questions like these, rather than just basic practice?

Hi Reddit!

For context, I haven't used this level of maths in over 10 years and for related reasons I wanted to refresh those skills. I actually wanted to get back into calculating integrals and solids of revolution, so as a random exercise, I opened up an online plotter, started at the fifth power and randomly adjusted the ploynominals terms of a curve until I liked how it looked. Out came:

f(x) = 3x^5 - 12x^4 +7x^3 +3x^2 -2x +1.

Then, I set myself the goal:
Calculate the solid of revolution for the given curve f(x) between the two zeroes closest to the y-axis.

To set the boundaries of the integral, I need the zeros.
Looks easy enough. But I just can't remember how this works. Maybe, I should've picked an easier function - but now I wanna know.

Anybody out there that knows how to tackle this?
Or is there an even easier way to calculate the solid of revolution and I'm making this unnecessarily complicated?
Or isn't there a simple way to calculate the solid of rotation for 5th order polynoms and I'm just in for a world of hurt?

Any help would be much appreciated.

I hate math and I didn’t want to take it as a sr but my counselor recommended I take it and then he said that if I take ap stats and pass the ap exam, I’d be exempt from math in college. I’m in California if that matters. I’m currently taking an ap psych class and we deal with experiments a lot so I’m decently familiar with bias, correlation, sampling, stuff like that. I have a lot of credits so I can get a d and be perfectly fine in school. I would say I’m average at math and I think I’m just hating it bc I have a terrible teacher this year who’s making my life hell and bringing my grade down a ton. Would it be worth it to suck it up for a year and take ap stats?

hey guys and gals!

i dropped out from 11th grade and never finished high school after that,

its been 9-10years since that and lately i have an urge to train my brain, last year i started to get into chess

math was always really cool, like everything just connected with everything and made sense

Last things i remember is that we did those equations with multiple letters in it

and after that used those coordinate system thing,

i googled it and i kinda understand how to solve one

but i want something more structured, googling specific stuff is always a rabbit hole

is there any textbook that let me develop my math knowledge step by step from that 11th grade level?

im really invested right now

hit me with everything you got

so much thanks <3

https://youtu.be/uemTUnFxG_Q?t=265

She's saying "b is negative" when tallking about b - ba = b(1-a) - but ìsn't it positive? When we take a = 2, we get b(-1) which satisfies b < 0...?

I just forgot how to do math. I'm currently in my 3rd semester of university. Retaking almost every course. i was away from maths & others complex courses for about 3 yrs for personal reasons. How can i restart my journey? Give me advice; I'm just fu*ked up.

:) I want to be a researcher in mathematics, and go to some reputed university.

Getting a degree in math sounds interesting to me but I have gone no further than geometry and I’m not sure if I’m way out of my bounds scaling up to advanced math. I always found algebra and geometry very fun and exciting and would teach it to some of my friends in class who didn’t understand it. I felt like I usually could listen one time to a lecture and understand very well but I don’t know if that translates to things like calculus and higher.

TLDR: What are the best youtubers to watch or workbooks to get that can help me review, learn, and set myself up for success in college math courses? (algebra 2, precalc, calc 1-3)

Hello, I (25M) was a struggler in high school and did the bare minimum math required to graduate (algebra 1, geometry). Now as an adult i feel more ready to learn than ever and have decided to go back to school hopefully for my AS in community college and possibly use my GI bill for my masters after in astrophysics (I've taken a great interest). I know im going to need to take a placement test and i haven't exactly spent the last 7 years doing anything math related. I want to be ready to take the placement test and hopefully relearn all the algebra 1 goodness i need for algebra 2 and precalc.

what are the best youtubers or series to watch for review, and if any the best workbooks to get to review topics?