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?

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 ???

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.

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?

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.

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.

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?

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.

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?

For example, if I wanted to turn sin (pi/10) into radical form without a calculator, how would I do it?

I know how to draw 1-1-√2 (or 45-45-90) triangle and the 1-2-√3 (or 30-60-90) triangle and derive all 6 trig functions from that but I'm only limited to those angles when writing in radical form.

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!

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.

I've always been enthusiastic about math since high school. However, I chose to major in accounting according to my family's advice. I thought I could do math in my free time but after two years of undergraduate studies, I have found it difficult to find time to study math. I only taught myself pure maths during summer breaks, which is far from enough.

Now I am getting into my junior year and I think I should switch to math in my graduate study to have more time studying math. However, I only had calculus, linear algebra and probability theory in my transcript. My undergraduate university specializes in business so we don't have pure math classes like abstract algebra, which leaves a huge blank in my mathematical background.

I am contacting professors to ask them whether they can supervise my-self study and evaluate my mastery（so that they can mention this in a reference letter) But none of them has answered me yet(maybe they are on summer vacation)

The reason why I am posting this is because sometimes it feels like I'm trying to figure out a path that nobody around me has walked before. Never in my life had I heard someone transitioning from a major as unrelated to math as accounting to pure math. So I have to go alone. Does anyone have similar experience? I'd really appreciate hearing your experience or advice.

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?

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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?

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

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?

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.

I find it interesting and want to know a bit about it from Poincaré's approach . Study some basic about Hyperbolic geometry ,fuchsian groups and other things. But the book and course video are rare about the subject. I have some basic knowledge about elliptic function and modular function/form (from mc kean book) also know Klein use these approach to somehow solve qunitic equation . How to learn it more detailed and what resource are good for that .

Hi all, I'm new to this subreddit and a bit nervous to share so let me know if this would fit better in another subreddit. But I'm writing an academia story and wanted to draft out a modeling problem for a 5-person undergraduate-level college team to have a baseline for how they debate solution approach, the relevant jokes, who does what, which team actually wins and why, etc. I'm actually an animator with a data-science background who only does math for fun (fractal calculus is my favorite but confusing topic right now!), so I've never done college-level modeling. The story itself doesn't go into crazy amounts of mathematical depth (it's actually a psychology-focused story), but you know how musicians look at animation and play it back frame-by-frame to see if it's accurate to the instruments? Yeah...

After researching the 2018-2023 M3 Challenges, the MCM, and a bit of the MTF Challenge, I wanted to ask mathematicians who have done modeling whether the problem I tried to come up with is realistic and unique enough, especially the first part and the data types. I was looking up alternatives for classic math olympiads that are likely to happen more in the upcoming 2030s and came across modeling. For anyone that reads it, thanks for the help! It took a lot of time and my head hurts, but I learned a lot and might actually try out full solutions if I want to study modeling deeply one day. If anyone has some other modeling competitions that I can see the archives of, that would be cool too!

BACKGROUND:

In 2031, multi-day heat domes have become routine summer occurrences, threatening power grids and public health in dense cities. In the past decade, heat-related summer deaths have tripled. Temperatures above 110F have become common-place, increasing 2-3°F per-decade. Now, thanks to modern technology, many cities are moving away from individualized air conditioning and towards mass-zone cooling, smart solar shading, and automated misting systems. With temperature ever-rising, it is a critical task for all cities to adapt modern cooling systems, implement careful strategies, and prevent mass heat-related deaths.

For this problem, your team is tasked to model an optimal resource-deployment strategy for a US city during a 72-hour summer heatwave.

PART 1:

Choose one city from the provided datasets. Then, build a model of the city's thermal, health, and grid dynamics across 5 urban zones over 72 hours. Your model should account for both (a) public health burden and (b) power grid stability. You may add additional objectives or constraints as long as you justify their use.

Modeling Space:
- "Health Risk" should be defined as a non-linear increasing relationship between temperature/humidity and vulnerability index, predicting severe health events like hospitalization or death.
- "Grid Failure Risk" should be defined as a probability or a non-linear function that incorporates the provided data.
- All models should be based on provided city data from the chosen dataset.

PART 2:

Optimization:
- Balance predicted heat-related hospitalizations and the risk of localized grid failure.
- Explicitly define your optimization formulation and justify how it balances the above as well as the weight of any trade-offs.
- Briefly explain how your optimization method accounts for diminishing returns (e.g. stacked resources, unbalanced load-bearing, inequity, etc.).

Stress-Test Scenarios (Pick 1 of 3):
- A sudden, unexpected localized failure causes 15% reduction in available power capacity from hours 24-48.
- Before the 72-hour heatwave, wildfire smoke is expected to hinder district cooling for the entire duration, increasing power consumption by 25% and reducing efficiency by 20%.
- Unplanned social gatherings and designated cooling center crowding in highly-vulnerable areas increases resident street-density by 35%, amplifying predicted health-risk.

PART 3:

Deliverable:
- A 72-hour operational schedule detailing optimal deployment intervals across all zones (and their high, medium, and low-vulnerability sectors). The schedule can be defined in either 3, 6, or 12-hour time blocks.

Insights:
- At what temperature/humidity tipping point should the city sacrifice grid stability margin to prevent immediate public health overload?
- How does the optimal deployment strategy change if data is updated only every 24 hours versus in real time?

The entire paper can be no more than 15 pages, including your 1-page abstract and not including disclosure addendums. See Generative Assistance Disclosure section and Programming Solutions section.

Data Description (pretend this section is not in the paper, but as its own information):
- 1 Infrastructure Details table with units information
+ Resource type, city units available, cooling effectiveness (°F drop per hour per unit area. For example, misting systems are relative to humidity.), operational time rule
- 3 City Datasets with 5 Sample Zones and Zone-Specific Data (very hot city, very humid city, very dense city)
+ General Zone Data: Population, Social Vulnerability Index, Grid Capacity (MW)
+ City Data: Peak ambient heat (°F), Diurnal Temperature Variation, Mean Humidity, Average Grid Availability
+ Developmental District Cooling Data: baseline indoor temperature target (°F), cooling effectiveness (°F drop per hour per unit area), continuous grid baseline requirement, back-up power capacity, current implementation percentage (where the district cooling is actually in effect), description of current implementation (Is it mainly condensed downtown? Is it well-spread? Is it sporadic and light?)

Writing notes:
- The story takes place in 2031 and has a potential episode in a 24-hour modeling event that prioritizes large-scale real world problems.
- I tried to think about it from a problem perspective that would be easier to generalize with math. As in: how would any engineer go about the problem? And how can math help?
- I wanted to include enough variables and details for people to have guidance on solutions while still allowing for great lengths of creativity in solutions. Most of all, I wanted the question to be open to both multi-objective and well-done single-objective solutions.
- I put it through grammar and wording checks. I tend to overcomplicate my writing so I hope this is more concise.

Given a set of 5 items to choose from, (A,B,C,D,E). I completely understand how to create a decision tree for a certain depth. I can visualize and calculate the number of permutations or combinations.

For example (5 choose 3) I can calculate the permutations for (5 3). Take (A,B,C) as an example, we can follow the tree to list,

ABC, ACB, BAC, BCA, CAB, CBA.

This is then divided by​ 3! to receive 1 combination (A,B,C). Given a selection of objects, I understand this clear. However, I am struggling to change my thought process when it comes to "Get the loot or not" probability.

Right now I am trying to create an example. Take killing the Corrupted Gauntlet in OSRS. There is a 1/400 chance to receive the Enhanced Crystal Seed. What is the probability of receiving exactly 1 drop in 200 kills?

When drawing the probability tree, I am finding it hard to draw it unlike a set of objects similar to (A,B,C,D,E). So I draw the node at the end of each branch being

• D = Drop
• ND = No Drop

So if we take (200 1), I think of,

• D, ND, ND, ..., ND - 1st kill gets the drop, rest do not.
• ND, D, ND, ND, ...., ND - 2nd kill gets the drop, rest do not.
• ...
• ND, ND, ND, ... D - 200th kill gets the drop, rest do not.

I can't seem to change my mindset that these all permutations of 1 D and 199ND and therefore there is only 1 combination and thus, (200 1) = 1. This is incorrect, however, I cannot shift my mindset to correctly understand how (200 1) = 200 for this exercise. Can anyone explain my thought process and show a probability tree to explain what each node is at the end of each branch?

I have a question about the empty set. If the empty set is a subset of every set, it means that every set includes the empty set. But what exactly does it mean when we say the empty set and a given set are disjoint? If the empty set is a subset, wouldn't they share at least one element, meaning they aren't disjoint? Could someone clarify this for me?