If mathematics is the science of patterns, what pattern does set theory study? My thoughts so far: set theory studies the pattern of relationships between members of a collection and between collections. Background: This isn’t a homework question, I’m starting on Introduction to Mathematical Thinking and ensuring I have a good grounding in the fundamentals […]

If $a \in S$ is some invertible element in a ring $S$, then a computation shows $$\pmatrix{a & 0 \\ 0 & a^{-1}} = \pmatrix{1 & a \\ 0 & 1} \pmatrix{1 & 0 \\ -a^{-1} & 1} \pmatrix{1 & a \\ 0 & 1} \pmatrix{0 & -1 \\ 1 & 0}.$$ If $R \to […]

I haven’t taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat’s little theorem, the law of quadratic reciprocity, etc. rely upon techniques found in abstract algebra? Thanks.

I want to ask if there is some book that treats Differential Geometry without assuming that the reader knows General Topology. Well, many would say: “oh, but what’s the problem ? First learn General Topology, and you’ll understand Differential Geometry even better!” and I agree with that, but my point is: I’m a student of […]

I notice some problems has many different proofs, do all theorems have multiple proofs, is there some theorems which has only 1 way to prove it? $n$ ways? infinite?

Question. How does one know that a theorem is strong enough to publish? Basically, I have laid out a framework in which many theorems may be proven. I’m only 18 and therefore lack knowledge of whether this framework and the theorems sprouting from it are trivial along with the theorems. What is a good indicator […]

I recently placed a question based on quadratics and received a few valuable answers. One of them was a comment in an answer with a link in it which I found useful. But unfortunately the webpage (of which the link was sent) was in Russian (which is totally a foreign language to me) and so […]

I’ve always been curious about this, why do we use fields as the only algebraic structure to put vector spaces over? It seems a bit arbitrary to me, so I was wondering what would happen it we replaced the requirement with something less structured like a ring (with unity, we don’t want to violate the […]

I have been concentrating on olympiad questions, and PUTNAM exams, Putnam is my main focus. Can you suggest a book from one of these: Problem Solving Strategies By Arthur Engel Putnam and Beyond by Andreescu Titu et. al First, can you tell me which one is better from the above? I only have one choice? […]

I just took a course in general topology about a month back, and I was wondering whether it was possible to explain why the Earth seems flat from our point of view but is in fact a sphere using the concept of a homeomorphism? Is it the fact that the sphere and plane are homeomorphic […]

Intereting Posts

Frattini subgroup of a finite group
Isomorphisms between group of functions and $S_3$
Limit of argz and r
Every minimal Hausdorff space is H-closed
Center of dihedral group
Eigenvalues of the transpose
The complement of a torus is a torus.
If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$
The axiom of choice in terms of cardinality of sets
Size Of Proper Classes
How to prove $f(\bigcap_{\alpha \in A}U_{\alpha}) \subseteq \bigcap_{\alpha \in A}f(U_{\alpha})$?
Can forcing push the continuum above a weakly inacessible cardinal?
Proof for power functions
Let $a$ be a quadratic residue modulo $p$. Prove that the number $b\equiv a^\frac{p+1}{4} \mod p$ has the property that $b^2\equiv a \mod p$.
How to make four 7 s equal to 4 and to 10?