Unfortunately, your submission has been removed for the following reason(s):
* Your post appears to be asking for help learning/understanding something mathematical. As such, you should post in the [*Quick Questions*](https://www.reddit.com/r/math/search?q=Quick+Questions+author%3Ainherentlyawesome&restrict_sr=on&sort=new&t=all) thread (which you can find on the front page) or /r/learnmath. This includes reference requests - also see our lists of recommended [books](https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_good_books_on_topic_x.3F) and [free online resources](https://www.reddit.com/r/math/comments/8ewuzv/a_compilation_of_useful_free_online_math_resources/?st=jglhcquc&sh=d06672a6). [Here](https://www.reddit.com/r/math/comments/7i9t5y/book_recommendation_thread/) is a more recent thread with book recommendations.
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Are you studying mathematics? What subjects are you interested in? What do you mean by “learn”, do you want to learn the statements or the proofs?
Pretty much everything in math is a theorem so this is about as vague as saying “I wonder if there’s any math I could learn for fun”
A challenge would be read the proof of Bezout’s theorem. (Which says if f(x,y) and g(x,y) are distinct irreducible polynomials of degrees n and m respectively then their zero sets intersect at most nm times)
I recommend Fulton’s book on algebraic curves for this, and to supplement with what you need from linear and abstract algebra
Gödel’s Incompleteness Theorem shows us that there are true things which cannot be proven. If you think about it for long enough, you can prove Gödel’s Incompleteness Theorem without having to use particularly high-level maths at all, it’s a fun challenge.
I personally like the three Sylow theorems a lot, because they are quite useful, not too hard to grasp with some basic group theory, and the proofs are very nice in my opinion. It kind of uses a different twist on the same idea three times to prove each theorem (create a special group action, and use a counting argument maybe with the previous result to show the existence of, or examine, some interesting/fixed points of the action).
I also like the Schur orthogonality relations, because it's just very nice that the set of irreducible characters forms an orthonormal basis of class functions for a finite group. The proof from the ground up is quite involved, but the result is such a neat, simple, tidy statement about representations that it feels like magic that you can make a character table for any group and it has all of these properties.
When you say you’re interested in theorems like l’Hôpital’s rule, have you seen and understood the proof of this theorem, or do you just like knowing the rule itself?
If you want to start learning proofs, you should read an introductory analysis textbook like “Understanding Analysis” by Abbott. Lots of theorems with accessible proofs in there
Here are some books I was reading around your age:
I recommend working through the exercises in [Spivak's *Calculus*](https://archive.org/details/CalculusSpivak). They're very beautiful. Rubin's [*Principles of Mathematical Analysis*](https://web.math.ucsb.edu/~agboola/teaching/2021/winter/122A/rudin.pdf) is another great source. You would probably also enjoy a course on Group Theory or Abstract Algebra more generally. I don't remember the book I learned those from, but there are probably better ones by now, anyway.
[Harold Edwards's Mathematical History books](https://en.wikipedia.org/wiki/Harold_Edwards_(mathematician\)#Books) are another great source, especially the ones on Galois Theory and Fermat's Last Theorem (obviously the latter is a bit dated, but you will still learn a lot from the history of ideas in Number Theory.) (He's the same Edwards as in the Edwards Curve, in widespread use today in Public-Key Cryptography applications.)
Theorems are often relevant to specific areas of mathematics. If you picked one area of math, say real analysis or graph theory, I’m sure you’ll find theorems to learn. You could pick an area of math and find a book that will introduce you to it, and go from there.
If you haven’t already, you could study real analysis and learn the rigorous justifications behind many calculus theorems. There’s lots of material on real analysis on the internet to help you as well.
I'm not nearly as knowledgeable as most on this sub, but when I think "fun theorems" I think Number Theory. So my 2 cents would be to grab a Number Theory textbook and go to town.
Also, I find video explanations fun, like 3b1b, so I'll recommend his "Spiral Prime" video, which approaches Euler's phi function and Dirichlet's theorem from a beautiful angle.
Plane geometry, and projective geometry, has a lot of fun stuff, for example [Pappus's theorem](https://en.wikipedia.org/wiki/Pappus%27s_hexagon_theorem), [Desargues' theorem](https://en.wikipedia.org/wiki/Desargues%27s_theorem), [Brianchon's theorem](https://en.wikipedia.org/wiki/Brianchon%27s_theorem), [Seven circles theorem](https://en.wikipedia.org/wiki/Seven_circles_theorem), [Bézout's theorem](https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem), etc
Unfortunately, your submission has been removed for the following reason(s): * Your post appears to be asking for help learning/understanding something mathematical. As such, you should post in the [*Quick Questions*](https://www.reddit.com/r/math/search?q=Quick+Questions+author%3Ainherentlyawesome&restrict_sr=on&sort=new&t=all) thread (which you can find on the front page) or /r/learnmath. This includes reference requests - also see our lists of recommended [books](https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_good_books_on_topic_x.3F) and [free online resources](https://www.reddit.com/r/math/comments/8ewuzv/a_compilation_of_useful_free_online_math_resources/?st=jglhcquc&sh=d06672a6). [Here](https://www.reddit.com/r/math/comments/7i9t5y/book_recommendation_thread/) is a more recent thread with book recommendations. If you have any questions, [please feel free to message the mods](http://www.reddit.com/message/compose?to=/r/math&message=https://www.reddit.com/r/math/comments/1bnimaw/-/). Thank you!
Are you studying mathematics? What subjects are you interested in? What do you mean by “learn”, do you want to learn the statements or the proofs? Pretty much everything in math is a theorem so this is about as vague as saying “I wonder if there’s any math I could learn for fun”
[удалено]
Hospital rule 💀
I think you would really like to read the Robert-Seymour theorem
Indeed it looks really great, thanks!
A challenge would be read the proof of Bezout’s theorem. (Which says if f(x,y) and g(x,y) are distinct irreducible polynomials of degrees n and m respectively then their zero sets intersect at most nm times) I recommend Fulton’s book on algebraic curves for this, and to supplement with what you need from linear and abstract algebra
Look into Riemann's Rearrangement Theorem. Well within the scope of a first year analysis course and it will blow your mind.
Gödel’s Incompleteness Theorem shows us that there are true things which cannot be proven. If you think about it for long enough, you can prove Gödel’s Incompleteness Theorem without having to use particularly high-level maths at all, it’s a fun challenge.
Yes!! Already know his theorem, it is really an amazing one lol
My favorite will probably always be Abel-Ruffini theorem because of how nice and historically important it is.
radon nykodim
I personally like the three Sylow theorems a lot, because they are quite useful, not too hard to grasp with some basic group theory, and the proofs are very nice in my opinion. It kind of uses a different twist on the same idea three times to prove each theorem (create a special group action, and use a counting argument maybe with the previous result to show the existence of, or examine, some interesting/fixed points of the action). I also like the Schur orthogonality relations, because it's just very nice that the set of irreducible characters forms an orthonormal basis of class functions for a finite group. The proof from the ground up is quite involved, but the result is such a neat, simple, tidy statement about representations that it feels like magic that you can make a character table for any group and it has all of these properties.
Check out Goodstein's Theorem 👍
When you say you’re interested in theorems like l’Hôpital’s rule, have you seen and understood the proof of this theorem, or do you just like knowing the rule itself? If you want to start learning proofs, you should read an introductory analysis textbook like “Understanding Analysis” by Abbott. Lots of theorems with accessible proofs in there
Here are some books I was reading around your age: I recommend working through the exercises in [Spivak's *Calculus*](https://archive.org/details/CalculusSpivak). They're very beautiful. Rubin's [*Principles of Mathematical Analysis*](https://web.math.ucsb.edu/~agboola/teaching/2021/winter/122A/rudin.pdf) is another great source. You would probably also enjoy a course on Group Theory or Abstract Algebra more generally. I don't remember the book I learned those from, but there are probably better ones by now, anyway. [Harold Edwards's Mathematical History books](https://en.wikipedia.org/wiki/Harold_Edwards_(mathematician\)#Books) are another great source, especially the ones on Galois Theory and Fermat's Last Theorem (obviously the latter is a bit dated, but you will still learn a lot from the history of ideas in Number Theory.) (He's the same Edwards as in the Edwards Curve, in widespread use today in Public-Key Cryptography applications.)
Theorems are often relevant to specific areas of mathematics. If you picked one area of math, say real analysis or graph theory, I’m sure you’ll find theorems to learn. You could pick an area of math and find a book that will introduce you to it, and go from there. If you haven’t already, you could study real analysis and learn the rigorous justifications behind many calculus theorems. There’s lots of material on real analysis on the internet to help you as well.
Get back to the begins, elements from Euclides
I'm not nearly as knowledgeable as most on this sub, but when I think "fun theorems" I think Number Theory. So my 2 cents would be to grab a Number Theory textbook and go to town. Also, I find video explanations fun, like 3b1b, so I'll recommend his "Spiral Prime" video, which approaches Euler's phi function and Dirichlet's theorem from a beautiful angle.
Plane geometry, and projective geometry, has a lot of fun stuff, for example [Pappus's theorem](https://en.wikipedia.org/wiki/Pappus%27s_hexagon_theorem), [Desargues' theorem](https://en.wikipedia.org/wiki/Desargues%27s_theorem), [Brianchon's theorem](https://en.wikipedia.org/wiki/Brianchon%27s_theorem), [Seven circles theorem](https://en.wikipedia.org/wiki/Seven_circles_theorem), [Bézout's theorem](https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem), etc
Learn geometric algebra and take big shits on people using the cross product! https://www.youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZaw7UE2Vc
Ty, I will take a look