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Results tagged with big-list
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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
278
votes
Examples of common false beliefs in mathematics
I don't know if this is common or not, but I spent a very long time believing that a group $G$ with a normal subgroup $N$ is always a semidirect product of $N$ and $G/N$. I don't think I was ever sho …
195
votes
Most harmful heuristic?
This isn't really a heuristic, but I hate "functions are formulas". For most students it takes a really long time to think of a function as anything other than an algebraic expression, even though nat …
174
votes
What are the most misleading alternate definitions in taught mathematics?
Here's another algebra peeve of mine. The definition of a normal subgroup in terms of conjugation is pretty strange until it's explained that normal subgroups are the ones you can quotient by. Again …
138
votes
Accepted
What is entropy, really?
Here is a simple story one can tell about the entropy
$$H = -\sum_{i=1}^n p_i \log p_i$$
of a discrete probability distribution. Suppose you wanted to describe how surprised you are upon learning …
121
votes
What are the most misleading alternate definitions in taught mathematics?
In my experience, introductory algebra courses never bother to clarify the difference between the direct sum and the direct product. They're the same for a finite collection of abelian groups, which …
115
votes
What are your favorite instructional counterexamples?
A polynomial $p(x) \in \mathbb{Z}[x]$ is irreducible if it is irreducible $\bmod l$ for some prime $l$. This is an important and useful enough sufficient criterion for irreducibility that one might w …
99
votes
Your favorite surprising connections in mathematics
From an essay of Arnol'd:
Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and t …
95
votes
Famous mathematical quotes
Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discove …
86
votes
Sophisticated treatments of topics in school mathematics
The angle addition formula $\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha) \tan(\beta)}$ for tangent gives one of the simplest nontrivial examples of a formal group law, nam …
73
votes
Sophisticated treatments of topics in school mathematics
It's common in calculus classes and textbooks to state that the antiderivative of $\frac{1}{x}$ is $\log |x| + C$, where $C$ is a constant. This is incorrect: $C$ need only be a locally constant funct …
70
votes
What are the most attractive Turing undecidable problems in mathematics?
As I mentioned in the other thread, Matiyasevich's theorem implies that it is undecidable whether a system of Diophantine equations over $\mathbb{Z}$ has a solution (Hilbert's 10th Problem). I have t …
63
votes
Favorite popular math book
Title: Godel, Escher, Bach: an Eternal Golden Braid
Author: Douglas Hofstadter
Short Description: It's mildly debatable whether this is in fact a book about mathematics, but any mathematician who …
61
votes
What do named "tricks" share?
How about the following (which I think applies to some of these tricks but not others): a trick is something whose usefulness is not fully captured by any particular set of hypotheses, so it would lim …
61
votes
Accepted
What practical applications does set theory have?
The purpose of set theory is not practical application in the same way that, for example, Fourier analysis has practical applications. To most mathematicians (i.e. those who are not themselves set th …
- 118k
59
votes
What are the most overloaded words in mathematics?
Normal
Normal distribution
Normal vector
Normal space
Normal extension
Normal subgroup
Normal operator
Normal convergence