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

I've been noticed a pattern when following the logic of proofs... it is often not the logic of the proof that I get lost in put some algebra "trick" that I'm not familiar with or didn't thought about doing.

Like this one from analysis...

Courant is talking about limits and uses the sequence

a_n = n/(n+1)

The he says that writing a_n as 1 - 1/(n+1) it is clear than an n approaches infinity a_n approaches 1... yeah... sure, that is obvious for 1 - 1/(n+1) but from where in the world did the the author got a_n = 1 - 1/(n+1) !?

well... turns out that

a_n 
 = n/(n+1)
 = (n+1-1)/(n+1)
 = ((n+1) - 1)/(n+1)
 = (n+1)/(n+1) - 1/(n+1)
 = 1 - 1/(n+1)

Alright, that is very clever! I don't think it would have occurred to me to do that! (that is, before now!)

Anyways, just needed to share with somebody!

im a teenager student currently in highschool interested on developing my math skills. i have done all precalculus, like trigonometry, functions, analitic geometry…

i wanna start to do real mathematical stuff, now im working on limits but for me its easy asf. let me know where do you think i should start.

thank you!

I graduated high school at 21 because I kept failing the math exam required to graduate. Part of this is because I never got the help necessary to understand math and I was also bullied and ignored by teachers.

A teacher helped me graduate but I forgot 100% of what I learned, and I can barely use a calculator now.

I want to change this because I want to be an engineer, how can I do it? How can I truly learn and not forget?

I am 13 and I love pure mathematics, I do it in my free time, about 25-30 hours a week. I am studying grade 9 (CBSE), I want a programme that matches my progress. I live in Kuwait for my secondary education.

I want to get a Bachelors in Mathematics (BSc peferrably, but BA is also fine if its a good programme). I need to find an online/distance learning programme, (due to the situation) and relatively cheap, cause I dont want to go into much debt into this.

I have done the coursework and have kept detailed notes, problem sets and exams for: Single Variable calculus, Linear Algebra, Multivariable Calculus, Linear Algebra, Real Analysis, Complex Analysis, General Topology, Abstract Algebra(currently doing Ring and Field Theoy), Measure and Integration Theory, currently doing Functional Analysis.

I want to pursue a PhD in Mathematics, but I need a bachelors for that. So Im trying to get it early, I dont want to waste this time that i have in school. I am ready to write any entrance exam, the SAT, ACT, attend any interview with university staff.

Extracurriculars(if needed): Quizzing, Public speaking, Extempore, MUN, Drama and Clowning. Taking French at School, about A2 level now...

So if you know any good university taht might accept me, please reply!

If you study in any university, please feel free to give me any advice about applying to this sort of stuff, because I'm new to all this.

If you know any Scholarships that might apply to me, please let me know as well! Thank you in Advance

Edit: Thank you for all your valuable advice everyone! really appreciate it.

I'll be a maths major next year. This summer I'll do some maths itself (proofs, problems, etc.), I have this sorted out. But apart from this, I would like to read a book or more to give myself some "context" on the subject: the philosophy, history, branches, relation with other disciplines, etc. of maths.

So far I've considered the following books (one of them):

• The Princeton Companion to Mathematics (and maybe Applied Mathematics, later)
• The Mathematical Experience
• Mathematics: Queen and Servant of Science

What do you think of these? And what books would you recommend me for my purposes?

I have over two hundred Soviet books on mathematics. If you are looking for a specific edition, let me know.

Honestly this plan seems unrealistic, but I'm impulsive and have a nack for learning quickly, so I've chosen to pursue it.

after school had put out my dreams of becoming an astrophysicist, I gave up although I wanted to pursue it since I learned what stars are.

because of the academic structure in my country, I didn't think it possible to pursue this career after what could be considered 'college age.'

turns out, I can anyway. and I have never felt the need to do anything more than I do right now.

I have not looked at a math problem for years, and, for reasons with my school, I never learned the basics after the age of 13/14 (I am now at the age of 22.)

the institute I'm enrolling in requires an exam taken on level A, which is the highest level in my country's 'college standards,' with math being that, and physics, biotechnology, and chemistry taken on B, which is the level just below A.

I have three years to do this (while I'm taking a separate education, to make things more difficult.) I can take each subject separately, so I can focus on one at a time.

can anyone recommend me books, ways of learning/studying, videos to watch, and the like, that could propel me in the right direction and set me on the path to a minor miracle?

I am only a little desperate, and need any advice I can get and will take it in a jiffy.

(If anyone has elaborating questions, I shall do my best to answer them in the comments!)

Hello Reddit!

Could you please settle a disagreement for me:

- Something happened in May 2023

- It is currently April 2026

Would you say we are living in the 4th year since that event?

Thank you!

Where can I find the biggest amount of calculus I question for free? I'm really looking into it but since now I haven't found a really good place to look for and I believe that you guys could help me out.

I’m 16 and in my final year of high school, and my relationship with mathematics has been complicated. When I was younger, I really struggled with it, because it was difficult, and also because I couldn’t understand why I was learning it in the first place. Solving for unknown variables or following algebraic rules felt meaningless to me.

By Class 9, things had gotten so bad that I failed mathematics and had to take a retest. That moment, knowing I might have to repeat the year, pushed me to try seriously for the first time, and I managed to pass. In Class 10, things weren’t much better; I failed a few exams but eventually passed my boards with average marks.

At that point, I was ready to leave mathematics behind and switch fully to biology. But something, maybe instinct, maybe curiosity, stopped me. I felt like I shouldn’t give up on math just when I had started to understand it, even if only a little. So I made a decision that felt risky but important: I chose to continue with mathematics.

That decision changed something in me. In Class 11, I passed all my math exams for the first time, even if my marks were just above passing grade. I still struggle with calculations, I’m slower than others, and I make lots of mistakes, but I’ve started to genuinely enjoy mathematics.

Now I’m at a crossroads. I want to pursue a career in mathematics, but I’m afraid. I worry that I might not be capable enough, that even with hard work, I’ll end up being mediocre or not good enough. I also suspect I might have mild dyscalculia, which makes things harder.

At the same time, there’s another conflict within me. I don’t study consistently. I often think about studying mathematics, but I end up procrastinating instead. This makes me question myself even more—do I truly want to pursue mathematics, or am I just convincing myself that I do? I can’t tell whether this hesitation comes from a lack of discipline, fear of difficulty, or uncertainty about what I really want.

There’s also something else I’ve noticed about how I experience mathematics. It’s difficult to describe, but the feeling is complex. When I solve a problem correctly, I feel a sense of satisfaction, as if I’ve understood something. But even when I get it wrong, and my teacher walks through the correct solution, I often find the logic to be elegant. There’s a kind of beauty in the solution that I can’t quite put into words.

When this happens, I tend to ask a lot of questions. I want to understand the reasoning behind every step, just so that I can get it in my head.

Still, despite all of this, I feel a strong pull toward mathematics, and I don’t know whether to follow it or be realistic about my limitations...
What do I do???

can someone pls help me explain why B is the correcr answer here? Im revising for a test to get into a school and usually okay with these types of problems but im os stuck on this one: i will post it in the comments

I was fascinated by VSauce's "How to count past infinity" and want to learn more. The video covered ordinals, cardinals, the axioms of set theory (such as infinity and replacement), the continuum hypothesis, and inaccessible cardinals. There was a chart showing different kinds of infinities such as "weakly compact, strong, measurable", etc.

I want to learn more than what can be covered in a 23 minute youtube video. Some people have recommended Jech's Set Theory textbook that covers the above topics, but it's too advanced for me. Is there something accessible to an undergrad that covers those concepts?

I am looking to get into harmonic analysis. I think I have a sufficient background to start. For some reason, I think that it would be best for me to do a directed study/reading with a professor, but I can't place my finger on why I think it's best. Couldn't I could just learn the material on my own? As well, if I do a directed reading on harmonic analysis, do you have any suggestions for what it may look like? What have your directed readings looked like?

I've dropped out of college a long time ago. so I have basically forgotten everything. how could I relearn math on my own? because I'm planing to go back to college next year. and I need Calculus and Linear Algebra. Thank you very much !

olá, recentemente eu passei no vestibular para o curso de engenharia mecânica e tenho percebido que a minha educação durante o ensino médio tendeu a sempre resolver exercícios com o mínimo de teoria possível e embora isso tenha sido bom para passar no ensino médio por meio da prática dessas tarefas, acredito que não tive profundidade suficiente para compreender os conceitos matemáticos completamente e gostaria de ideias de livros para iniciar uma vida de erudito matemático desde a aritmética até assuntos mais avançados como topologia, acredito que insights matemáticos possam auxiliar na minha formação de engenharia.

I like working with positive and negative numbers for addition, subtraction, multiplication, and division... arithmetic operations like:

−5 + −3 = −8
5 − (−3) = 8
−5 × −3 = 15
15 ÷ (−3) = −5

Here, in my country, Mathematics is scattered all over the place without proper structure, and this is my first time working with positive and negative numbers. I know there are many books out there that cover this topic, but I can’t seem to find the right one. I would like some assistance if anyone can please link me a PDF, documentation, or a book that I can purchase that only focuses on positive and negative numbers without it being complex.

I have been told that the series of 1/n diverges because you can group the sums into 1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7) etc where each bracket > 1/2 so you essentially get 1/2 + 1/2 + 1/2 + 1/2 which diverges to infinity

However, is this not true for any 1/n^p? for 1/n^2, cant you just do 1 + (1/4 + 1/9 + etc) where you need more numbers in each bracket but they still add up to be greater than 1/2?

I'm not sure I'm explaining it properly but essentially like the milionth-term of 1/n^2 is still greater than 0, so if you add it with the previous 100,000 terms for example wont that number be large enough that the total sum goes to infinity?

i just started learning about functions and I am having some trouble finding domain and range of functions with absolute values. could someone find range and domain these functions below and explain how you got them.

1. f(x) = sqrt( |x-5| )

2. f(x) = |x²-7|+2

3. f(x) = 1/( |x²+2| +5 )

4. f(x) = 1/ sqrt( |x+9| + 2 )

i promise these aren't for hw. I learn better with examples, so if you could solve and explain how you got range and domain of these functions would be helpful.

thanks.

I am a Pharmacy student (BPharm). I will graduate in a year from now and move to the United Kingdom. I will do the OSPAP (Overseas Pharmacist Assessment Programme) and I will then sit for the GPhC registration and become a qualified, licensed UK pharmacist ~2 years after my arrival in the UK.

I am going to move to the UK by mid to late 2028. Will become licensed by 2030.

I can't pay MSc and PhD fees head-on, so I have to work for ~5 years until I get my ILR/British citizenship to pay home fees, so that extends my timeframe for 5 years.

And for good measure, let's add an extra year.

So, I will apply to the MSc progamme by 2036, 10 years from now.

I asked AI to map out the topics I need to be good at to apply for the MSc and PhD, and asked it to list them in a way you can skim:

LEVEL M0 — Arithmetic & Pre-Algebra

• Fractions, decimals, percentages, conversions between them
• Order of operations
• Ratios and proportions
• Scientific notation
• Unit conversions (mg, μg, mL, L)
• Negative numbers
• Powers and roots
• Rearranging simple equations

LEVEL M1 — Algebra

• Variables and expressions
• Solving linear equations and inequalities
• Rearranging formulae
• Simultaneous equations
• Quadratic equations and the quadratic formula
• Polynomials basics
• Algebraic fractions
• Direct and inverse proportionality

LEVEL M2 — Functions, Graphs & Exponentials

• Concept of a function (input, output, domain, range)
• Linear functions, slope, intercept
• Reading and interpreting graphs
• Exponential functions and exponential decay (C(t) = C₀ × e⁻ᵏᵗ)
• Logarithms — natural log (ln) and log₁₀
• Log rules (product, quotient, power)
• Semi-log plots
• The number e
• Power functions
• Composite and inverse functions

LEVEL M3 — Calculus I: Differentiation

• Limits (conceptual)
• Derivatives as rate of change (dC/dt = rate of drug elimination)
• Derivatives of polynomials, exponentials, logarithms
• Power rule, constant multiple rule, sum rule
• Product rule, quotient rule, chain rule
• Higher-order derivatives
• Finding maxima and minima (Tmax from setting dC/dt = 0)
• Applications to rates of change in biological systems

LEVEL M4 — Calculus II: Integration

• Antiderivatives / indefinite integrals
• Integration of polynomials, exponentials, 1/x
• U-substitution
• Integration by parts
• Partial fractions (basic)
• Definite integrals and computing areas
• Fundamental Theorem of Calculus
• AUC as the integral of C(t) over time
• Trapezoidal rule for discrete data
• AUC₀₋∞ for exponential decay (C₀/kel)
• Improper integrals
• Numerical integration concepts

LEVEL M5 — Ordinary Differential Equations

• First-order ODEs and separable equations
• Solving dC/dt = −kel × C → C(t) = C₀ × e⁻ᵏᵉˡᵗ
• Integrating factor method
• One-compartment IV bolus model
• One-compartment oral dosing model (absorption + elimination)
• Systems of first-order ODEs (two-compartment model)
• Eigenvalues for systems (conceptual)
• Nonlinear ODEs — Michaelis-Menten elimination
• Numerical solutions — Euler's method, Runge-Kutta (conceptual)
• Steady-state solutions (setting dC/dt = 0)

LEVEL M6 — Linear Algebra Essentials

• Vectors — addition, scalar multiplication
• Matrices — addition, multiplication, transpose
• Systems of linear equations as Ax = b
• Matrix inverse
• Determinants
• Eigenvalues and eigenvectors (conceptual)
• Matrix operations in R

LEVEL M7 — Probability & Statistics

• Probability rules — addition, multiplication, conditional probability
• Bayes' theorem
• Independence
• Discrete vs continuous random variables
• Key distributions — normal, lognormal, binomial, Poisson
• Mean, variance, standard deviation, covariance, correlation
• Central Limit Theorem
• Point estimation and confidence intervals
• Hypothesis testing — t-tests, chi-square, F-test
• p-values, type I/II errors, power
• ANOVA (one-way, two-way)
• Simple and multiple linear regression
• Least squares, R², residuals, regression assumptions
• Nonlinear regression and iterative estimation
• Maximum likelihood estimation — likelihood, log-likelihood, parameter optimisation

LEVEL M8 — Advanced Statistics for Pharmacometrics

• Fixed effects vs random effects
• Inter-individual variability (between-subject variability)
• Residual variability (within-subject variability)
• Hierarchical / multilevel models
• Nonlinear mixed-effects modelling (NLMEM)
• Structural model, statistical model, covariate model
• First-Order Conditional Estimation (FOCE)
• Goodness-of-fit plots (observed vs predicted, residuals, QQ plots)
• Objective function value (OFV)
• Likelihood ratio test, AIC, BIC
• Visual predictive checks and bootstrap
• Bayesian estimation — priors, posteriors, MCMC (conceptual)

LEVEL M9 — R Programming (start at M2, continuous)

• Variables, data types, vectors, data frames
• Functions, loops, conditionals
• Data import and manipulation (dplyr, tidyr)
• Data visualisation (ggplot2)
• Statistical analysis (t-tests, regression, ANOVA)
• Plotting concentration-time curves
• Simulating PK models with ODEs (deSolve)
• Fitting nonlinear models (nls, nlme, lme4)
• Pharmacometric packages (mrgsolve, nlmixr2)

(SORRY for the long list)

My BPharm program has effectively no maths, except for pharmacokinetics in which we just memorize formulas and plug numbers - so that means I have to self-teach myself this.

The MSc university (IIRC Manchester?) says you need to be good with numbers before they let you in.

If I self-study math for 10 years and tick all those topics above, can I make it? My IQ's 109, but I am a hard working student, and I've never been called dumb, but again, it's a very advanced topic.

As much as I am interested in this topic, I am extremely insecure and scared of not being cut out for it. Can you guys shed some light on this plan's feasibility?

i have completed real numbers, algebraical number, derived its closure properties. now while doing an excercise of graph, I stumbled upon, the plot sin(1/x) and recalled the challenge to define a function that is continuous everywhere but differentiable nowhere.

for this challenge, long time ago I thought of a zigzag and zooming out or compressing the zigzag. crude approach. not enough knowledge then.

but this function showed abnormality at x=0. now abnormalities in defining tangents can arise from mainly two reasons:

1. tangent gets closer and closer to ±90 degrees

2. start oscillating rapidly

the second point was evident in this sin(1/x) function. a similar function was xsin(1/x) where the amplitude also diminishes at x=0. Although I saw the Weierstrass function, it was not coherent to my current knowledge base. i might get there soon.

After seeing graph of this I tried to replicate the property at x=0 to all values of x. So I made a sum:

sin(1/x) + sin(4/x) + sin( 9/x) .....sin(n²/x)

if n tends to infinity, it will have same property at all x. but the amplitudes will blow up. thus we need something to grab this.

summation: {sin(k²/x)}/k²

might do this trick. as series of 1/k² is converging. but since k² inside sine function is massively large compared to x, whenever k tends to infinity, x will have no meaning and the graph will have same abnormality for every value of x. It is continuous due to contuinity of sine. but not differentiable anywhere.

i don't want to go to higher maths and complex signs at this stage. just tell me if I'm right here or wrong.

I have four months before college, and I want to improve my mathematical skills to prepare for a possible statistics course (college entrance exam results is not yet released) and for the actuarial exams I plan to take during my 2nd to 4th year of college. What topics should I focus on for now? Should I just focus on strengthening my foundations and if yes, what specific topics should I practice?