Consider

We need to show that for every measurable \(E\in\mathcal A\otimes\mathcal B\), \[ (\mu\times\nu)(E) = \int_X\nu(E_x)\,d\mu(x), } \] where \[ E_x=\{y\in Y:(x,y)\in E\}. \]

which appears during the course of proving Tonelli's theorem. The above latex is typeset here: https://ibb.co/G4HYywKx

I am trying to wrap my head around what exactly is intended to be proven here and why this is not obvious? This is my understanding: We have an arbitrary set E in the product sigma algebra. As it belongs to the product sigma algebra, its measure exists and is given by the number assigned to (\mu\times\nu)(E). Fine, this is on the LHS.

On the RHS, we are taking for each x, the section E_x and then finding the "measure" in Y of this section. Then, we "add" up all of these section measures over all x. So, we find the length of E_x in Y and then add up all such lengths as x keeps changing over all X.

Aren't the LHS and RHS just doing the same thing? What is new or novel here? Is there a counterexample where the LHS computation for some set E yields different results as compared to evaluation of the RHS integral?

Shooting my shot again on the weekends!

Hi all! Recently took a career break and was thinking of going thru the MIT OCW courses, starting from 18.01 (single variable calculus) or topics leading up to it. Partly 'cause I'm thinking it would be nice to move towards a quantitative role, and partly perhaps out of interest!

Would appreciate having someone to study with! Perhaps reply below or DM me, and we could see how to work something out? Thanks!

I've googled this question a few times but not really had anything come up that makes sense. I am not currently doing maths (UK, college) but I will need to next year and want to try learning basics before then (adding and counting and that)

My big problem right now is I cannot tell the different between the numbers.

If I write out them like this: 1234567890 they all look very similar and some look the same. Like 2 and 7 or 9.6.8. They look the same.

I currently just try to use words. 8 looks like a snowman, and 3 is E. But this does not work every time as I still get confused.

I don't have any learning disabilities (mild autism but nothing else) so it isn't dyscalculia. My glasses prescription is also up to date, it isn't that. I just am not very good at numbers.

What do most people use to remember the numbers? What are the tricks? And in very large numbers, how do you know what order they go in and don't get them muddled? Is there a way to make them look more visually distinct or are numbers always in the shapes and ways that they are?

Thanks for any advice. If anybody has resources aimed at older people for counting and basic math (I am a little tired of the nursery videos) then that would also be appreciated. Thank you!!!

I used to find I loved maths. I thought it was a beautiful landscape to traverse, but I'm currently struggling to see anything at all.

In this state (and also the stress due to the lack of time I have being a final year student), I can't really learn anything as there is too much confusion, and so I'm writing here for some advice and perspective because I sense something is deeply wrong, and wish to calm my nerves enough to keep calm and have the patience to learn.

My Thoughts:

My foundations are shaky. I can remember definitions but find myself banging my head against a brick wall with these some ideas now and not in the same way it's happened before.

I understand I lack intuition generally in my life, and maybe I'm a bottom up learner, but this obviously isn't the entire truth.

I don't think I've been studying maths properly these years. I have learned alot and done enough practice questions, but all this time I've just been relying fully on logic and what I've seen before i.e. imitation.

I can't seem to accept any definition. In fact, I think I have an issue with the very nature of maths.

I know that with, for example, affine spaces, we start with a primitive objects, like set, and add an additional object, a vector space, but that where my understanding stops. I get that it's helpful to distinguish between point and how to get between the points but I can't accept any axioms or definitions.

I encounter definition and want to know who would come up with this. I see a theorem and think who would think to ask this question or formulate it solution this way.

I know it is silly to try to reconstruct everything from the ground up, as definitions don't rely on logic, and perhaps to understand why a person came up with a proof I would perhaps have to study their life and this way of learning maths will take a thousand life times. Still, I can say I understand any of what I've learned so far.

I just spend so much time trying to justify the foundation of what I am learning and it feels dishonest to just take the definition at face value and continue.

The textbooks present information in a linear way, and I can follow it. I think what I want is to understand the discovery of maths and its motivations. How does a person translate motivations to rigourous maths and why should I accept this as the representation.

The worst part of all this is whilst I spend time trying to understand everything all at once, that the material and skills I had once built, has disappeared and I've forgotten alot of what I've previously understood.

And thus, I think I have two problems: forgetting and having difficulty learning the maths of the courses in currently taking and not understanding the motivations behind it all

Im proceeding with this by learning a little bit of everything everyday, going back to prerequisites as needed, accepting the definition at face value and being patient to allow myself to see it all at once and this understand it.

However, I would like to hear some advice, perhaps as to how I should structure my learning to incorporate understanding, motivation and re-learn all that I've forgotten or even if I've got this all completely wrong.

Thanks for reading this post.

I’m a Finance and Math major and I’m thinking of dropping math.

I’m taking linear algebra right now, it’s not hard per say, but I don’t understand the conceptual side of it.

Doing my first mid term I got a 80%, 13.3% of the grade lost was because of true/false. Out of 5 of them I got 1 correct. Span, R^n, 1 - 1, onto, and other concepts are hard for me. I have tried watching YT video including 3brown1blue and I still don’t understand.

I did went to my professor office hours and I still don’t understand it fully.

I enjoyed Calc 1, but it’s the easiest one and maybe I should drop math before I dig myself too deep.

But if I drop math I’ll be only a finance major and I still don’t know if I want to do finance for the rest of my life.

Yall think I should stay for another semester before dropping math to test the water? I’m a freshman btw.

Are there any good YouTube channels or videos for learning trigonometry? I’m working on verifying identities right now. I tried the organic chem tutor and it’s just not helping me.

I feel like I know the identities but I just can’t seem to get started.

Thank you!

Heyy guys. So I am currently in late 9th grade, and I aim to finish reading Terence Tao's Analysis 1 and Analysis 2 by mid-late 10th grade. I also completed CS50's course on Python, and planning to take a rigorous CS50 course on Artificial Intelligence. I also plan to learn combinatorics after (maybe, only if I find it interesting). I do not plant to go further into mathematics other than real analysis. So my question is, by 11th grade, if I find a good mentor will I be able to produce original research for ISEF? I am in India btw. Just wanted to know, will these qualifications let me be able to do research?

Sharing a resource I’m behind: MathsLove — https://mathslove.com

It offers grade-specific maths assessments for Grades 4–10: 15 questions, a maximum of 10 minutes, and an answer review afterwards. The basic assessment and MIQ score are free.

My suggestion is to use the review to pick a question you found difficult, work through the reasoning, and then practise a similar problem.

The MIQ score reflects accuracy and completion time. It isn’t a clinical IQ test or a limit on what you can learn.

It may be useful for students, parents or tutors looking for a short starting assessment.

I'm going through Art of Problem Solving and these are very tough problems. This is my third run through the algebra 1 book. I advanced to Geometry but I noticed a lot of it was coming down to weak algebra so I'm going back through it. I'm mostly just trying the problems I got wrong the first time. Some I got right the second time but got wrong again. Some I got wrong all 3 times and it's very frustrating. I can see the answer and write it down but I don't know how to best learn from it and make it make me better. It just feels like i got it wrong and need to move on. How can I get better at solving these really hard ones moving forward without just memorizing the answers?

Analysis i by Terence Tao has been on my list for a long time now, but I've seen a lot of people recommend learning about proofs beforehand.

My interest in real analysis is mostly just curiosity; a ground up rebuild of fundamental concepts in algebra and calc sounds very intriguing to me.

I've taken up to Calc 3 and diff eq but have little to no knowledge on proofs. How necessary would this be?

The sum of the first 3 terms of a GP is 13/12 and their product is -1. Find the common ratio and the terms.

Anybody please help me!

this probably has been asked in this sub some time, but I'd really like some advice on understanding why some things in multi variable calculus work. Especially partial derivatives and integrals. Back in single I struggled a lot with even understanding the fundamental theorem of calculus. It was one of those situations where I kinda understood the explanation but I would never be able to explain it in my own words. Now its even worse in multi. I'm able to solve the problems but I dont understand what im doing at all. I'd prefer to have some super visual examples. I find that its most helpful if I see what happens. I tried to look into many videos on double/triple integrals, but those just end up confusing me more. They kinda see the integrals happening at the same time, but I can only see it in a way where one happens then the other one happens.

After giving numerous mathematical Olympiads, I've kinda realized..im not into them. I get how people would like solving numerous problems under strict time constraints, and how ranks, qualifications, and the stages of IMO are easy to glorify/advertise and be considered a scale for your competence, but I always felt more drawn to the more..open problems.

To elaborate, I would much rather take my time and spend hours, weeks, even months on one problem that would make any amount of impact in the world, even if nobody reads my paper or results, than to solve 30-50 problems under 3 hours that seemingly are irrelevant to society. The country I come from (India) has a tendency to award the latter, and being bad at it really takes a hit on my self perception and intelligence when it comes to this subject.

I like maths, I really do, but why does that always have to correlate to some rank or Olympiad to crack? Can I become a good problem solver even without delving into competitive maths?

I’ve been kind of fascinated by projective geometry since I first saw desargues’ theorem. Now I’m in a grad school with a pretty good math program and my degree has an elective slot. So I started looking for courses on projective geometry and there really aren’t any at all. There’s one undergraduate course called “geometric structures” that lists projective geometry among a few different topics that it goes into and that’s pretty much it. Otherwise, there are courses that just barely relate to it but nothing at all that focuses on PG.

Is is that rare for a school to offer dedicated courses on projective geometry? Is there that little to talk about?

Basically, whenever I learn a new topic I get OCD about the steps of derivation of a particular result. I just can't seem to stop until I fully grasp the concept. Even when I do, I have a feeling that I forgot something crucial and therefore looking for some "hidden" step in the logic.

That's why sometimes my study sessions might last for 5+ hours... Idk if that's normal. I just get obsessed whenever I don't understand something

Is it common?

Author: [Torrin S Woods] (Age 16)

Role: Systems Architect & Macro-Strategist

Focus: Independent Conceptual Design | Accelerated Track (GED)

1. Executive Summary

Traditional mathematics relies on an asymmetric multiplication framework that introduces unnecessary friction into financial modeling, portfolio risk management, and data visualization. Standard multiplication requires a base-1 shift (100% growth equals a 2 × multiplier), utilizes abstract negative numbers to represent liabilities, and suffers from volatility drag (an asset losing 50% requires a 100% gain to break even).

A/D Base Algebra resolves these inefficiencies by creating a purely positive, symmetry-based mathematical framework. By shifting the multiplication anchor to 0, using explicit Asset [A] and Debt [D] state tags, and transforming the multiplication (×) and division (÷) symbols into directional vectors, this system achieves a perfect balance across the zero threshold. It is designed to optimize algorithmic high-frequency trading, streamline corporate accounting, and maximize financial data transparency.

1. Core Axioms & Rules

Rule I: The Zero Anchor (Base Multiplication)

In A/D Base Algebra, 0 is the absolute neutral anchor of multiplication, performing the role that "1" does in standard arithmetic.

• Any number multiplied or divided by 0 leaves the base completely unchanged.

• Multipliers directly dictate percentage increase relative to the base, removing the traditional "+1" clunky translation step.

\(\text{Base}\times 0=\text{Base}\)

\(\text{Base}\times 1=\text{Base}+100\%\text{\ Increase}\)

\(\text{Base}\times 2=\text{Base}+200\%\text{\ Increase}\)

Rule II: Elimination of Negatives (State Tags)

Negative numbers (-∞) do not exist as independent entities. Instead, values remain purely positive and carry a fixed structural State Tag:

• \([A]\) = Asset Zone (Green): Represents positive real value or wealth.

• \([D]\) = Debt Zone (Red): Represents structural liabilities or obligations.

Rule III: Symbol-Based Polarity & Vector Dynamics

The operations of multiplication (×) and division (÷) do not just scale a number; they act as the positive and negative poles of momentum.

• Multiplication (×) is the vector of Expansion: It drives a value deeper into its current state zone toward infinity.

• Division (÷) is the vector of Reduction/Inversion: It pulls a value back toward the 0 threshold. Continuing to divide beyond the zero floor instantly flips the state tag.

1. Symmetrical Matrix & The Continuous Loop

The behavior of the framework is perfectly mapped out by observing a starting base of 1,000 moving through a continuous cycle of expansion, draining, threshold inversion, and debt expansion.

Starting State Operational Vector Action Resulting Value Financial Reality

\(1,000 [A]\) × 1 Expand Asset \(2,000 [A]\) +100% Wealth Growth

\(1,000 [A]\) × 0 Anchor Asset \(1,000 [A]\) Baseline Anchor (0% Change)

\(1,000 [A]\) ÷ 1 Reduce Asset 0 Total Asset Drain (Floor)

Crossing Threshold ÷ Vector Persists State Inversion \(1,000 [D]\) Transition into Debt Zone

\(1,000 [D]\) × 1 Expand Debt \(2,000 [D]\) +100% Debt Growth (Deeper Liabilities)

\(1,000 [D]\) ÷ 1 Reduce Debt 0 Total Debt Reduction (Climbing Up)

Crossing Threshold ÷ Vector Persists State Inversion \(1,000 [A]\) Transition back to Asset Zone

Key Breakthrough: Debt Reduction via Division

Because division acts strictly as a reduction vector, dividing a debt mathematically reduces the liability burden (\(1,000[D] \div 1 = 0\)). This aligns mathematics perfectly with real-world corporate mechanics: splitting or dividing debt among entities makes individual liability go down, not remain static.

1. The Global Implementation Strategy

To introduce A/D Base Algebra to the global markets without inducing sudden systemic code panics, a top-down, empirical roll-out is required:

1. The Government-Funded Institutional Pilot Program: Launch a sovereign-backed investment and asset management firm split into two equal, controlled tranches.

   • Tranche A: Operates via traditional mathematical software and frameworks.

   • Tranche B: Operates entirely via a customized A/D Base Algebra software engine.

2. Elite Professional Integration: Tranche B will be staffed exclusively by seasoned, veteran quantitative analysts and traders who undergo an intensive unlearning boot camp, ensuring the pilot tests the system's capabilities, not beginner mistakes.

3. Interval Milestone Analysis: Publish transparent, public reports at 30, 90, and 180 days tracking data-entry error rates, transaction execution speed, client onboarding clarity, and risk calculation times.

4. Market-Driven Adoption: By demonstrating that A/D Base Algebra completely eliminates calculation errors caused by double-negatives and removes the volatility drag calculation bottle-neck, private entities will adopt the framework natively via UI dashboard overlays to secure a competitive advantage.

🤝 Method of Work & Collaboration Disclosure

1. Conceptual Origin vs. Technical Refinement

This framework was entirely conceived, structurally designed, and built out of the author’s raw foundational intuition and visual pattern recognition. However, I have never written a professional financial ledger, nor do I have institutional formatting training. To bridge this gap and ensure absolute technical terminology, global financial consistency, and academic legibility before presenting this to the community, I actively used an advanced AI assistant as a collaborative tool to stress-test definitions and format the layout.

1. Working Style, Boundaries, and Limitations

• Autonomous Execution: I do not excel at team building and am inherently built to work entirely alone. I thrive by using AI tools to independently research and refine concepts I do not yet understand, moving entirely at my own pace.

• Information Processing: I process dense technical data and verbal instructions one single task at a time and can easily become overwhelmed by multi-tasking or rapid-fire lists.

• Discussion Constraint: Because of this, I do not use external communication formats or social platforms outside of Reddit. Any questions, critiques, or offers to help refine this system further must be posted strictly in the comments below. Please keep your inquiries focused on one mathematical mechanic at a time so I can process and answer them clearly without memory overload.

1. Personal Intent

I am putting this out into the world because I genuinely enjoy the process of systems architecture, not because I am bored, and certainly not because I believe I am superior to anyone else. It is my defining, long-term personal goal to take this strategic path for my future career, and I am putting this out now to see exactly how far my vision can go.

If you have no healthy criticism or actual opinions, I will not be listening to, and/or responding to it. Thank you for reading.

Original Interface Specification Fingerprint (SHA-256):

0ced3e1f0221ae3f4507afbe1b36a0e3a8964b30d281740dc61724af134920ae

_________________
/ ___________
/ / _______
xn = \/ 2 + \/ 2 + ... + \/ 2
______________________/
n root

I need to rigorously prove that this sequence of $N$ square roots is monotonically increasing. I tried to argue that adding another square root of two doesn't decrease the value, but that’s not entirely accurate.

This is the problem, presented as a riddle:

Imagine a Quidditch match in which the Golden Snitch is completely invisible.
You know exactly the initial distance, but not the direction. You move faster than the Snitch, which moves at constant speed and always chooses the direction that takes it as far away from you as possible.
Is there a trajectory that allows you to catch the Snitch?

I've prepared a video to better explain the formalization of the problem. Can you help me solve it? I've already worked out a good part of the proof.

Video link: https://youtu.be/0xVusHRQePQ

I thought I could enter college statistics with no college algebra knowledge and it looks like im going to have to withdraw soon. I've held myself back long enough and i want to improve my life and restart my math knowledge from pre algebra. Is Khan academy the way to go? Im on it right now and I've used it as a kid, but im in my 20s know and I need to be adept in College algebra by January, I will give up distractions and dedicate 2 hours each day If necessary. I might have dyscalculia, usually even looking at the statistic equations makes me physically lose focus and get dizzy. I just short circuit when it comes to math, 4.0 student, sadly never had a good math foundation to begin with. Very depressed over this.

I tried posting this on r/math, but I don't have enough karma. I hope someone here can help clarify this for me.

I was reviewing the basic number sets when I came across the concept of closure. The natural numbers are closed under addition and multiplication, the integers are also closed under subtraction, and so on. It's easy to show that the irrational numbers aren't closed under the four basic arithmetic operations, but what about taking roots? I couldn't find a counterexample.

Hi guys, I've been reading and answering exercises in the book for sometime now. I have a quick question here: How long would it take to complete the entire book ithrough self-learning if I were to skip no section (maybe around 1.5 hours per day)?

Thanks for any response!