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## Hot answers tagged mathematics-education

464

For vector spaces, $\dim (U + V) = \dim U + \dim V - \dim (U \cap V)$, so $$\dim(U +V + W) = \dim U + \dim V + \dim W - \dim (U \cap V) - \dim (U \cap W) - \dim (V \cap W) + \dim(U \cap V \cap W),$$ right?

246

Everyone knows that for any two square matrices $A$ and $B$ (with coefficients in a commutative ring) that $$\operatorname{tr}(AB) = \operatorname{tr}(BA).$$ I once thought that this implied (via induction) that the trace of a product of any finite number of matrices was independent of the order they are multiplied.

226

Many students believe that 1 plus the product of the first $n$ primes is always a prime number. They have misunderstood the contradiction in Euclid's proof that there are infinitely many primes. (By the way, 2 * 3 * 5 * 7 * 11 * 13 + 1 is not prime.) Much later edit: As pointed out elsewhere in this thread, Euclid's proof is not by contradiction; that is ...

215

The closure of the open ball of radius $r$ in a metric space, is the closed ball of radius $r$ in that metric space. In a somewhat related spirit: the boundary of a subset of (say) Euclidean space has empty interior, and furthermore has Lebesgue measure zero. (This false belief is closely related to Gowers' example of the belief that there are no non-...

192

Here's my list of false beliefs ;-): If $U$ is a subspace of a Banach space $V$, then $U$ is a direct summand of $V$. If $M/L, L/K$ are normal field extensions, then the same is true for $M/K$. Submodules/groups/algebras of finitely generated modules/groups/algebras are finitely generated. The Krull dimension of a subring is at most the Krull dimension of ...

169

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 shown an example in a class where this isn't true.

164

These are actually metamathematical (false) beliefs that many intelligent people have while they are learning mathematics, but usually abandon when their mistake is pointed out, and I am almost certain to draw fire for saying it from those who haven't, together with the reasons for them: The results must be stated in complete and utter generality. Easy ...

163

Topology is the art of reasoning about imprecise measurements, in a sense I'll try to make precise. In a perfect world you could imagine rulers that measure lengths exactly. If you wanted to prove that an object had a length of $l$ you could grab your ruler marked $l$, hold it up next to the object, and demonstrate that they are the same length. In an ...

145

Here are two things that I have mistakenly believed at various points in my "adult mathematical life": For a field $k$, we have an equality of formal Laurent series fields $k((x,y)) = k((x))((y))$. Note that the first one is the fraction field of the formal power series ring $k[[x,y]]$. For instance, for a sequence $\{a_n\}$ of elements of $k$, $\sum_{... 142 The textbook presentation of a topology as a collection of open sets is primarily an artefact of the preference for minimalism in the standard foundations of the basic structures of mathematics. This minimalism is a good thing when it comes to analysing or creating such structures, but gets in the way of motivating the foundational definitions of such ... 132 a student, this afternoon: "this set is open, hence it is not closed: this is why [...]" 125 Some false beliefs in linear algebra: If two operators or matrices$A$,$B$commute, then they are simultaneously diagonalisable. (Of course, this overlooks the obvious necessary condition that each of$A$,$B$must first be individually diagonalisable. Part of the problem is that this is not an issue in the Hermitian case, which is usually the case one ... 118 I once thought that if$A$,$B$,$C$, and$D$were$n$-by-$n$matrices, then the determinant of the block matrix$\pmatrix{A & B \\\ C & D}$would be$\det(A) \det(D) - \det(B) \det(C)$. 116 I have given talks about mathematics to non-mathematicians, for example to a bunch of marketing people. In my experience the following points are worth noting: If the audience does not understand you it is all in vain. You should interact with your audience. Ask them questions, talk to them. A lecture is a boring thing. Pick one thing and explain it well. ... 114 "Any subspace of a separable topological space is separable, too." Sounds natural. 111 Here are a few more: (Everything between quotation marks is a false belief.) Basic logic: Among students: "If A implies B then B implies A" (or "if A implies B then not A implies not B"). Even among mature mathematicians a frequent false belief is to forget that the conclusion of a theorem need not hold once the conditions of the theorem fail. Another ... 99 I think, there are different types of false beliefs. The first kind are statements which are quite natural to believe, but a moment of thought shows the contradiction. Of this type is the sin-example in the opening post or a favorite of mine (also occured to me): The underlying additive group of the field with$p^n$elements is$\mathbb{Z}/p^n\mathbb{Z}$. ... 98 It may seem hard to add a new answer to all this, but here's mine. How to motivate the open set garbage of topological spaces: Answer: Don't. There are many ideas in mathematics that can be easily derived from some real situation, and I would count approximation (ie limits), metric spaces, and neighbourhoods as among these. I think that it is quite ... 98 I used to believe that a continuous algebra homomorphism from$k[[x_1,\dots, x_m]]$to$k[[y_1,\dots,y_n]]$, with$m > n$, could not be injective. Konstantin Ardakov set me straight on this. 98 It's sort of like the inverse function theorem, and that is why it is so strong. If you have$n$functions vanishing at the origin of$k^n$and want to know if they give a local coordinate system, you ask if their differentials are independent at the origin. Or equivalently if their differentials generate the cotangent space at the origin. So in a [not ... 97 The matrix$\left(\begin{smallmatrix}0 & 1\\ 0 & 0\end{smallmatrix}\right)$has the following wonderful properties. (Feel free to add or edit; I can't remember all the reason I loathed it when I was learning linear algebra. It's funny how unexciting they all now seem, but it's a counterexample for almost every wrong linear algebra proof I tried to ... 91 Here are two group theory errors I've seen professionals make in public. 1) Believing that if$G_1 \subset G_2 \subset \cdots$is an ascending union of groups such that$G_i$is free, then$\bigcup_{i=1}^{\infty} G_i$is free. Probably the vague idea they have is that any relation has to live in some$G_i$, so there are no nontrivial relations. 2) ... 91 $$2^{\aleph_0} = \aleph_1$$ This is a pet peeve of mine, I'm always surprised at the number of people who think that$\aleph_1$is defined as$2^{\aleph_0}$or$|\mathbb{R}|$. 87 The Fabius function, everywhere$C^\infty$, nowhere analytic. see... sci.math post references: J. Fabius, "A probabilistic example of a nowhere analytic$C^\infty$-function". Z. Wahrsch. Verw. Geb. 5 (1966) 173--174. K. Stromberg, PROBABILITY FOR ANALYSTS (Chapman & Hall, 1994), pp. 117--120. 86 From the Markov property of the random walk$(X_n)$we have $$P(X_4>0 \ |\ X_3>0, X_2>0) = P(X_4>0\ |\ X_3>0).$$ To paraphrase Kai Lai Chung in his book "Green, Brown, and Probability", "The Markov property means that the past has no after-effect on the future when the present is known; but beware, big mistakes have been made through ... 85 The Banach fixed point theorem. 84 Here's one of my favorite axiomatizations of topology! To make a set$X$into a topological space, you introduce a relation, "touches," between the elements of$X$and the subsets of$X$. This relation must have the following properties: No point touches the empty subset. If$x$is an element of$A$, then$x$touches$A$. If$x$touches$A \...

83

The surface area $|\partial S|$ of a (bounded, smooth) body $S$ is the derivative of the volume $|S_r|$ of the $r$-neighbourhoods $S_r$ of $S$ at $r=0$: $$|\partial S| = \frac{d}{dr} |S_r| |_{r=0}.$$ Thus, for instance, the boundary $\partial D_r$ of the disk $D_r$ of radius $r$ has circumference $\frac{d}{dr} (\pi r^2) = 2\pi r$. More generally, one ...

82

The story I heard from a senior colleague when I was at this stage was: "Twenty years ago I had no children and five theories on how to bring up children. Now I have four grown-up children and no theories."

81

The field of $p$-adic numbers has characteristic $p$.

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