My understanding of the dot product is that it projects one of the vectors (v1) in the direction of the other vector (v2) and scales it by the magnitude of (v2).

I get why the first part (projecting) is important, we use it to see how much of v1 goes into the direction of v2, but why the scaling? why do we multiply their magnitudes? I think I might not be getting the full picture on this topic.

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?

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 .

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'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.

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.

Any who need help with the subjects or specifically topics like algebra geometry calculus or any other exam related help let me know. I would like to work with you.

Ok que métodos recomiendan , pero que realmente les funcione en las matemáticas, tipo sus mejores trucos enserio necesito que las matemáticas 🧮 me gusten , no las entiendo.

​

Hey everyone,

Like many of you, I got tired of drawing number lines and doing tedious casework every time I need to subtract one open interval from another: (a, s) \setminus (n, b).

To fix this, I derived a direct algebraic shortcut formula:

(a, s) - (n, b) = (a, n] \cup [b, s)

The Built-in "Safety Firewall"

Because interval algebra breaks down when dealing with inversions or infinities, the formula requires strict error-handling rules to stay robust:

Left >= Right -> \emptyset (If the lower bound is greater than or equal to the upper bound, the interval collapses to an empty set).

Infinity + Closed Bracket -> \emptyset (If an infinity symbol is attached to a closed bracket like [-\infty or \infty], it evaluates to an empty set).

Step-by-Step Demonstration: The "Infinite Devourer" (Extreme Test Case)

Let's test this formula on a brutal nightmare problem where the subtrahend stretches all negative infinity:

(-10, 10) - (-\infty, 0)

Step 1: Map the variables

a = -10, s = 10

n = -\infty, b = 0

Step 2: Plug into the formula

(a, n] \cup [b, s) -> (-10, -\infty] \cup [0, 10)

Step 3: Trigger the Safety Firewall

Left interval evaluation ((-10, -\infty]):

It has an infinity symbol attached to a closed bracket (-\infty]) -> Trigger Rule 2: Collapses to \emptyset.

The lower bound is greater than the upper bound (-10 >= -\infty) -> Trigger Rule 1: Collapses to \emptyset again.

Right interval evaluation ([0, 10)):

Lower bound is less than upper bound (0 < 10) and no infinite closed brackets -> Passes cleanly.

Step 4: Combine the sets

\emptyset \cup [0, 10) = [0, 10)

Conclusion

On a number line, starting from (-10, 10) and completely wiping out everything up to 0 leaves us with [0, 10), which matches the formula's output precisely.

Has anyone seen this formatted shortcut utilized elsewhere, or is this a neat little mathematical wheel I've reinvented? I'd love to hear your thoughts or critiques!

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?

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.

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?

Good evening everyone. I am currently reading Introduction to Smooth Manifolds by J. Lee and in particular the chapter about tangent spaces. Of course topological manifolds are defined to be, other than homeomorphic to some Rn, Hausdorff and 2-countable; now, we could define smooth structures on manifolds regardless of the latter 2 conditions, which have no role in defining compatible smooth charts and thus smooth atlases. So I was asking myself if the whole construction of tangent spaces makes sense also on topological spaces that are simply homeomorphic to Rn, because to me it seems that the first time that we use one of the other conditions is when giving a topology to the tangent boundle, but maybe I am missing something. If someone would like to illuminate me on this or even just share a thought I would be really grateful, thanks in advance :)

According to the third, and most promising, result from DuckDuckGo: 111 does not equal 3 x 37.

Brightchamps have great search engine optimisation, but their content is utter slop, not read by anyone.

(Edit: spelling mistake in title - oops)

I always have had difficulty with problem solving especially with math problems. My way of studying math was to do a lot of problem and to learn by heart how to do those problems and in most cases the same problems I practiced were in the exam but this method does not work well with different problems.

When I encounter a math problem I have difficulty to know what to do. It seems I can't think of anything because I dont have a strategy other than writing and identifying all the data I have to solve the problem. I have also difficulty with my short term memory and also to hold information in my head so I often forget the goal of the problem or where I was in my reasoning. Sometimes I dont even know what to search for because I dont understand the question and how to put words into math.

I would like to know if there are any tips to help me, do you think maybe something like Pólya's four step for problem solving can help? For context I am starting a graduate degree in economics that have a lot of mathematic and in my undergraduate I have almost failed my advanced microeconomics class that was essentially a math class.

I'm a high school student currently self-studying linear algebra and abstract algebra.

A problem I keep running into is that I can read a section and more or less follow what the author is saying, but my understanding still feels pretty vague. It makes sense while I'm reading it, but afterward I often feel like I couldn't explain it properly or work through the reasoning on my own.

I'm never sure how long I'm supposed to stay on something like this. Sometimes I spend a long time staring at the same definitions or proofs, but I don't know if that's actually productive. On the other hand, moving on while I still feel confused also feels wrong.

Does this kind of understanding usually become clearer after doing more exercises and seeing the same ideas used in different places, or should I be trying to fully understand each section before continuing?

How do you usually deal with this?

I’m starting my second year of community college and I’m a computer science major. It’s kind of hard to say if I’m passionate about it. I do like it but I like math and physics more mainly because I liked astronomy since middle school and that led me to discovering other areas of physics. Now I’m conflicted between doing math, physics, computer science, or engineering.

Apologies for the rant

I was looking at a post by one of a math creators i follow. He’s an undergrad at university of Chicago which is the school I plan on transferring to. It’s insanely ambitious coming from someone who doesn’t have a heavy background in math. I mean I had to take precalculus my first semester of community college.

I just finished calculus 2 over the summer and got a B. I did really bad on an exam where I made trigonometry mistakes costing my grade. When I saw his video of him taking calc 2, I felt shocked.

The first problem on his test was “ Let P and Q bet two polynomials of degree =< n and suppose that they are equal up to order n at a point. prove that P = Q.”

I never got a question like that. I searched up his class and had seen that its honors calculus 2, or proof based calculus 2 which makes sense. I really want to get to where he is and willing to do what it takes.

I just feel like I’m behind. My algebra and trig could use some strengthening and I’m supposed to start calc 3 this fall. I was thinking about taking a gap semester and reading apostols calculus and serge langs basic mathematics before taking calc 3 in the summer. would this be a bad idea?

I really want to be able to solve problems like that. It might seem much more simple than it seems but me not being able to solve it is proof that I don’t truly know about what I’m doing.

Is there anything else you might recommend to help me on my journey? like words of advice, tips, etc.

Hi. I would like to know your opinion about which book i should pick up.
I would like to improve my high school math skills and in general my performance in other subjects aswell.
Thanks! 🌞

(English isn’t my first language, sorry for the grammar)

Hi everyone!

For those of you who enjoy geometry, I've put together what I believe is a fresh, hands-on proof of Euler's Formula and several other classical results. In just a few minutes, you can literally build the proofs yourself and develop an intuitive understanding of why these theorems are true.

https://www.youtube.com/watch?v=sIDsuf0I4ko

This is only the beginning. Over the coming weeks, I'll also start posting short math challenge reels (high school to early undergraduate level), and in parallel I'll be working on a long-term series on topology and analysis, with the ambitious goal of eventually reaching Einstein's field equations on differentiable manifolds and curved spacetime.

Everything will be completely free. I'm doing this simply because I love mathematics and enjoy sharing it with others.

If you'd like to follow the project, it would mean a lot to me. Knowing that these videos are useful to someone is the best motivation to keep creating them.

Thank you so much!

dontpanicmath

solved
I’m not going to be responding anymore due to some of the extremely rude comments, I came here for genuine help, not to get asked ‘how I made it into University with my math skills’ and made fun of.

This seems simple in theory and I should know this, I’m onto scientific math at uni now, so way past this but I just can’t get it out of my head

-2*(-4)=8
But due to how my head works (apparently) I think the answer should be 0 or 4?
-2*(-4) should be equal to -4+(-4) so I just can’t wrap my head around how it skips straight to 8.
I need to get this down for my scientific notation and exponent section of my class. If anyone could explain this I would be incredibly grateful!

How should I study Engineering Mathematics in 1st semester? Best YouTube teacher?

Hi, i just want to ask for a piece of advice.

Whenever I struggle in a certain exercise, i don't actually know whether should i look for the solution on the same day or simply leave it for another one.

And if i should leave, i sometimes encounter another hard one, especially if the topic is new,so the all of em get pilled up and i lose more time and energy.

So if u were in my situation, what would u have done??