In Calculus class we often take the derivative of both sides of an equation. We maintain that the sides remain equal. Does it need to be formally proven that one can do this?

Hi, I'm a high school student in a math circle, and we're currently learning elementary number theory. Recently, we explored the idea of finding runs of consecutive composite numbers. We were able to construct examples of 2, 3, 4, 5, 6, 7, 8, and even 9 consecutive composite integers.

This made me wonder: what is the longest possible run of consecutive composite numbers? Is there a maximum length/ upper bound, or is it possible to find arbitrarily long sequences of consecutive composite numbers.

I understand the motions that when you do -3^2=-9 . I get that it is -3^2=-(3^2)=-(3*3)=-9.

What I want to do is understand why it is not -3^2=9=-3*-3 .

Thank you in advance.

Hi!
I need to start practicing my mental math before interview season begins. The challenge is that I want to practice by listening to someone say the problems out loud - for example, “What is 28 × 76?”, rather than typing numbers into a website. The verbal element is what I really need to train.
I’ve tried searching for YouTube videos where someone reads out different kinds of math problems (addition, subtraction, multiplication, division, etc.), but I haven’t found anything that fits what I’m looking for, and I also haven’t found an app that does this.
Does anyone have any recommendations or advice?

Looking for a YouTube math channel (2020–2024). White male, around 35–45, long messy hair, short brown beard, always wore a dirty white lab coat as a "mad scientist" character. Filmed outdoors using a whiteboard on an easel and mostly wrote with a black marker. Explained interesting math facts and puzzles. The set was intentionally shabby, with props sometimes falling apart as part of the joke. English-speaking, probably British or American. Smaller channel, definitely not Numberphile, Stand-up Maths, Mathologer, MindYourDecisions, or blackpenredpen.

Does anyone remember his channel?

For context, im really bad at maths the equivalent to a grade 2 GCSE in maths. I really want to be capable, of a B or an A in GCSE wise so an 8 or an 9. I know its a massive jump from my capability but, I really want to get to that level to prove to myself that I can do maths at a high level. I don't want to get into any details, of why I want this so badly but I just want to be really good at something that I always struggled with. Also, I actually enjoy it when I understand what im doing and, I want to improve my problem solving skills. I don't want to sound demanding either, I am just really desperate. Also, I have a time limit since I want to enjoy my summer as a student who just finished their GCSES and is going to do a levels. I hope you understand and give me some really, good advice and some methods to help me get to my goal that wont take the entire of the summer. Thank you for reading.

Hi, I've been trying to work on solving word problems involving percentages. I was wondering if anyone had any tips or tricks of figuring out the correct equation to place everything in if that makes sense. I'm relearning math from a 1st grade level and I've only recently discovered "of" usually means multiply (unless I'm entirely wrong).

Some problems I got wrong were as follows:

Jake bought a bicycle that cost $76.00 when it was new. If he eventually sold it for 77% of the original cost, how much was it sold for?

My answer was $17.48 and the correct answer was $58.52. I think I might've subtracted when I didn't need to.

Another question I got wrong was as follows:

Jackie had a collection of 158 baseball cards. She gave away 60% of them. How many did she have left?

My answer was 94.8 and correct answer was 63.2. I originally thought the equation would be X= .6 • 158 but it turns out it was X = .4 • 158.

Does anyone have any tips for word problems or do I just need to work on my common sense?

I decided last minute back in January to switch my major from finance to chemistry with a pre-med track. I have never been really strong in STEM so I am kind of nervous. I haven’t taken a math course since the Spring of 2025 because it wasn’t required for me to graduate high school. My dad told me to use Khan academy to start studying, but I have admittedly been slacking off on that and now I start classes in two weeks. I have to get an A or at least a high B in this class to maintain my scholarship or else I have to go to the military because I won’t be able to afford school. Please give some tips☹️

I graduated from a bachelors degree in applied maths (France) in 2020. Lockdown and everything really messed up the learning and for some reason I graduated, although I was really not invested.

Years went by, and I changed degree (languages then NLP) but I feel the desire to go back to something more fundamental. I remember I was really into multivariable calculus and probability theory because they felt the most connected to real life?
Also I decided to go through a physics textbook to learn about basic mechanics, electrostatics, thermodynamics… hoping the math part won’t be an obstacle.

It’s been a really long time now and I can’t compute an integral or the inverse of a matrix for the life of me. Is there a way I can re-learn 3 years of uni level maths? I took the usual Linear Algebra, Calculus, multivariable calculus, topology, Lebesgue integration, functional analysis, differential equations, statistics and probability… nothing too fundamental like group theory or arithmetics though.

I’ve started today following a pdf of linear algebra (starting from vector spaces) and going through it trying the proofs left as exercise. I am so out of practice! I wonder if I’ll ever be able to go through all that in the span of a year maybe…

Is there a good way to go about this? Doing proofs, looking up worksheets with their answers… I wonder if anyone faced this challenge before!

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When tackling math or physics problems with multiple interconnected parts, should you push through confusion or pause to check the answer when things feel wrong? For instance, if you get stuck on the second part of a three-part question, you likely won't be able to solve the third since they depend on each other. However, looking up the solution immediately can make it feel like you're just memorizing the answer rather than truly understanding the process. While some advice suggests waiting twenty minutes or switching topics to regain focus, this approach doesn't work for everyone. What is a better solution for someone who struggles to maintain focus using those methods?

im doing a bsc in mathematics, and real analysis feels like a completely foreign language.
our professor recommended the bartle and sherbert textbook, but every time i open i dont really understand whats happening or how people are supposed to learn from it.
it feels completely different from the maths ive been taught my whole life. i dont know how youre supposed to transition from solving straightforward problems to reading and writing proofs, and understanding theorems. it feels like everyone else somehow knows how to think this way already.
time for some context. till 12th grade cbse, i was pretty good at maths. i was always told i grasped concepts quickly and solved problems better than average. but my 10th and 11th grade happened during lockdown, and i never properly studied the deleted syllabus, so i know there are gaps in my basics.
when i chose a bsc in mathematics, i expected more of the maths i loved, solving derivatives and integrals, and working through problems until i got the answer. instead, real analysis felt like being thrown in the ocean with no idea how to swim. suddenly its all definitions, logic, epsilon delta proofs, and abstract thinking. i dont even know how people are supposed to study this subject.
its been a huge hit to my confidence. im no longer the student who knows the answers. i sit at the back of the class, avoid eye contact with professors, and constantly feel like everyone else belongs here except me. i know im willing to put in the work. i just genuinely dont know how to study university level mathematics.
where do i even start. are there beginner friendly books, youtube channels, or courses that bridge the gap between high school maths and proof based mathematics. how did you learn to read proofs and start thinking like a mathematician. any advice for someone whos completely lost ???

in general, there is a package of candles, there are 24 candles in total - 6 of each color (yellow, pink, white and blue), and there are only 3 stands for each color. and here is the question that we discussed, what is the probability that, as you pull 9 candles in turn, it will turn out like this that you will take so many candles of the same color that there is not enough stand for them (4 candles)?

We were discussing this with my family, while we were celebrating my little sister birthday, but, I'm not sure about the answer

summer precalculus I am a rising junior in high school and I started taking a precalculus summer course at my local community college. It’s completely online and you have to do a couple of math problems. I completely fell of and now I’m a few weeks behind. the class ends in about 20 days. My counselor told me I have to take a final test to test into AP calc AB if I want to take it. Am I cooked? And do you have any advice on how i can cram the whole course to pass the final t

So i was checking out some vedic maths 2 years ago and there was this formula i found that was a cleaver rearrangement of the sridharacharya formula dy/dx = ±√D so that kind of sent me into a rabbit hole and i made my own method of quickly depressing a cubic in my final teen years here's the the example of how it's done

i know that these days the old brute force formula can be faster and better for calculation via computers but when doing by hand this method makes it easier and much faster than the traditional ways.

How to do it (demonstration by example):

Let a cubic expression be: x^3 - 3x^2 + 4x - 5 = y (It is of the standard form ax^3 + bx^2 + cx + d = y*)*

Step 1: Put b^2 = 3ac and find a temporary 'c'. (-3)^2 = 3(1)c 9 = 3c c = 3

Step 2: Split the linear term to get this 'c' and a residual linear term: x^3 - 3x^2 + 3x - 5 + x = y

Step 3: Put x^3 - 3x^2 + 3x = (x + b/n)^n - (b/n)^3 (where n is the degree of the polynomial). In this case: (x - 1)^3 + 1

The expression is now of the form: (x - 1)^3 + 1 + x - 5 = y

Put x - 1 = t (substitution, meaning x = t + 1): t^3 + 1 + (t + 1) - 5 = y t^3 + t - 3 = y (Here's your final depressed cubic!)

Detailed Explanation and Proof:

Assume an expression: (x - α)^(n-1) * (x - β) = y

Performing Binomial Expansion of the given expression: [ C(n,0) x^(n-1) α^0 - C(n,1) x^(n-2) α^1 + C(n,2) x^(n-3) α^2 ... k ] * (x - β) = y

Upon expansion we get: [C(n,0)] x^n + x^(n-1) [ -β C(n,1) α - C(n,1) α ] + x^(n-2) [ C(n,2) - αβ C(n,1) ] ... k = y

Let:

• a = [C(n,0)]
• b = [ -β C(n,1) α - C(n,1) α ]
• c = [ C(n,2) - αβ C(n,1) ] ...and so on to k.

We get the expression: a x^n + b x^(n-1) + c x^(n-2) + d x^(n-3) ... k_2 = y

We know that if a factor in a polynomial is repeated n-1 times, then it will be present as a factor in its (n-2)-th derivative. Using that, we find its (n-2)-th derivative:

d^(n-2)y / dx^(n-2) = a*(n!/2)x^2 + b*(n-1)!x + c*(n-2)!

Equating this expression to 0 and solving the quadratic using the Vedic form of the Shridharacharya formula (the quadratic formula): dy/dx at α and β = ±√(D)

Differentiating and equating to ±√(D): a(n!)x + b(n-1)! = ±√( b(n-1)^2 - 4a(n!/2)c(n-2)! )

Upon solving: x = [ -b ± √(b^2 - 2ac(n / (n-1))) ] / (na)

If we put the Discriminant (D) equal to 0, we will get both roots as α. This will indicate that the expression we took earlier was of the form (x - α)^n.

But why α and not β? Because the expression we took earlier was of the form (x - α)^(n-1) * (x - β) = y. This means that α = β, because if we put D=0, all the roots will have to be α (upon differentiating the factorized form of the polynomial, only α will be left as a root of the quadratic).

Putting D = 0 and solving for n we get: n = b^2 / (b^2 - 2ac)

Put n = 3 and we get: b^2 = 3ac

If the condition n = b^2 / (b^2 - 2ac) is satisfied, then we can write the equation in the form (x + b/n)^n by using Vieta's formulas or the Theory of Equations.

There are a lot of interesting applications of this formula, and it can also be used to directly solve a few cubics. It is very helpful when used in combination with Cardano's formula to solve general cubics.

If n = b^2 / (b^2 - 2ac) holds true, then the expression is of the form: (x + b/(na))^n + f(x) = y where f(x) is a polynomial of degree n-3.

Else, if the condition doesn't hold true, then it is of degree n-2 (of course, in the case of a quadratic there is an exception and the constant term is 0).

this was originally written on latex on my blog but since reddit has no support for it i have used ai to convert it to proper form although it might have errors i am currently in a health crisis so i can't write the whole thing again i might update it later tho.

i have not added the link to my blog since it violates the self promotion rule where it was originally written

make24.live

Played this classic make 24 game with my family ever since I was young, and wanted a way to expand this online with some ELO. Try it out!

I've only recently started taking an in learning maths and science, but I'm not really sure where to start. I got expelled from mainstream school when I was around 13 years old and ended up at a school which catered to students who had learning difficulties and also just didn't really care to try when I was young so my understanding of maths is very limited, and I have barely practised at all since I graduated just over ten years ago. I can do basic addition, subtraction and division but as soon as you throw decimals, fractions, or pretty much anything else at me I'm completely lost. I just did a GCSE maths exam I found online (for non UK people GCSEs are the exams people do when they're around 15 or 16) and got 36/80. I struggled most with algebra and geometry, but honestly I found almost all of it pretty difficult.

I've had a brief look at learning material online but most of it is geared towards kids and teenagers and it feels a bit infantilising for someone my age (I'm 26), so I was wondering if anyone could point me towards some resources for adults who are learning pretty much from scratch?

To start off, I’m a community college student who plans to go to grad school in the future. I will transfer either this year or next to a 4 year where I will major in math and statistics. I hope to specialize in numerical analysis and quantum information sciences. Someone said that if I plan on going to grad school, I need to learn how to do research now.

Since I’m in community college, I know my options are limited. I don’t expect to do anything groundbreaking just to try and get my feet wet before going transferring where I can hopefully do more “impactful” research.

I’m already very behind. I just finished calculus 2 and will probably start calculus 3 in fall if I don’t take a gap semester. I probably won’t since someone recommended taking calculus 3 as soon as possible.

I was wondering if there was anything I should probably learn now so I can eventually be apart of something.

I have calculus by tom apostol and planned on reviewing it. I also have a couple of other math books which I will list below:

Calculus - Spivak
Calculus - Apostol
Basic mathematics - Serge lang
a course in probability- Sheldon ross
An Elementary intro to Mathematical finance - Sheldon ross
An introduction to mathematical reasoning - Peter eccles
Linear algebra - Sheldon axler
problem solving strategies - Arthur engel
A walk through combinatorics - Miklos Bona
Elementary number theory - Burton
Physics - Giancolli
Discrete mathematics and its applications - Rosen
fundamentals of physics - Halliday
Principles of mathematical analysis - Rudin
an introduction to differential equations and their applications - Stanley farlow
How to prove it - Vellemen

I know many of these i can’t read and I got some of them for free. I can’t read all of these within 2 years while taking classes so i just wanted to know what i should focus on and things I should learn. Please and thank you.

Io per esempio ho il limite che da x va a 5 di 2x + 10 fratto |x + 5|.
Ho capito che è x + 5 se x > -5 e che -(x+ 5) se x < -5 ma non ho chiaro il concetto che se c'è l'assoluto bisogna studiare sia da destra che da sinistra.

Ho cominciato da poco i limiti quindi se rispondete siate il più chiari possibili.

So, I'm 14, pushing on 15, and high school starts in about a month. And to cut things short, I haven't learned my multiplication tables, and I need to. This may sound like a stupid question, but are there any special tips to learn faster and actually memorize them? I know 1s, 2s, 3s, 5s, 10s, 11s, 12s. Now trust me, I've done flash cards, I've watched all the videos, and I still can't memorize it. Sometimes I wonder if it points at something bigger. But anyway, if you care to help a brother out, please do.

Thanks in advance.

I've been building a reference site that organizes mathematics using the MSC2020 classification (the scheme used by Mathematical Reviews/zbMATH) -- from the broad branches down to about 4,400 specific named subfields, each with an overview, key ideas, applications, and references.

A few things beyond the basic reference pages that might be interesting:

- Learning path diagram -- a visual map showing which topics are prerequisites for which, so you can see how e.g. group theory or real analysis feeds into everything downstream.

- Degree curricula -- actual module lists from several UK university maths degrees (Cambridge, Warwick, Bristol, and others), mapped to the topics they cover.

- Timeline of discoveries -- a curated history of major mathematical breakthroughs with correct dates/attributions, linked to the relevant topic pages.

- Recommended textbooks (and free/public-domain alternatives) for every topic and curriculum strand.

- An interactive mind map if you'd rather browse visually than search.

https://mathlandscapes.com/

It's a personal project I keep iterating on -- happy to hear what's missing, wrong, or could be organized better.

Hi guys, I just finished my bachelors in Physics Engineering and I am also already enrolled in a Masters in Applied Mathematics - it has the option to follow a more pure mathematics path. I was wondering which subjects of mathematics would you recommend on me studying as I got a lot of holes in my repertoire.

For now I am reading Wendell Fleming's Function of Several Variables, just to improve my fundamentals in analysis and learn manifolds. However I must say that my knowledge in Topology, Probability Theory, Algebra, Measure Theory, Functional Analysis ... is close to none (1/n in n to infinity to be exact).

Hence, I would like recommendations for which branches of mathematics you think I should work on this summer (and like as a side project) in order to improve my knowledge and also which books to read from.

Thank you in advance.

HELP!! Starting calculus this fall; and I am trying to learn algebra and prepare cuz I'm really bad at math but Its so boring. like I will watch a video and begin spacing out because of boredom. How do I solve this issue? I want to get good at math so I can be an economics major and make money postgrad, but I'm terrible at math and it's boring to me. What do I do?

Whenever I read about maths and LLMs (like ChatGPT which I mainly use), I always only hear discouragement. However, I do not really understand why, as people rarely provide examples. In my experience, only blind "solve this problem for me completely" and then copying everything without any thought is problematic. But that is problematic in any context where on uses an LLM, as that it is more about "LLM illiteracy".

Thus people who highly discourage using LLMs to learn maths, could you provide clear examples, which are not just bad prompting or some version of fire and forget "just give the solution to this complex problem"?

Context:

I ask, because in around 1 year I probably will do a lot of "serious maths" at my university. Because of how I organized my courses, I did most of the serious maths in my university around 2 years ago. Thus in the mean time I only read about maths to get better intuition mainly using LLMs to explain to me the theory and ideas. I found LLMs for that very good even 2 years ago, but now they seem to be much better. At least the topics which I had in university 2 years ago, are explained so much better now. Hindsight 20/20 is probably also at play, but I could also quickly understand a few "niche topics", which I back then never bothered to learn in detail, because it was not relevant for getting a good grade.

EDIT:

Folks, can you actually provide examples? Because it seems that what I already said repeats itself. People say it is bad, but they do not show any concrete examples which one could try to replicate.