# Elementary+Short+Useful

Imagine your-self in front of a class with very good undergraduates who plan to do mathematics (professionally) in the future. You have 30 minutes after that you do not see these students again. You need to present a theorem which will be 100% useful for them.

What would you do?

For example: 30 min is more than enough to introduce metric spaces, prove existence of partition of unity, and explain how it can be used later.

P.S. Many of you criticized the vague formulation of the question. I agree. I was trying to make it short --- I do not read the questions if they are longer than half a page. Still I think it is a good approximation to what I really wanted to ask. Here is an other formulation of the same question, but it might be even more vague.

Before I liked jewellery-type theorems; those I can put in my pocket and look at it when I want to. Now I like tool-type theorems those which can be used to dig a hole or build a wall. It turnes out that there are jewellery-type and tool-type theorems at the same time. I know few and I want to know more.

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How many years of undergraduate education do those students have? What can we assume that they know? (It is a big difference between one who's in the second half of her third year and one that just started two months ago.) –  Willie Wong Apr 3 '11 at 18:28
I find it hard to square the "no prerequisites" condition with the "partitions of unity" example. Or are we talking about ideal undergraduate students, who like ideal gases are only an approximation to the reality? –  Captain Oates Apr 3 '11 at 20:34
In my opinion, the "try to be realistic" injunction (which I approve of in all pedagogical questions; note that a lot of experienced teachers do see some of the more ridiculously ambitious pedagogical suggestions promulgated in some answers here and have a good laugh at the naivete of the authors) is hard to square with the vagueness of the question. The term "very good undergraduate" alone is a currency whose value will rise and fall according to where you go. It is tempting to close the question as "too localized" for this reason, but I'll think about it a bit more... –  Pete L. Clark Apr 3 '11 at 23:03
I too find the partitions of unity example unrealistic. I do think this and some of the examples below could be made to work if one wasn't obliged to give a proof, but perhaps only an intuitive idea, and then explain why it was useful -- sort of like a colloquium talk for undergraduates. –  Todd Trimble Apr 3 '11 at 23:10
Indeed, Anton, you can do all sort of things in 30 minutes... but unless the students already somewhat familiar about the subject you are talking about, it is rather unusual that you can introduce three new objects, two concepts, and a theorem to anyone and as a result get them to understand the significance of anything. –  Mariano Suárez-Alvarez Apr 4 '11 at 17:10

Moore closures, their relation to collections of Moore-closed sets and a characterization for closure under finitary operations.

One can then discuss why Moore-closed sets form a complete lattice and a lot more, if one feels so inclined.

This is certainly something students will encounter over, and over, and over again in different guises. Moore-closures are certainly among the most useful trivialities I know.

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I suppose I just find it no more implausible than taking 30 minutes to introduce metric spaces and partition of unity, and to convince students who've never encountered even those definitions of the significance of what you're talking about. I second the sentiment of Willie's and Yemon's comments (to the original question): from the dismissive response you're giving to many answers just for involving a concept like, say, ultraproduct, I confess that it is not at all clear to me what you're after for these 30 minute talks. I'll try one more answer :-) –  Ed Dean Apr 3 '11 at 21:27
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Something that I found very interesting and very useful is Singular value decomposition. It shows that every operator is "almost diagnosable", and is skipped in a lot of basic linear algebra courses I have seen.

I has many application, for example - solving sum of least squares of example. You can give a 30 minute talk on this in various levels as well.

There are prettier theorems (Stokes, Uniformization, and many more) but I think with the 3 constraints (interesting, useful, little background) this is a good topic.

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Singular value decomposition has already been suggested. mathoverflow.net/questions/60457/elementaryshortuseful/… So has Stokes mathoverflow.net/questions/60457/elementaryshortuseful/… –  Willie Wong Apr 8 '11 at 18:50

Every matrix can be represented by the linear combinations of four orthogonal matrix

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Example of use, maybe? –  Michal Kotowski Apr 10 '11 at 9:49

It's really an example of a jewellery-type and tool-type theorem at the same time. It can be introduced and proved for students that even don't know about functions of complex variables in 20 minutes. And other 10 minutes can be spend to say how many applications and generalizations these results have in theory of functions and applied mathematics.

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Having come across this question by searching within the mathematics-education tag, I will try to answer it from the perspective of someone in the field of Mathematics Education.

Theorem: $n^2 - n$ is even for all natural numbers $n$.

It is quite possible that very good undergraduates (I am imagining freshmen) will laugh at seeing such a "theorem" written on the board; it is almost certain that professional mathematicians will scoff. Nevertheless, this is a talk that I have given in the past to graduate students in Math Education who wish to teach secondary school mathematics in the future. Under some reasonable interpretation of the parameters given in this question, I should think these two groups alike enough to outline the talk here.

After writing the theorem on the board, I then write down a collection of headers, each of which is intended as suggesting a method of proof. Once the headers are written out, I give the students three minutes to prove the theorem using one method that they are sure they can carry out, and to attempt a proof using another method they are less sure of. Below I will write the headers, followed parenthetically by the sort of remark I might say aloud as I write them down, and then a brief indication of the proof.

Cases: (Probably you don't need more than two) The cases I am thinking of are even and odd; check what happens when $n = 2k$ and then check what happens when $n = 2k+1$.

High School Algebra: (Factoring) Write $n^2 - n = (n-1)n$ as the product of consecutive integers, hence once of them must be even; so the product is even.

Number Theory: (This might not mean so much to you all as freshmen; we'll return to it later!)

Arithmetic: (I'm thinking of adding up a certain arithmetic sequence) Consider the sum of the first $n-1$ natural numbers; this gives some natural number $k = (n-1)n/2$. Multiplying both sides by $2$, we find that $n^2 - n = (n-1)n = 2k$ is even.

Geometry: (How would you represent $n^2$ with a geometrical picture?) Consider an $n \times n$ array of squares; remove the $n$ squares along the diagonal. The number of squares remaining is $n^2 - n$ and one sees symmetrically that they have been split into two groups of equal size. Hence the total is even.

Combinatorics: (I'm thinking of forming two person committees...) The number of two person committees in a group of $n$ people is some integer $k = (n-1)n/2$. Cf. Arithmetic.

Mathematical Induction: (For students familiar with induction, you might give this a shot) The base case is clear; suppose $k^2 - k$ is even and note $(k+1)^2 - (k+1) = k^2 + k = (k^2 - k) + 2k$ is the sum of two even numbers, and hence even.

The point of the above is to demonstrate that even a seemingly simple statement can be proved in a number of different ways. Such a demonstration, more than any particular theorem, is likely to be useful for all students (as specified by the OP). I usually have students discuss their answers and then use the theorem we've proved to talk about something else that ought to be useful for everyone: generalization.

The proofs above made frequent use of the following fact: $(n-1)n = n^2 - n$.

How would you generalize the following statements?

Statement A: If $n \in \mathbb{N}$, then $2$ divides $(n-1)n$.

Statement B: If $n \in \mathbb{N}$, then $2$ divides $n^2 - n$.

The former statement suggests (in my mind) that $k$ divides $k$ consecutive numbers; the latter statement suggests (in my mind) that $k$ divides $n^k - n$.

Consider when $k = 3$.

Then the statements become:

Statement A: If $n \in \mathbb{N}$, then $3$ divides $(n-1)n(n+1)$.

Statement B: If $n \in \mathbb{N}$, then $3$ divides $n^3 - n$.

Not only are these statements true, they coincide: $(n-1)n(n+1) = n^3 - n$.

This overlap breaks down for $n>3$, though, and we find that only A is true for $n=4$. (Perhaps a good point at which to mention how a single counterexample can disprove a for all statement.)

From here, the talk suggests that A is a good segue into modular arithmetic, while B practically begs us to find the $k$ for which it holds. Of course, we can answer this question using Number Theory (as mentioned early on!) and, more precisely, by appealing to Fermat's Little Theorem.

I believe the talk outlined above, with its messages about the possibility of finding multiple proofs and the interesting directions in which a simple proposition can be generalized, is a practical and doable thirty minute talk for first-year students in mathematics. I have done nothing close to using Groebner bases or making use of ultraproducts, but I have tried to heed the OP's request to try to be realistic.

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Nice lecture :) –  Anton Petrunin 4 hours ago