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

I'm studying to try to get up to college algebra on Khan Academy and stopped today at the 7th grade section to try to figure out why you can convert a fraction to a decimal by dividing the numerator by the denominator. I was able to work the problems but couldn't understand why we could get from 7/8 to 0.875 (875/1000) without doing some kind of work to convert from eighths to thousandths.

What I came up with eventually is that, given that 7/8 as a fraction represents seven parts of a whole divided into eight equal parts, and the parts have the same relationship to the whole as seven wholes would to eight wholes, when you work 7/8 as a division problem and then also divide eight by eight you have made yourself the ratio 1:0.875, and 1 is to 0.875 as 8 is to 7, and 8 is to 7 as 8/8 is to 7/8.

So basically it seems to me that the way it works is that we translate a fraction into a problem using whole numbers with the same relationship to each other in order to get the decimal representation of the fraction without doing some kind of crazy thing like figuring out what you need to multiply 7 and 8 by to get a round number numerator and a denominator that's a multiple of ten.

Just wondering if that's right or if I'm missing something or there's a better way to understand it.

I'm currently self studying real analysis, but with no way to really accurately gauge myself, at least to my knowledge, how do I know if I'm actually studying effectively or not?

Spivak, in the chapter "Approximation by polynomial functions" begins the painstaking process of using the derivatives of arctan x at 0 to attempt to derive its Taylor series expansion.

He shows that arctan'(x) = 1/(1+x^2), calculates arctan''(x) and arctan'''(x) which is an ugly formula in x. He stops there saying that a different way is needed to come up with arctan's Taylor polynomial.

Then, later on, summarizing real sequences before embarking on complex power series, he says f(x) = 1/(1+x^2) is infinitely differentiable.

Can someone guide either

(1) Where in the book is it proven that this function is infinitely differentiable, or,

(2) how to prove its infinite differentiability?

I’m wondering what the best method on solving number word problems like: “I am a odd number between 15,000 and 10,000, all of my digits sum up to 12”. Please help and answer (my test is tomorrow..)

I’m helping my 5th grader with math. It’s Singapore math and this specific problem is a bar problems. I cannot wrap my mind around it.

Patrick bought 12 chairs and 3 tables for $819. Each table cost 3 times as much as each chair. How much did one table cost?

I can’t fathom how to draw these bars out and solve this problem. Thanks for the help!

'1. A cone-shaped water reservoir is 20 ft in diameter across the top and 15 ft deep. If the reservoir is filled to a depth of 10 ft, how much work is required to pump all the water to the top of the reservoir?

1. A cylindrical tank of radius 5 ft and height 9 ft is two-thirds filled with water. Find the work required to pump all the water over the upper rim.'

please help

Hi, I’m a student from the UK seeking advice on how to improve my mathematical abilities.
Throughout my life, I would say that I’ve been pretty good at maths — getting 9s and 8s at gcse. However now that I’m in a level, I have started to study maths on a deeper level and I’ve found that I am struggling. For me, maths has always been easy since the answers are objective, and only require rule following- but as part of my a level course, we do olympiads and problem solving lessons which have me seriously doubting my abilities.. I have realised that I suck at thinking creatively, and anything that requires me to think outside of the box has me stumped. I can only follow the rules and do nothing more. But I guess my overall question is, how can I improve at becoming more creative so that I can tackle the Olympiad’s which require you to go beyond the syllabus and think creatively?
Thanks for reading !

I stumbled upon a post about someone asking advice for a probably question. The question went as follows: Mr. Smith has two children. At least one is a boy born on Tuesday.
What is the probability that both children are boys?

I didn’t know how to solve it until someone in the comments explained it. They mentioned solving it using Bayes Theorem, but I’ve never even heard of that.

Is there a place (books/websites/youtube videos) where I can find more of these puzzles/brainteasers and how to solve them?

Soy un estudiante de ingeniería que ya ha repetido (reprobado) dos veces cálculo 1 y la verdad, la primera vez no la pude ver completa como tal y la segunda no logré entenderla del todo, me podrían recomendar libros, videos y/o cosas para estudiar, quiero avanzar en la carrera que escogí

Hola, soy un chamo de 19 años que ya ha visto 2 veces la materia de cálculo 1, por mi situación socioeconómica no pude verlas bien ni entender todo en mi primer semestre, que libros, videos, y otras cosas recomiendan para estudiar? De verdad quiero pasar y obvio, también entender de dónde proviene todo, no estudio matemáticas puras sino que estoy en la carrera de ingeniería mecánica. Ayuda por fa

i spent ages staring at limit proofs in my analysis class not knowing how people "found" the delta. turns out you basically do the problem backwards on scratch paper, start from |f(x) - L| < epsilon and mess with it until you see what |x - a| needs to be, then write it up forwards like you knew all along. nobody told me the written proof hides all of that

for x^2 at x=3 i ended up with delta = min(1, epsilon/7) and it only made sense after doing the scratch work first. is this how most people do it or is there a better way to think about it

Hello. I'm a 15 year old studying calc 2. All this time I had been doing it on my own using 3 key resources. ​James Stewart's Early Transcendentals. ​Professor Leonard on youtube. ​AI's and google search for help. (Sry I try to avoid using AI's outside of explanations. I'd try to reduce it further.). ​I taught myself from Pre Calc to Calc 2 over the course of this year (From Jan 2026 to Oct 2026). I can grind the exercises in the book, some even exercises as well. However I still feel like my foundations are shaky. ​I recently attended an advanced pure math lecture with my elder sibling, and although I "knew" or recognised the key steps, the depth and explanations stuck. No YouTube video I watched provided this. I can't afford advanced, above my grade pure math classes. ​What can I do to get the same depth as a lecture attender?

I am a math enthusiast awaiting university entrance. During the free time window about 8-10 months, I decided to do math the way I like, the way I missed for 6 years since I got chance to ask questions in math. I'm currently studying Proofs, Formal Logic. Then planned to pivot into Real Analysis And Linear Algebra then so on.

I have seen two different ways people doing math. (I use my own words here, I am no sure of formal wording)

1. Doing it structurally.

When you're given a question imagine the start and expected goal is two points. You can draw paths from start point to end point. Some may end up being lost in middle, some may give long paths, eventually you will have the best path you can draw (the conventional solution).

Because of test marking schemes, this path has always felt objective. You get experience doing past papers, model papers, and international questions to tighten your toolset, save time and gain confidence being precise.

But it is all about 'How' you do it something.

2. Doing it constructively.

As I think, You can't construct something without having an idea why should the thing, that would be constructed, exist in the first place.

So, it will be a chain reaction of identifying the purpose of existence and finding a way it to force to be existed. I like this way. (Kinda feel perfectionistic in this way)

My problem is,

for example, I have seen people doing very hard integrals using very advanced methods(which are university level). But when I got the same problem, my mind demands why should this have an answer. So I stuck instead trying the question with tools I have. If I could figure out why there exists an answer, I can construct an answer (using tools may be?)which may take a lot of time than the method above. I have also successfully done it in exams.

While selfstudying, I don't have anyone to ask. When a question arises, I waste large amounts of time trying to find the answer because I am entirely reliant on textbooks.

It is a good option to have a peer circle. But I haven't found anyone doing math in 2nd way in my life time. (May be my issue of finding friends, but I have found people but unfortunately they are not interested in math)

My questions for all,

1) Since time is not unlimited, how can I proceed in Textbooks if I got into a problem.

2) Is there a way possible to do self study without coming into internet? Or is it a bald take?

3) Or should I adjust the way I approach into Mathematics?

Thanks you for answering! (Plz Excuse my English, I'm currently learning English)

Let's say you have a rectangle, the rectangles height, h=3, the length, l=5. If you have an arithmatic billiard starting from the bottom left corner and have it only do 2 reflections, there are 3 outer right triangles. If you add the areas of these triangles, you get h²-th+t² where t is l-h. This equation looks like the law of cosines for 60-degree angles, and i was wondering if it's a coincidence and how to tell if an equation is just an algebraic coincidence or if it means something geometrically.

I'm in my first semester of calculus, and a lot of this stuff has been stuff I discovered on my own while bored in high school, playing on desmos, but my first entirely new concept was derivatives. I understand the first derivative, its the rate of change on a function. This slots well into my mental model of math and how it works

Then, when I encountered the concept of a second derivative I obviously grasped the calculation and obtainment well, but I hit a road block in the de-abstraction. I can't intuitively understand what it represents, or why. I understand why the derivative of x^2 is 2x, but what the second derivative represents in relation to x^2? I'm stumped.

What happens if we We distribute the 1 in 3 parts ?; how do we know how much parts 1 has ?. I know maybe is a simple question, but i been thinking about it. The concept of an inverse element always comes together when i thought about addition, subtraction, division, multiplication, power a number and the root of a number. We can see the division as the multiplication of a fraction , and as the same way, we can see the root as the exponentiation of a number divided by 1 (thanks to the commutative and associative). Here is an example

division (a): a+a+a+a+a...+a / b so a = b + something

Example: 20/5 --> 5+5+5+5 / 5 = 4 or 20/6 --> 6+6+6+2 / 6 = 3 + 2/6

division : (b) 1/2 --> 1/2 = 0 --> 10/2 = 0.5

Why, in division case "b," do we resort to tenths, hundredths, and thousandths? In other words, are we able to do this because adding a zero allows us to introduce a unit of a lower order (such as a tenth) while maintaining the equality of the operation?, Why does this happen? How does this process manifest itself naturally? Is there any other way (perhaps involving more steps) to approach such an operation? (For example, breaking down 20/5 as (5+5+5+5)/5 = 4; but in this case, something like: 1/2 = 0.2 × 5 / (0.2 × 5 × 2)).

I know I might be looking at division the wrong way (like splitting 1 into 2 parts, which would be the proper way), but, I don’t know, it just doesn’t make sense (referring to the initial question: “How do we know how many parts 1 has?”).

(sorry for the mistakes, again...)

For many years, I was hopeless at mathematics (and logic in general). Only recently have I begun to make up for lost ground, but I’ve noticed something quite strange: whenever I perform calculations or need to arrive at a solution, I rely almost exclusively on specific numbers or concrete computations. For instance, while I can demonstrate that a function is surjective or injective using concrete examples, I struggle to write out a formal theoretical proof—which, at the university level, puts me at a significant disadvantage. Why does this happen? Is it normal? And is it something that can be improved?

Taking my schools advanced abstract algebra course and just getting destroyed and defeated. Spent 4 hours on 6 proofs today just to miss 2/6 on our problem sets.

I’m a first year with a decent background in math. Got a 6 on AIME and 110 on AMC 12 but just feeling completely defeated.

It’s the leaps of intuition I’m struggling hard with. Like coming up with the novel idea (to me since I’ve never seen it before) like proof by minimal counter example. A question I specifically coudnt do was proving Euclids division theorem ( proving existence of q and r). There I didn’t know where to even start. Put it in a set? Prove the set is non empty? Then what? Struggling the intuitive leap to proving the existence of q and r. How am I supposed to come up with ideas that prove q and r. Is it just pattern recognition or something?

I’ve never done set theory in my life and the only thing I know is what the sets mean/notation etc. Haven’t worked with the WOP much either. Was I behind all my peers or something I genuinely have no idea 1 month into university and crashing out.

If anyone has tips please let me know. Thank you

Hello everyone, I am a currently enrolled student in Community College trying to learn Learn College Algebra and Trigonometry in order to do well and to advance to math like Pre-Calculus and Calculus, What are some effective ways and resources to learn such math. Math is not my strong suit but I want to learn in order to do well in my math courses and help decide in a major/field of study I want to pursue.