What are the most misleading alternate definitions in taught mathematics?

Question

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent definitions where one is more natural or intuitive? (I'm meaning so greatly more intuitive so as to not be subjective.)

Alternatively, what common examples are there in standard lecture courses where a particular symbolic definition obscures the concept being conveyed.

Answer 1

Many topics in linear algebra suffer from the issue in the question. For example:

In linear algebra, one often sees the determinant of a matrix defined by some ungodly formula, often even with special diagrams and mnemonics given for how to compute it in the 3x3 case, say.

det(A) = some horrible mess of a formula

Even relatively sophisticated people will insist that det(A) is the sum over permutations, etc. with a sign for the parity, etc. Students trapped in this way of thinking do not understand the determinant.

The right definition is that det(A) is the volume of the image of the unit cube after applying the transformation determined by A. From this alone, everything follows. One sees immediately the importance of det(A)=0, the reason why elementary operations have the corresponding determinant, why diagonal and triangular matrices have their determinants.

Even matrix multiplication, if defined by the usual formula, seems arbitrary and even crazy, without some background understanding of why the definition is that way.

The larger point here is that although the question asked about having a single wrong definition, really the problem is that a limiting perspective can infect one's entire approach to a subject. Theorems, questions, exercises, examples as well as definitions can be coming from an incorrect view of a subject!

Too often, (undergraduate) linear algebra is taught as a subject about static objects---matrices sitting there, having complicated formulas associated with them and complex procedures carried out with the, often for no immediately discernible reason. From this perspective, many matrix rules seem completely arbitrary.

The right way to teach and to understand linear algebra is as a fully dynamic subject. The purpose is to understand transformations of space. It is exciting! We want to stretch space, skew it, reflect it, rotate it around. How can we represent these transformations? If they are linear, then we are led to consider the action on unit basis vectors, so we are led naturally to matrices. Multiplying matrices should mean composing the transformations, and from this one derives the multiplication rules. All the usual topics in elementary linear algebra have deep connection with essentially geometric concepts connected with the corresponding transformations.

Answer 2

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, in my opinion I think normal subgroups should be introduced as kernels of homomorphisms from the get-go.

Answer 3

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 in my opinion gets confusing.

Of course, they're quite different for infinite collections. I think students should be taught sooner rather than later that the first is the coproduct and the second is the product in $\text{Ab}$. This clarifies the constructions for non-abelian groups as well, since the direct product remains a product in $\text{Grp}$ but the coproduct is very different!

Answer 4

I increasingly abhor the introduction of the finite ring $Z_n$ not as $\mathbb{Z}/n\mathbb{Z}$ but as the set $\{0,\ldots,n-1\}$ with "clock arithmetic". (I understand that if you want to introduce modular arithmetic at the high school level or below, this is the way to go. I am talking about undergraduate abstract algebra textbooks that introduce the concept in this way.)

Two problems:

1. Using clocks to motivate addition modulo $n$: excellent pedagogy. Be sure to mention military time, which goes from $0$ to $23$ instead of $1$ to $12$ twice. But...using clocks to motivate multiplication modulo $n$: WTF? Time squared?? Mod $24$??? It's the worst kind of pedagogy: something that sounds like it should make sense but actually doesn't.

   Of course soon enough you stop clowning around and explain that you just want to add/subtract/multiply the numbers and take the remainder mod $n$. This brings me to:

2. Many texts define $Z_n$ as the set $\{0,\ldots,n-1\}$ and endow it with addition and multiplication by taking the remainder mod $n$. Then they say that this gives a ring. Now why is that? For instance, why are addition and multiplication associative operations? If you think about this for a little while, you will find that all explanations must pass through the fact that $\mathbb{Z}$ is a ring under the usual addition and multiplication and the operations on $Z_n$ are induced from those on $\mathbb{Z}$ by passing to the quotient. You don't, of course, have to use these exact words, but I do not see how you can avoid using these concepts. Thus you should be peddling the homomorphism concept from the very beginning.

   As a corollary, I'm saying: the concept of a finite ring $Z_n$ for some generic $n$ is more logically complex than that of the one infinite ring $\mathbb{Z}$ (that rules them all?). A lot of people seem, implicitly, to think that the opposite is true.

Answer 5

A simple example is the two definitions for independence of events:

1. A and B are independent iff $P(A\cap B) = P(A)P(B)$
2. A is independent from B iff $P(A\mid B) = P(A)$

Some presentations start with Definition 1, which is entirely uninformative: nothing in it explains why on earth we bother discussing this. In contrast, Definition 2 says exactly what "independent" means: knowing that B has occured does not change the probability that A occurs as well.

A reasonable introduction to the subject should start with Definition 2; then observe there is an issue when P(B)=0, and resolve it; then observe independence is symmetric; then derive Definition 1.

Answer 6

One that I particularly dislike is the definition of an action of a group G on a set X as being a function $f:G\times X\rightarrow X$ that satisfies certain properties. I cannot understand why anybody gives this definition when "homomorphism from G to the group of permutations of X" is not only easier to understand but is also how one thinks about group actions later.

Answer 7

I normally won't bother with a 5 month old community wiki, but someone else bumped it and I couldn't help but notice that the significant majority of the examples are highly algebraic. I wouldn't want the casual reader to go away with the impression that everything is defined correctly all the time in analysis and geometry, so here we go...

1) "A smooth structure on a manifold is an equivalence class of atlases..." Aside from the fact that one hardly ever works directly with an explicit example of an atlas (apart from important counter-examples like stereographic projections on spheres and homogeneous coordinates on projective space), this point of view seems to obscure two important features of a smooth structure. First, the real point of a smooth structure is to produce a notion of smooth functions, and the definition should reflect that focus. With the atlas definition, one has to prove that a function which is smooth in one atlas is also smooth in any equivalent atlas (not exactly difficult, but still an irritating and largely irrelevant chore). Second, it should be clear from the definition that smoothness is really a local condition (the fact that there are global obstructions to every point being "smooth" point is of course interesting, but also not the point). The solution to both problems is to invoke some version of the locally ringed space formalism from the get-go. Yes, it takes some work on the part of the instructor and the students, but I and a number of my peers are living proof that geometry can be taught that way to second year undergraduates. If you still don't believe there are any benefits, try the following exercise. Sit down and write out a complete proof that the quotient of a manifold by a free and properly discontinuous group action has a canonical smooth structure using (a) the maximal atlas definition and (b) the locally ringed space definition.

2) "A tangent vector on a manifold is a point derivation..." While there are absolutely a lot of advantages to having this point of view around (not the least of which is that it is a better definition in algebraic geometry), I believe that this is misleading as a definition. Indeed, the key property that a good definition should have in my opinion is an emphasis on the close relationship between tangent vectors and smooth curves. Note that such a definition is bound to involve equivalence classes of smooth curves having the same derivative at a given point, and the notion of the derivative of a smooth curve is defined by composing with a smooth function. So for those who really like point derivations, they aren't far behind. There just needs to be some mention of curves, which in many ways are really what give differential geometry its unique flavor.

3) The notion of amenability in geometric group theory particularly lends itself to misleading definitions. I think there are two reasons. The first is that modulo some mild exaggeration basically every property shared by all amenable groups is equivalent to the definition. The second is that amenability comes up in so many different contexts that it is probably impossible to say there is one and only one "right" definition. Every definition is useful for some purposes and not useful for others. For example the definition involving left invariant means is probably most useful to geometric group theorists while the definition involving the topological properties of the regular representation in the dual is probably more relevant to representation theorists. All that being said, I think I can confidently say that there are "wrong" definitions. For example, I spent about a year of my life thinking that the right definition of amenability for a group is that its reduced group C* algebra and its full group C* algebra are the same.

4) Some functional analysis books have really bad definitions of weak topologies, involving specifying certain bases of open sets. This point of view can be useful for proving certain lemmas and working with some examples, but given the plethora of weak topologies in analysis these books should really give an abstract definition of weak topologies relative to any given family of functions and from then on specify the topology by specifying the relevant family of functions.

I'm sure I could go on and on, but these four have proven to be particularly difficult and frustrating for me.

Answer 8

One of my biggest annoyances is professors or books which fail to adequately distinguish between prime and irreducible elements of a ring, Herstein if I remember correctly being a (ha ha) prime example of this. The fact that these are the same in Z, where people first learn about unique factorization, doesn't help matters.

Answer 9

My biggest issue is with the coordinate-definition of tensor products. A physicist defines a rank $k$ tensor over a vector space $V$ of dimension $n$ to be an array of $n^k$ scalars associated to each basis of $V$ which satisfy certain transformation rules; in particular, if we know the array for a given basis, we can automatically determine it for a different basis. Another way to say this is that the space of tensors is the set of pairs consisting of a basis and a $n^k$ array of scalars, identified by an equivalence relation which gives the coordinate transformation law. For some strange reason, people seem to call this a coordinate-free definition. While it is in a sense coordinate-free (the transformation between coordinates lets you break free of coordinates in a sense), it is very confusing at first sight. People who use this definition will they say that certain operations are coordinate-free. What they mean by this, and it took me a long time to figure this out, is that you can do a certain algebraic operation to the coordinates of the tensor, and the formula is the same no matter which basis you work with (e.g., multiplying a covariant rank $1$ tensor with a contravariant rank $1$ tensor to get a scalar, or exterior differentiation of differential forms, or multiplying two vectors to get a rank $2$ tensor).

The much nicer definition uses tensor products. This is a coordinate-free construction, as opposed to the coordinate-full description given above. This definition is nice because it connects to multilinear maps (in particular, it has a nice universal property). It also helped me see why tensors are different from elements of some $n^k$-dimensional vector space over the same field (they are special because we are equipped not just with a vector space but with a multilinear map from $V \times \cdots \times V \to V \otimes \cdots \otimes V$. The covariant/contravariant distinction can be explained in terms of functionals. This allows you to talk about contraction of tensors without worrying having to prove that it is coordinate-invariant! Finally, once you have all that under your best, you can easily derive the coordinate transformation laws from the multilinearity of $\otimes$.

Answer 10

I'd say the standard definition of singular homology is pretty bad.

It's a historical relic in some sense -- topologists were so concerned by naturality, whether manifolds have combinatorially distinct triangulations and issues such as that, that they decided those preoccupations were more important than imparting a solid foundational intuition as to what a homology class is.

In my experience, people who see Poincare's proof of Poincare duality first vs. the people who see a singular homology exposition usually have a far better command of what is actually going on, to the point where they view Poincare duality is something light and natural, while most students that see it through the eyes of singular homology more often see it as something distant and intractible.

And all that effort is for what? So students can know Poincare duality is true on topological manifolds, when all the examples they've seen are smooth manifolds.

edit: my preferred way to describe Poincare's proof is to modernize it a tad. Your set-up is a triangulated manifold $M$, then you construct the dual polyhedral decomposition (a CW-decomposition) so that the (simplicial) $i$-cells of $M$ are in bijective correspondence with the (dual polyhedral) $m-i$-cells of $M$. This is much more straightforward than living in the simplicial world. Then you show that (up to a sign change) the chain complex for the simplicial homology is the chain complex for the cohomology of the dual polyhedral decomposition. The fussiest bit is keeping track of the orientations in the orientable case.

Answer 11

Another simple example is the definition for equivalence relations:

1. R(.,.) is an equivalence relation iff R is reflexive, symmetric, and transitive.
2. R(.,.) is an equivalence relation iff there exists a function f such that R(a,b) iff f(a)=f(b).

Most presentations start with Definition 1, which contains no hint as to why we bother discussing such relations or why we call them "equivalences". In contrast, Definition 2 (along with a couple of examples) immediately tells you that R captures one particular attribute of the elements of the domain; and, since elements with the same value for this attribute are called "equivalent", R is called an "equivalence".

A reasonable introduction should start with Definition 2, then go on to prove Definition 1 is a convenient alternative characterization.

Answer 12

One often sees the cumulants of a probability distribution defined by saying the cumulant-generating function is the logarithm of the moment-generating function: $$ \sum_{n=1}^\infty \kappa_n \frac {t^n}{n!} = \log \sum_{n=0}^\infty \operatorname{E}(X^n) \frac{t^n}{n!} = \log\operatorname{E}\left( e^{tX} \right). $$ This fails to explain one of the basic motivations behind such a concept as the cumulants of a probability distribution.

The variance $\operatorname{var}(X) = \operatorname{E}\left( (X - \operatorname{E}(X))^2 \right)$ is simultaneously

• $2$nd-degree homogenous: $\operatorname{var}(cX)=c^2\operatorname{var}(X)$;
• translation-invariant: $\operatorname{var}(c+X) = \operatorname{var}(X)$;
• cumulative: $\operatorname{var}(X_1+\cdots+X_n) = \operatorname{var}(X_1)+\cdots+\operatorname{var}(X_n)$ if $X_1,\ldots,X_n$ are independent.

The higher-degree central moments also enjoy the first two properties (with the appropriate degree of homogeneity in each case), but the third property fails for $4$th and higher-degree central moments. (That it works for the $3$rd-degree central moment has been known to surprise people. It's trivial to prove it.)

All of the cumulants have the three properties above (with the degree of homogeneity equal to the degree of the cumulant).

For example:

$$\text{4th cumulant} = \Big(\text{4th central moment}\Big) - 3 \cdot \Big( \text{variance}\Big)^2.$$

This is $4$th-degree homogeneous, translation-invariant, and cumulative.

Each cumulant above the $1$st degree is the unique polynomial in the central moments having those three properties and for which the coeffient of the $n$th-degree central moment in the $n$th cumulant is $1$.

Is this not a more intuitive and motivating characterization of the cumulants than is the "definition" that speaks of the logarithm of the moment-generating function?

Answer 13

A function is a collection of ordered pairs such that ...

Answer 14

Similar to gowers's answer about group actions, a module over a ring R is an abelian group M together with a function $f:R\times M \to M$ that satisfies certain properties. It may set the beginner's mind at ease to hear, "They're just like vector spaces except over arbitrary rings instead of only fields," which is misleading in itself but is a good mnemonic for remembering the definition. However, I usually find it more intuitive to think of a module over R as a homomorphism from R to the endomorphism ring of an abelian group, and with this definition no mnemonic is necessary.

Answer 15

I know that this comment will be somewhat controversial, but I strongly believe that the standard (algebraic) textbook definition of d of a differential form is unpedagogical.

I much prefer the route taken in Arnold's GTM book on classical mechanics: Just define d of a form as the thing that makes Stokes theorem true!

Then one derives the algebraic formula for d of a form. Everything is motivated at every step, and the student isn't confronted with a confusing algebraic definition of unknown origin.

Answer 16

Inspired by some of the comments, I would nominate the definition of infinite product topology in terms of its open sets, found in, e.g., Munkres' otherwise excellent Topology. "The product topology on $X = \prod_{\alpha \in J} X_\alpha$ is the topology generated by the sets of the form $\pi_\alpha^{-1}(U_\alpha)$, where $U_\alpha$ is an open subset of $X_\alpha$." One then proves that one can also use the basis of sets of the form $U = \prod_{\alpha \in J} U_\alpha$ where $U_\alpha$ is open in $X_\alpha$, and $U_\alpha = X_\alpha$ for all but finitely many $\alpha \in J$. This just makes it look like an annoying and unnatural modification of the box topology.

Better in my opinion is to view $X = \prod X_\alpha$ explicitly as a function space (not as some sort of tuples, though they are really functions underneath), and to use the terminology of nets. Then it becomes clear that the product topology is just the topology of pointwise convergence, i.e. a net $f_i \to f$ iff the nets $f_i(\alpha) \to f(\alpha)$ for all $\alpha \in J$.

Under this definition, Tychonoff's theorem, which previously seemed pretty obscure, has an obvious application when combined with Heine-Borel: given any set $J$ and a pointwise bounded net of functions $f_i : J \to \mathbb{R}$, there is a subnet that converges pointwise. This is maybe the most useful application, especially in functional analysis. (Indeed, I understand this was actually Tychonoff's original theorem, that an arbitrary product of closed intervals is compact.) For instance, it makes Alaoglu's theorem clear, once you see that the weak-* topology is just a toplogy of pointwise convergence.

It's nice then to compare this with the Arzela-Ascoli theorem, which says that if $J$ is a compact Hausdorff space and the functions $f_i$ are not only pointwise bounded but also continuous and equicontinuous, then a subnet (in fact a subsequence) converges not only pointwise but in fact uniformly.

Answer 17

What about definitions that are elegantly concise to such an extent that they confound intuition? A classic of the genre:

1. A forest is an acyclic graph;
2. A tree is a connected forest.

(Presumably most of us would be less surprised to hear a forest defined as a disjoint union of trees.) But perhaps there is something to be said for a shocking definition: I shall never forget this, and probably I will always remember the moment I first saw it.

In a similar vein, I once saw a video of John H Conway giving a lecture on ordinals. He began, conventionally enough, by defining the notion of well-ordered set. But the definition he gave was an unconventional one:

A set $S$ equipped with a transitive relation $\mathord{(\leq)}\subseteq S\times S$ such that every non-empty $T\subseteq S$ has a unique least element $m\in T$ such that $m\leq t$ for all $t\in T$.

Notice that this implies reflexivity (take $T$ to be a singleton); totality (by existence of the least element of a two-element set); and antisymmetry (by uniqueness of the least element of a two-element set). So it’s equivalent to the usual definition. And it is certainly memorable! But I doubt I would have understood it if I wasn’t already familiar with the usual definition.

Answer 18

Since this is a big list, I might as well comment 5 years later.

David Corwin mentioned tensor products, and the top post is about linear algebra, so I thought I would mention that, in my opinion, coordinate definitions in general tend to obscure meaning. Before going on, I'll mention that I'm not saying coordinates are bad! I just think that introducing ideas with coordinates tends to be very unrevealing.

A few examples which I find are obscured by coordinates are:

1. Derivatives. The easiest example is the differential of a map $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$. This is often given by the Jacobian matrix, and while the Jacobian is very useful in computation, it was not at all clear to me how it generalises the ordinary derivative, until I saw the proof that it satisfies the coordinate-free definition of a derivative at a point. Namely, the map $D_af:\mathbb{R}^m \rightarrow \mathbb{R}^n$ such that

$$\lim_{\lvert x \rvert \rightarrow 0}{{\lvert f(a+x)-f(a)-D_af(x) \rvert}\over{\lvert x \rvert}} = 0.$$

2. Tensor products and Tensors. David Corwin already covered this.

3. Local coordinates on manifolds. I think a lot of elegant definitions and properties are lost when using local coordinates, for example, the tangent space becomes very unwieldy and unnatural when interpreted in a local coordinate setting (although it does become more intuitive).

4. Matrices and linear maps. I recommend reading the top post. But I'll mention that I am personally most bothered by determinants: they made no sense at all to me until I learned of the definition of the determinant as the top exterior power of the associated linear map!

Answer 19

I see the problem crop up: a certain mathematical object has many characterizations, any one of which can be taken as the definition. Which do you use when you are introducing the subject?

The first one that comes to mind is the basis of a vector space. Perhaps this is not the best example for the title question of this thread of discussion, but I know that this confuses some students. When I last taught linear algebra, we taught them at least four characterizations. It's not really that any of the characterizations is obscuring or misleading. Rather, each one highlights some important property(-ies). Of course, the better students enjoy seeing all of the characterizations, and they appreciate every one. The less facile students get flustered because they want there to be just One Right Way of thinking about them.

A similar issue arises with the characterizations of an invertible matrix or linear transformation, though at least with a matrix it seems most reasonable to define an invertible matrix as one that has an inverse, namely another matrix that you can multiply it by to get the identity matrix.

The issue comes up in spades when introducing matroids.

Answer 20

Ok I'm joining very late but let me tell this.

I think the following definition of relation, that is often given, is misleading:

"Wrong" definition: A relation $R$ between the set $A$ and the set $B$ is an arbitrary subset of the cartesian product, $R\subseteq A\times B$.

In fact, I think it is a "wrong" definition:

• it obfuscates the fact that the datum of source set and target set is important and itself part of the definition. For example, if you defined a function as a relation (in the sense of the definition above) satisfying the functional property ($\forall x\in A\exists ! y\in B: (x,y)\in R$) then the notion of codomain will not be well defined (or not explicitly defined), and one could be led to think that $x\mapsto x^2$ as a function $\mathbb R \to [0,+\infty)$ is literally equal to $x\mapsto x^2$ as a function $\mathbb R\to \mathbb R$.
• It allows you to define, given $A$ and $B$ (in that order), the set $\mathsf{Rel}(A,B)$ of relations between $A$ and $B$, but not (immediately) the class of all relations (or the set of all relations within a given universe $U$).
• It makes less clear that there should be a category $\mathsf{Rel}$, of sets with relations as morphisms, of which the $\mathsf{Rel}(A,B)$ are the hom-sets.

The right definition should of course be:

"Right" definition: A relation is a triple $(A,B,R)$ where $A$, $B$ are sets and $R\subseteq A\times B$.

Now the notion of codomain is well defined (or explicitly defined). And also the category $\mathsf{Rel}$ is well defined.

Answer 21

Convolution: whether it is convolution of functions, measures or sequences, it is often defined by giving an explicit formula for the resulting function (or measure, etc.). While this definition makes calculations with convolutions relatively easy, it gives little intuition into what convolution really is and often seems largely unmotivated. In my opinion, the right way to define convolution (say, of two finite complex Radon measures on an LCA group $G$, which is a relatively general case) is as the unique bilinear, weak-* continuous extension of the group product to $M(G)$ (the space of measures as above), where $G$ is naturally identified with point masses. Then one can restrict the definition to $L^1 (G)$ and get the well known explicit formula for convolution of functions. Of course, a probabilist will probably prefer to think of convolution as the probability density function associated to the sum of two independent absolutely continuous random variables. And there are other possible alternative definitions (see this Mathoverflow discussion). But the formula definition is really the hardest one to get intuition for, in my opinion.

Answer 22

Induced representations defined in terms of tensor products of $G$ modules and in terms of vector-valued functions on $G$. It would be nice, if more textbook in representation theory stress this more heavily. Both definition have their advantages and disadvantages, I guess, but I personally feel more comfortable with the interpretation in terms of functions.

Answer 23

I remember being confused by the multivariable calculus approach to vectorfields. Sometimes thinking of functions just as functions and sometimes as feilds. It should be possible to convey the idea of having a space of directional derivatives attatched to each point without having to talk about vector bundles.

In general I can accept that there is more going on behind the scenes than there is time for in a course, but simply knowing that there is a more general and "right" way of doing things is very helpful to me. This also tends to make the course much more interesting.

Answer 24

The entire branch of point-set topology as taught in most textbooks has completely unintuitive definitions that obscure the entire subject. For instance, the definition of a topological space in terms of open sets tells you nothing about the meaning of point-set topology. It would be much more clear if topological spaces where originally defined in terms of topological closure operators since closure operators since intuitively we have $x\in\overline{A}$ if the set $A$ touches the point $x$ in some way. Other unnecessarily obscured concepts in point-set topology include the definitions of product topology, subspace topology, Hausdorff spaces, regular spaces, compact spaces, and continuous functions. Furthermore, some definitions are obscured when the spaces are not required to be Hausdorff. For instance, the notions of compactness, paracompactness, regularity, and normality do not have much meaning without the Hausdorff separation axiom. If one has a non-Hausdorff space where every open cover has a finite subcover, then one should call that space quasi-compact and not compact. It is a shame that general topology is taught in such a meaningless fashion.

Answer 25

A function $f: \mathbb R\to \mathbb R$ is continuous if $f^{-1}(G)$ is open for all $G$ open in $\mathbb R$ is less intuitive than the delta epsilion definition of a continuous function.

Answer 26

When we write tensor products, it's optional to indicate the ring over which we do it; we can write $M\otimes N$ or $M \otimes_R N.$ But for elements, we always write $x\otimes y$ without reference to $R.$ You must keep it in mind and that can induce lapses. For example, $v\otimes u^2 - u\otimes uv$ may be $\ne0{:}$ it depends on the base ring, and that doesn't appear [in the notation].

Sometimes the problem is not the concept so much as the notation we use for it.

Answer 27

A discrete probability distribution is often defined as one for which the number of values that a random variable with that distribution can take is finite or countably infinite.

I prefer to define it as one for which one has $\displaystyle \sum_x \Pr(X=x) = 1,$ where the sum is over all values $x$ for which $\Pr(X=x)>0.$

(One should not define it as one for which the support is finite or countably infinite. For example, suppose a probability distribution assigns positive probability to the singleton of every rational number between $0$ and $1,$ and the sum of those probabilities is $1.$ Then every real number between $0$ and $1$ (inclusive) is in the support, since every interval about every such number gets positive probability.)

Answer 28

"Prime number" is sometimes defined as a number with exactly two positive divisors, which are itself and $1.$ The deficiency of this characterization is only that it doesn't motivate the definition in the following way. $$ \begin{array}{cccccccccc} & & & & 60 \\ & & & \swarrow & & \searrow \\ & & 4 & & & & 15 \\ & \swarrow & \downarrow & & & \swarrow & & \searrow \\ 2 & & 2 & & 3 & & & & 5 \end{array} $$ One could continue factoring by pulling out $1$s, but that is uninformative in that it doesn't distinguish the number being factored from any other. The definition is motivated by the fact that the number $1$ cannot play the sort of role in this process that either composite or prime numbers play.

(For Euclid this was not problematic since he didn't consider $1$ to be a number.)

Answer 29

I find simplicial homology very difficult. In particular, I find the idea of a simplicial complex very hard to comprehend, except in the case of an abstract simplicial complex. Although it's not equivalent, I much prefer the idea of what Hatcher calls a $\Delta$-complex, although I still have some trouble with that definition.

Answer 30

The first sentence in a probability talk is likely to be, Let $X$ be a random variable. As a non-probabilist who dabbles occasionally in probability, I find the notion of rv difficult to absorb; and then they have the expectation operator (integration), characteristic function (with a different meaning---indicator function means what characteristic function means in measure theory), and in general, it is a distinct language.

Answer 31

$\pi=3.14$ cm. Tongue-in-cheek of course, but this can supposedly be found in books.