The Question : 107 people think this question is useful I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study mathematics. I request people to tell me

The Question : 105 people think this question is useful Quite often, mathematics students become surprised by the fact that for a mathematician, the term “logarithm” and the expression $\log$ nearly always mean natural logarithm instead of the common logarithm. Because of that, I have been gathering examples of problems whose statement have nothing to

The Question : 104 people think this question is useful There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the “yes/no” questions among them mathematicians are typically not working in both directions but rather have a pretty clear idea of what the answer should

The Question : 104 people think this question is useful This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I’ll show you what proofs I have and I’d like to know

The Question : 116 people think this question is useful The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here. What are the various real life applications of topology?

The Question : 111 people think this question is useful There are several unknown numbers in mathematics, such as optimal constants in some inequalities. Often it is enough to some estimates for these numbers from above and below, but finding the exact values is also interesting. There are situations where such unknown numbers are necessarily

The Question : 125 people think this question is useful As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition. My first and favorite experience of this is Gabriel’s Horn that you see in intro Calc course, where the figure has finite volume

The Question : 125 people think this question is useful This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the advisability of participating, the career trajectories of former participants, or other such things. This is

The Question : 121 people think this question is useful What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in roughly $5 \pm\epsilon$ minutes. Let’s define ‘general audience’ as approximately an average

The Question : 121 people think this question is useful What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved thanks to some out-of-the-box flash of genius, and the proof is actually short

The Question : 118 people think this question is useful I am looking for examples of problems that are easier in the infinite case than in the finite case. I really can’t think of any good ones for now, but I’ll be sure to add some when I do. The Question Comments : It’s been

The Question : 129 people think this question is useful Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually wrong. (e.g. missing square puzzle) Do you know the other examples? The Question Comments : all of the answers