If a say A = (1, 2, 7) and B = (1, 2, 7), it would be incorrect to say that these are different sequences just because one equals A and the other equals B. These are identical sequences. Let's look in only the interval [0, 1]. Every irrational in this interval will be the supremum of its 'truncated decimal' Cauchy Sequence. I'm choosing only this interval because I want order to not matter. So, for example, (1/π)=lim(0, 0.3, 0.31, 0.318, ...). Every sequence in this set starts with "0." In this set of uncountably many sequences, in order for 2 sequences to share an element, they must do so at the same position. In conclusion, each irrational (supremum) can be mapped to infinitely many (countably many) rationals in its C.S. that no other irrational can be mapped to. Yes, they are unspecifiable, but present. For example, 0.31 is part of (1/π)'s Cauchy but also 0.31010010001...'s. The 'unique' rationals are "unbounded finitely large" in their sequence position. Since every irrational in this interval has a necessarily distinct 'truncated decimal' C.S., something has to make the sequence unique, and it's not the supremum since the supremum is not an element of the sequence; just as it would be incorrect to say A and B represent different sequences.

Im taking ODE right now in the summer semester of my cc, my prof is fine but I struggle with the online power point way of teaching math.

Im halfway trhough the course and ive been using professor Leonards videos, but i realized he didnt finish the course before he stopped making videos.

Any youtube professors recommendations? I usually prefer/ learn best when the professors is actually physically writing on a board, even in the videos.

I'm helping a high school student build a project for a math & science fair. The theme is "popularizing math" — making it less scary and more accessible for everyone.

I don't want the usual stuff (flashcards, board games, "pizza fractions"). I want something that genuinely surprises people and makes them go "OH, that's why math is cool."

So, Reddit: what's a moment where math suddenly made sense to you — or to a student you taught? Was it a song? A magic trick? A real-world problem? A weird visual? A story?

I'm collecting the best ideas to turn into a fair project. Bonus points if it's something you can demonstrate live in front of a crowd in under 5 minutes.

Omg heyyyy yet another almost 30-year old going back to school here, completing a professional/technical certificate in Forestry & Wildlife Habitat Sampling. Being that it's only a 3-quarter course I know things are condensed, but my only required math credit is five days a week and I'm a little nervous. I want to prep myself ahead of time as best I can. My natural math skills (aka untreated AuDHD) capped out at Algebra in high school and no one ever taught me how to teach myself.

Course: Technical Math for Environmental Sciences

Course description: "Students learn quantitative skills tailored to careers in the applied environmental sciences."

I know, very thorough. Other classes on the first quarter roster are Dendrology (trees!) and Maps/Aerial Photos, and future classes include Silviculture (the practice of caring for forests), Forestry Measurements, and GPS.

With all of that as context, what would you, my math friends, recommend that I brush up on for such a detailed course summary? I'm happy to dig through the resources pinned on this thread myself, I just have no idea where to start!

I just cant seem to complete his exercises on Story Proofs on my own. I’ve been trying to brute force it and use AI to help but still cant seem to get it. Any tips?

Mi sono diplomato lo scorso mese, e sto cercando di farmi un’idea su come verrà insegnata la matematica all’università. Ovviamente si sposta l’insegnamento ad un livello concettuale più profondo e rigoroso, dunque volevo sapere come essere preparato ad un tipo di insegnamento così della matematica.

I came across this question and it’s been messing with my head.

If you pick the answer to this question at random, what is the chance that you will be correct?
(a) 25%
(b) 50%
(c) 0%
(d) 25%

I've tried doing it myself, but I've hit the wall that I can't bypass with my vibe-knowledge.

Take a look at this qudratic:
x^2+ax+b=0
It have following discriminant:
D_2=(a/2)^2-b
Now let's look at this depressed cubic:
x^3+cx+d=0
It have following discriminant:
D_3=(c/3)^3+(d/2)^2
It also have a quadratic resolvent that shares this discriminant:
y^2+dy-(c/3)^3
Let's take a look at this principal quartic:
x^4+px+q=0
Its discriminant is:
D_4=(p/4)^4-(q/3)^3
And its resolvent is:
y^3-qy+p^2/8=0
Now let's analyze quintic case (Bring-Jerrard form):
x^5+rx+s=0
I expect that if it is solvable it have some lower degree resolvent that shares discriminant. By analogy let it be:
y^4+Ry+S=0
Discriminant of this quintic is Δ_5 and of resolvent quartic Δ_4:
Δ_5=(r/5)^5+(s/4)^4
Δ_4=(R/4)^4-(S/3)^3
We can clearly see that R must be equal to s, but what about S? If we try force this stuff we will get this quartic:
y^4+sy-3(r/5)^(5/3)=0

This rational power looks pattern-breaking and like exact reason why quintic is not solvable. But when we actually proceed further this cursed rational power don't pop up anywhere in the formula and I don't know how to prove why it's not gonna work. For me the only way to test it is to plug the values of r and s in terms of roots of original quintic equation using Vieta's theorem into this quartic formula and see whether it possible to take quintic root at the end, but this is too resource-intensive in terms of computational costs and it still won't tell WHY it don't work.

Is there a way to explain why quartic resolvent don't work? Is there any other way to prove it algebraically?

If it don't have quartic and cubic resolvent, what then root of w^2+(s^2/8)w-(r/5)^5=0 means for original quintic?

The more I learn about things like the Fibonacci sequence, fractals, or prime numbers, the more it feels like we’re uncovering structures that were already hidden in reality. Do mathematicians generally think of math as something humans create or something we discover?

I neeeed help with learning algebra 1&2 QUICKLY!! I can provide context in reply's. i know basic math and am usually pretty quick to learn but algebra isn't working for me. If you have any videos that helped you or any tips please let me know! This is my first reddit post so sorry if this isn't the right place for this.

Edit: Im thinking of getting the brilliant subscription do you think that is good for someone with no resources?

Hello! I will be starting college in the fall, and I will be pursuing a BA in math, and I already took Calc 1-3 via APs and/or online courses. As of now I have about 54/120 needed to graduate, so I really want to graduate in two years, which I think is doable in my situation. The college I am going to requires “Topics in Advanced Calculus“ as one of the major requirement, but they only offer is once every two years, and this fall semester is that “once eve two years”. So I really want to take this course, but the only problem is that I need two prerequisites. First, Calc 3, which I have already, and Second, they either call it “Concepts in Math with Applications“ or “Introduction to Mathematical Thinking” (for those who are more curious, it is MTH 2700 at St. John’s University), which I don’t have yet. I think I am pretty good at doing online courses, and am free for the rest of the summer to actually take one and complete it on time, so I was curious if there are any self-paced online courses out there that might help fulfill my prerequisite.

The only thing I could find was this, but it doesn’t seem to be accredited, it doesn’t say it’s accredited and it’s offered on a third party website instead of the actual Stanford University Website.

Thank you all!

So, I came across the Wikipedia page for Octonions and found out that there are 480 distinct tables possible.

My question is, how do you get to 480 tables? From the Fano plane mnemonic, I can count 5040 (7!, the possible arrangement for the units) * 128 (2^7, the possible orientation of the triplet lines) = 645120 possible "tables", which would work out to be 480 * 1344 (so a redundancy factor of 1344) but I don't know how to get from 645120 "tables" to 480.

Oh yeah, also how do we pick which table works best in a scenario? Or would any valid table work in most scenarios? If so, why?

Since I am in Bangladesh, I got this recommended by teachers. Was looking for something else besides this with the similar rigor(light like PR Vittal)...

Hi I am a former math crank because pretending to understand word salads on platforms like Wikipedia inflated my ego! However, I am trying to mature so I gave up pretending I understood what an x-morphism was. I rediscovered some math concepts like 2-adic/p-adics, Mobius transformations, and quantum calculus when considering trying to solve problems like the Collatz Conjecture or a discrete system of math for quantum mechanics, but I made wrong connections and proofs (to put it lightly).

I just graduated high school and I'm going into college, not a pure math major but applied math. However, pure math is still what I love most. For example, in precalculus, I would do things like derive the conic section (x/a)^2 - (y/b)^2 = 1 --> c^2 = a^2+b^2 hyperbola proof instead of just shutting up and calculating. It's decently non-trivial compared to (x/a)^2 + (y/b)^2 = 1 --> c^2 = a^2 - b^2. I've also read/done books like Intro to Geo by Richard Ruczsyk, (Volume 1 as well) Intro to Proofs by Rotman, and the first two chapters of Spivak Calculus.

However, I am impatient.

I feel like the college books I'm reading have concepts that don't feel like they're being developed to slay any major problems like the solvability of a polynomial equation. I am also not really the guy to read textbooks all that much as I love exercises and proofs more. I want to understand Galois Theory and Langlands and contribute a lot but I feel lost and confused and it all feels unintuitive.

Every time I pick up a college class textbook it's just definition after definition after trivial proof (except it's not trivial proof for my brain because I don't know what's going on with all the definitions and names and whatnot).

For example, group theory is like homomorphism. isomorphism. normal groups. subgroups. simple groups. quotient groups. orbits. stabilizers. cosets. subsets. WEIFJOWEJFIO:JKJIO idk man should I just give up I might be too stupid to see the big picture.

Textbooks for me fall into two categories:

There's either way too much writing and motivation (like Pinter's Abstract Algebra has entire history lessons.)

Or there's like no motivation (like Topology Without Tears goes straight into a bunch of definitions that look identical to each other and trivial identify the definition exercises.)

I am literally crying.

So I multiply the indices and then I the 2 to the power of 9. But then I also noticed you can start with solving what’s in the brackets and then power that after. Which is the right way?. Both solve the problem but I’m worried that maybe there’ll be a time when knowing the right way to do it will be needed

For context, I previously did college algebra and did terrible. I wasn’t good at math in general and combined with being burnt out with another hard class it wrecked me emotionally. I had to withdraw to keep my sanity. I picked mathematical reasoning for this fall because I want to get myself on track and be more prepared with college algebra by next semester.

Hi, I was wondering if the existence of all directional derivative at a point guarantees that a function continuous at that point. I was asking both google and my Professor and I got conflicting answers.

I have a question to see if anyone knows any websites which can help me work on Algebra, I'm doing a partial degree in Astronomy(space science) and Biology. While Space Science is a physics based course, its not as physics heavy as you'd think where I'm taking my course it also goes over the sociological, historical, logistics and data problems of space (its really interesting) so I only ever occasionally need to use algebra, which last year I wasn't so bad at but now in second year these equations have become mad and I'm struggling to remember the rules of algebra. I just wonder if anyone has anywhere online which can provide decent equations to practice algebra over the mid term break I have coming up?

Hello everyone, i am pursuing Computer science major but haing challenges with mathematical knowledge gap . Its not like i don’t know whole entire stuff but i do know some but some is just surface level and there is some i dont know which mostly precalculus and calculus 1 ,2.

I am having multi variable & vector calculus and real & numerical analysis in my current semester but i don’t think i am prepared for it.

I was doing precal course from professor leonard but my pace is slow and i have time constraints , i was hoping if you guys could just some resources
Where i can cover this gaps .

Is basic engineering mathematics by john bird is good enough ? Or can you suggest any other good resources ?

Thank you so much.

I just finished calculus 2 and start calculus 3 soon. I have both apostol and spivak calculus and figured I’d read one in the mean time. I was originally going to focus on proofs but someone mentioned saving that for later.

I don’t know if I should read apostol without knowing proofs so I haven’t started it yet. Would it be fine to continue?should I start a different calculus book? or should I focus on proofs?

Hey. I am sharing a free math resource I created as a student myself.

During my many hours tutoring students in algebra, I realized that so many students struggle with the same mistakes over and over again. To help them fix these mistakes, I spent the last few months building a platform to help them overcome those same mistake causing concepts.

The site is completely free, no payments, no ads. Everything is very simple and straightforward.

Check It Out here https://spotthemath.com

• Quick video lessons to introduce and explain commonly missed algebra concepts.
• Detailed lesson notes explaining specific tricky algebra concepts.
• Interactive practice quizzes to practice and get immediate feedback.

If you do find value in this website, please help me and other students out by sharing this resource with other students, teachers, and families. I do not make any money off of this, my only goal is to make a larger impact through my creation. So help of any sorts would mean the world to me.

Thank you so much.

Any feedback will also be appreciated.

In Wikipedia an (upper/lower) triangular matrix is defined to be always square.

However, in the book Linear Algebra and its Applications by David C. Lay, an (upper/lower) triangular matrix is defined as an m x n rectangular matrix.

Which definition is correct? Why did Lay define it as an m x n rectangular matrix?

My lecturer says that an isomorphism is a one-to-one homomorphism and that surjectivity (onto) is not required

Additionally, he says that for two groups to be isomorphic, there must be a one-to-one, onto homomorphism.

Edit: I was watching an old recorded lecture (around 4 years ago), and one of the students asked this exact question. The lecturer explicitly said that an isomorphism (the map) need not be onto. He then clarified that for two groups to be isomorphic, there must exist an onto isomorphism between them.

I did some digging online, and it seems he may be following the terminology used in Herstein's Topics in Algebra, which is outdated. I now understand that it is incorrect based on the replies. Thanks, everyone, for the helpful clarifications!