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2

The small object argument is a very important technique based on transfinite induction allowing for the construction of approximations/factorizations in various algebraic and topological contexts.


3

A promiment theorem that comes to mind is Every vector space has a basis. More precisely, for every vector space $V$ over scalar field $F$ there is a (possibly very large!) basiss $B \subset F$ such that every vector $\mathbf{v} \in V$ can be represented uniquely as a finite sum $\mathbf{v} = \sum_{i=1}^k x_k \mathbf{b}_k$ with $x_1,\ldots,x_k \in F ...


1

By understanding transfinite induction by one of your answers earlier today, to this Question that I asked, I am also trying to understand the following result. Let $\mathcal{E}$ be the class of elementary amenable groups, then using transfinite induction, we can give another description for the class of groups $\mathcal{E}$. Let $\mathcal{A}$ be the class ...


2

I think that Zorn's Lemma made it possible for mathematicians who are not familiar with set theory to use transfinite induction. For example, every commutative ring has a maximal ideal, you can prove that with transfinite induction, or use Zorn's Lemma. Learning transfinite induction, and the theory of ordinals, is a bit time consuming. Many mathematicians ...


1

A historically significant example is that of the Cantor-Bendixson rank which is in fact the result that motivated Cantor to invent transfinite induction, coming from the problem of describing the set where a Fourier series can converge/diverge.


4

Here's an example due to Erdos and Hajnal: Theorem: There is a partition of plane into countably many pieces such that the distance between any two points in the same piece is irrational. Corollary: Every non (Lebesgue) null subset $X$ of plane contains a non null subset $Y$ such that the distance between any two points of $Y$ is irrational. Open ...


1

There are several proofs of basic theorems of general topology which are proved by transfinite induction —for example, metrization theorems. You can find the very beautiful arguments explained in Hu's book on general topology.


2

Showing that there are exactly continuum many Borel subsets of the real line uses transfinite induction on some measure of complexity of the set relative to the open sets.


3

The characterization of all real functions such that $f(x+y)=f(x)+f(y)$ for all $x,y\in\Bbb R$ relies on transfinite induction on a well-ordering of the real numbers. That question with its variations pops up here at this site from time to time.


1

Kaplansky's Zero-Divisor Conjecture Let $K$ be a field and $G$ a torsion free group, then is the group ring $KG$ a domain? All the research up until now have been affirmative. This problem has been dealt in the book "The algebraic structure of group rings" by D. Passman. It is one of the toughest and least approachable problem in the whole field ...


0

The Oxford University Maths department seem to have their full set of lecture notes with example sheets on here: https://www0.maths.ox.ac.uk/courses/material/


1

"Proofs without words: Exercises in Visual Thinking" is a book dedicated to visual proofs. The book has proofs about Geometry, Trigonometry, Calculus and also Sequences and Series. In case you run out of proofs for the class there is also a sequel of this book "Proofs without words II: More Exercises in Visual Thinking". Since, these are "exercises" no ...


-1

How about Galileo's paradox? Consider all the square numbers. Since every square number has an integer square root and every integer can be squared to give a square number, then even though not all integers are square numbers and both are infinite, the number of square numbers must equal the number of integers!


1

The proof of Thales' Theorem (that any triangle constructed using the diameter chord of a circle and a third point on the circle that doesn't coincide with the endpoints of the diameter is a right triangle) is a pretty nice one: The triangle $\triangle OAB$ is an isoceles triangle because $OA$ and $OB$ are both radii of the circle and thus by definition ...


2

My answer relates less to strictly mathematical proofs of specific concepts, but emphasizes drawing out the math behind stuff. There are 3 main topics which I would split my answer into: Math toys, Cool math tricks, and Math in everyday life Math Toys Spirographs: The mathematical basis of spirographs are simple and definitely within the grasp of high ...


1

I didn't see this problem until I was a freshman in college, reading Edwin Moise's Calculus I book. It was almost 40 years ago, so I am probably wrong about the details, but, here goes. He described how to prove $|x| + |y| \ge |x + y|$ using cases. Then he asked us to prove that $|u - v| \ge |u| - |v|$ and he hinted that it could be done without resorting to ...


0

These 3 triangles are similar, the two smallest pave the large one and their area is quadratic in the length of their hypotenuses...


1

A somewhat forgotten problem from Paul Erdős is proving his following conjecture: Let $f(r)$ equal the maximum of the sum of the side lengths of $r$ squares inscribed in a unit square such that they have no interior points in common. Erdős conjectured: For any positive integer $k$, $$f(k^2+1)=k.$$ As far as I know, this is not proven.


3

Cutting a bagel into linked halves is impressive to any audience. Just make sure to make a full twist: cutting along a Moebius band would produce a bigger and thinner bagel rather than linked halves.


6

I rather like the bean machine as a physical introduction to probability, normal distribution and the concept of randomness: Now that I think of it, I rather like mechanical devices in general that demonstrate mathematical truths/ideas. Leibnitz's mechanical binary calculator is another that springs to mind.


5

When I was at high school I loved geometrical arguments, especially those which were both simple and profound. Later on, I discovered Combinatorial Geometry, which combines both Euclidean Geometry and Combinatorics. I think that a good idea would be to demonstrate the proof that No matter how the plane is 3-colored, it contains a monochromatic unit distance ...


0

I did a swift check on Rudin's book on Principles of Mathematical Analysis: The book covers in my opinion (and Rudin's as well:) a 1-year undergraduate course in analysis (at least that is what is taught in German universities) and a bit beyond (for example chapter 10). All your topics (measure theory, integration, differentiation) are subject of one or ...


0

In ordinary language, a “biased” result is considered useless, but in Mathematics (Statistics), a result can be biased, but still useful.


0

From the Matrix revolutions: "Everything that has a beginning has an end, Neo." -Agent Smith Ummmm... what about the natural numbers? And the ordinal numbers? And lets not forget the cardinal numbers. Hell, even $\mathbb{R}_{\geq 0}$ is a counterexample. In fact, given any totally ordered set $T$ that lacks a greatest element, we can always adjoin a ...


0

The adage, “If you lose an hour in the morning, you’ll be looking for it the rest of the day.” Admittedly, the inconsistency is due simply to metaphorical usage, but still...


0

In ordinary language, a “set” usually means an indexed set (aka “multiset”).


0

In the ordinary language phraseology used in Chemistry, they speak of the “temperature dependence of the rate constant”. This amounts to a “variable constant”, a contradictory notion. The mathematical term for a “variable constant” is “parameter”.


0

from the poem “Kubla Khan”: “...through caverns measureless to man” There are unmeasurable sets, but no cavern is unmeasurable.


1

The test instructions, “Draw a circle around the correct answer.” really means, “Draw a Jordan curve around the correct answer.” - or, even more accurately, “Draw an approximation of a Jordan curve around the correct answer.”


0

the common expression, “If I’ve told you once, I’ve told you a million times.”


-1

In ordinary language “in general” means “most of the time, but possibly with exceptions”, whereas in Mathematics it means “always, without exception”. However, because (introductory) Mathematics textbooks rely so heavily on ordinary language, the ordinary language usage can show up there, as in, “The projection of A x B onto A is, in general, not ...


0

from “Animal Farm”: “...some are more equal than others”


2

Developing products of several terms is painful in the $\text{cis}$ notation as the number of terms grows exponentially. Compare $$e^{ia}e^{ib}e^{ic}=e^{i(a+b+c)}$$ to $$(\cos(a)+i\sin(a))(\cos(b)+i\sin(b))(\cos(c)+i\sin(c))=\\ ...


0

Complex Analysis uses a lot the logarithm. As you said with $e^{i\theta}$ you can talk more easily for the argument. The important thing is that a branch of the argument exists iff exist a branch of the logarithm. Therefore, exponential goes with logarithm. Also of the expressions $z^a$ are exponential forms...


1

Calculating integrals like $$\int_0^\infty\frac{\cos(x)}{1+x^2}dx$$ is difficult without Euler's identity. The physics of waves and optics is greatly simplified. Fourier transforms are easier to compute and work with, and have a more intuitive meaning, I think. There are a lot of applications in quantum mechanics. Almost all trigonometric identities are ...


1

It becomes trivial to derive the multiple-angle or half-angle formulae u\sing Euler's identity. The geometric derivation is long-winded and painful to visualise, but it's easy if you use the fact that $\frac{e^{i\theta}+e^{-i\theta}}{2}=\cos(\theta)$ and $\frac{e^{i\theta}-e^{-i\theta}}{2}=i \sin(\theta)$, and just apply Euler's identity and the rules for ...


2

It makes De Moivre's Formula much more obvious. Also n-th roots. Are these engineering students? It's very convenient for dealing with alternating current. There's bound to be more, but these spring to mind first. Also this


3

Have you checked out Paul Lockhart's "A Mathematician's Lament" and "Measurement"? He describes his approach to teaching K12 math and - in the latter book - walks you through some elementary (but non-trivial) geometry.


0

The fact that we can give finite result even to divergent sums like for examples the sum of all natural numbers: $$\sum_{n\in \Bbb N}=1+2+3+4+\dots =-\frac 1{12}$$ This fact really astonished me and continues to have a sort of fashion on me :)


25

Here is one example that I find aesthetically pleasing, and which I have found effective in 8th-grade classrooms. Suppose you desire to cut out a triangle from the middle of a piece of paper, not by punching the scissors through and cutting the perimeter, but rather by folding the paper and then cutting straight through the folded paper. The natural ...


3

There's so many interesting subjects, that I can't even organize them systematically. Here are some that come in mind in arbitrary order: The geometric series convergence illustrated by a perspective view of a railroad track or blocks of buildings along a street. Classic geometry constructions by a straightegde and compass. Plane tilings. Ellipses ...


4

In my $10^{th}$ mathematics book there was something given about the seven bridge problem. There was no proof given in the book so I tried it but I failed. I got a new book "Mathematical Circle(Russian experience)" in this book I found a chapter on graph theory. In that book I found the proof of the seven bridge problem, it was a very clean proof. Another ...


2

My two cents : Pigeonhole principle Real line Formal Languages (how to construct a word from a finite set, the alphabet) Simple harmonic motion physics and how the speed and acceleration of an object are derived as first and second derivatives of the displacement equation.


11

One mathematical fun fact that has left a strong impression on me when I was in middle school was the fact that you could state rules about random phenomena. I remember I wasn't familiar with limits, but a teacher had decided to show us funny things about random walks on Excel (yep Excel... :)). He basically made random walks on $\mathbb{R}^2$ with step ...


2

Not a direct answer to your question, since it's not about proofs, but it may help with the question behind your question. The two books that turned me toward a career in mathematics were Steinhaus's Mathematical Snapshots (http://store.doverpublications.com/0486409147.html), which I read as a high school freshman, and Polya's Induction and Analogy in ...


11

There are many interesting proofs of the Pythagorean theorem at Cut the Knot. This is my favourite (as you might guess from my gravitar) Let the length of the hypotenuses be $c$, and the length of the legs be $a$ and $b$, with $b$ being the longer (if either is longer). The area of the outer square is $c^2$, but it is also equal to the sum of the areas ...


14

I really liked the proof that you can raise an irrational number to an irrational number and get a rational number. Consider $\sqrt{2}^{\sqrt{2}}$. If this is rational, we are done (I think some answer on here shows that $\sqrt{2}$ is irrational). If it is not, $\left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}} = 2$, so we are definitely done.


25

I'll expand on my comment, now that I have some time. For high school students, I really like discrete-math type ideas, particularly combinatorics. First, the vast majority of students are never exposed to these ideas (save binomial coefficients, and these, if introduced, are just strange symbols used to expand $(a+b)^n$, in my experience). And second, they ...


9

I like the proof that the harmonic series diverge. The french mathematician Nicole Oresme gave an accessible proof of this. The idea behind his proof, which is grouping terms by power of one halves, was used by Cauchy for his condensation test. There is a nice visualisation on Wikipedia of the process involved in the proof. Oresme proof is explained in ...



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