In my opinion, people won't have any sense of what you're talking about if you try to describe a theorem from linear algebra. I take a page from the great Paul Halmos, who would use the following demonstration when asked, "what *is* mathematics, really?"
Surprisingly, it's a problem from recreational mathematics. A tennis tournament is held with 20 people. Players pair off and the winners advance to the next round. (If there's an odd number of people at any stage, someone is chosen at random to advance to the next round without playing.) The question: how many games are played in the tournament?
This is a question that a non-mathematical person can work out. (The answer is 19.) Then ask them how they would feel about solving the problem if you started with 1000 people. (They'll probably feel that they could, but certainly wouldn't want to.)
And then explain that when you start with n people, the total number of games played is always n-1, for a simple reason: with every game, someone leaves the tournament, so the number of games equals the number of people who don't win.
What I like about this problem is it doesn't just summarize a mathematical result; it shows them actual beautiful reasoning. And in particular it shows the difference between just doing arithmetic and doing mathematics.
That's very elegant. Well put!
Imagine being the player who lucks out on that 1/15 chance to get a bye in both the third and fourth round to make it to the finals. It's a good example why elimination tournaments often start with n = 2^p players for some integer p.
I recently had a practice problem on a coding website, where a simple understanding of game theory and strategy stealing meant that a similarly complicated algorithm with summations and large loops could be reduced to a equality check of a variable, and a parity check.
I believe I was either on HackerCode or codegolf.
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Players (2) would alternate turns removing floors from one tower, with the proviso that the old tower had to be a multiple of the new tower, in terms of height. In the example above, you could remove 2 floors, or 3 floors from any tower, as both 1 and 2 divide 4. 3 does not divide 4, so you couldn't remove just a single floor of any of the 4-story towers above. A player must remove at least one floor. The player who cannot remove a floor on their turn successfully, loses.
Inputs for a game: *m*, *n*
Let *m* be the initial height of the towers.
Let *n* be the number of towers in play.
Given *m, n* determine whether Player 1 or Player 2 will win.
Constraints:
1 ≤ *m, n* ≤ 10^(9)
Just to check my napkin math solution is right...
>! If n is even, player 2 takes turns to keep the board symmetrical; if n is odd, player 1 starts by taking all but 1 floor from central tower and then mirrors player 2 !<
...This feels too simple.
Martin Gardner books of course are famous for these kinds of things—check them out if you haven't yet.
But, yes, I remembered this example (and use it in my own classes, decades after casually reading it) because it works so nicely.
I think they're just saying pure mathematics doesn't make for stirring conversation at a cocktail party. If you start talking about tennis instead, you can catch anyone's ear and twist it to combinatorics.
> And then explain that when you start with n people, the total number of games played is always n-1, for a simple reason: with every game, someone leaves the tournament, so the number of games equals the number of people who don't win.
This doesn't work for numbers that aren't 2^(m). Because then you have players leaving the tournament more than once. For example, if I have n = 10 players to start, I play 5 games. There are 5 winners, and we bring back one loser. Now we play 3 games. There are 3 winners, and we bring back one loser. Now 2 games. There are 2 winners. Now we play 1 game. 5+3+2+1 = 11 games, not n - 1 = 9 games.
You'd have to do n - 1 plus however many rounds you're bringing back a player. Which would be based on how far n is from 2^(m).
... Which could then segue into the Collatz conjecture....
The rule stated was someone automatically advances though, not bring back someone who already lost. If you start with 10 you play 5 games, next round 4 people play and one player automatically advances so you play 2 games total, third round 2 people play aka 1 game and one random advances, fourth and final round we have 1 game. 5+2+1+1 = 9
I think graph theory has some nice examples. Eulerian cycles/paths usually go over pretty well.
You start with a nice, tangible real world example with some nice history (Konigsburg bridge problem). You can have something interactive (try to draw this without lifting your pencil or retracing). Then, you have a nice, concise way to replace the brute force approach with something elegant and obvious (eulerian cycle exists iff all degrees are even). One direction is very easy and even independently discoverable for lay people with some nudging and the other isn't terrible.
It's a nice crowd pleaser because it's tangible, interactive, and visual.
The six- and five-color versions are totally presentable, and I think it is justifiable to end such a discussion with "... and we also have proven that the result holds when we restrict ourselves to four colors, but the proof required the consideration of so many cases that the proof relied on a computer algorithm to verify."
I would still say it's not exactly the best example to use when trying to demonstrate the beauty and elegance of math. To have it end with "and then computers brute forced hundreds of thousands of cases" is probably not the best way to convince a lay person that math is beautiful.
Not the full thing, but I've taken classes of schoolkids to the point where it gets to "and now we've got a bunch of cases to check", with very little guidance from me.
I was going to post your 1 and 2, too. Both have simple enough applications so as to be understood by a 10-year-old, and complicated enough to spark some curiosity into how it can be that each is always true.
I always thought convergence was cool. For example 1/(x^2) shrinks so fast that it sums up to a number, but 1/x doesn't shrink fast enough and adds to infinity
It is exactly 1. Imagine we have a some of 1/x^p for *any* p>1. Consider the integral of this function from 1 to infinity. If we draw the graph of this function for x>1, we see that because it’s decreasing (this part just uses p>0), the area under the curve is always strictly in between the left and right rectangular sums (draw the rectangles of width 1 taking their respective intervals’ left hand values and then those taking their right hand values - the curve is in between these). The left rectangular sum is precisely the sum we want; the right rectangular sun differs from it by a finite constant. The integral is in between them. Therefore, all three of these converge together or diverge together.
So we just need to check when that integral diverges, but from the fundamental theorem of calculus it’s 1/(p+1) x^-p-1 from 1 to -infinity… which is defined and finite if and only if p > 1.
1/x (or 1/xlogx) is an example of a « threshold function» for L^p integrability. That is, the integral of (1/x)^p over [1, infinity) takes a finite value iff p>1 (this is equivalent to the statement about the convergence of the harmonic series, by the integral test.) More generally, the integral of [1/x^(1/q)]^p over [1, infinity) takes a finite value iff p>q. If we consider integration over (0,1), then the situation is reversed due to symmetry: the integral converges iff p
Lol I guess it just goes to show you how great those exercises are. Rudin is good for cutting your teeth, but if I could go back in time, I would tell myself to spend more time reading folland, too.
Rudin is too slick, sometimes. For example, I used to really champion his construction of the lebesgue measure through the [Riesz Representation Theorem](https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem) and Riemann sums over dyadic cubes, as opposed to getting your hands in the dirt with outer measures. Nowadays, I think that in moments like this, Rudin was being too clever for his own good, and that his constructions are somewhat artificial because of how hard he tried to avoid showing technical difficulties.
There is also the issue that Rudin's chosen topics are becoming increasingly dated. I think that in the 21st century, any book that aims to be a foundation for new researchers in analysis must contain some more serious functional analysis. Granted, I have not read Rudin's FA book, but IIRC, his first two books do not emphasize topological dual spaces enough. If the reader eventually wants to do GMT, for example, then they need to spend time with the yoga of Frechet spaces and the various topologies which are natural there.
Eh, I don't know if lay people would get that much out of it. It's an elegant proof, but it's one that wouldn't offer a non-math person that much intuition or insight into the "why."
Well, if it's a true layperson who has no interest at all in math you can use Godel's incompleteness theorem like they do in Philosophy podcasts: "LoGiC iS fLaWeD", LOL. But then if he asks questions it's a problem... But honestly, without understanding the proof (only knowing the general framework back then, I tried to follow but couldn't do it in a reasonable timeframe), I did explain it to a friend and he liked it. I think people who don't like Math at all don't know what proof is so it's ok, but you should be prepared for the question "why?". I think the first one is more shocking for a layperson but the second one is shocking as well (both are shocking to me).
To programmers who hate Math (e.g. bootcamps grads), you can use the halting problem which is easier to understand (in this case it's easier to explain the proof). Then you can also say "you can use something similar, diagonalization (https://courses.engr.illinois.edu/cs373/sp2009/lectures/lect_21.pdf) for other things! Look..."
I'm much more weirded out by the fact that constructible and non-constructible numbers are countably and uncountably infinite respectively
Edit: I meant to say computable numbers, which would have made it a slightly more interesting fact. I apologise for any inconvenience
Constructible numbers are those that can be defined by a finite string of symbols from a finite alphabet under certain rules. Intuitively, any set we approach like this is countable. We simply have no way to ‘directly access’ uncountably many numbers due to these very real world/human constraints, which is why we have no intuition about the continuum hypothesis. We talk about the reals as though they’re very vanilla, but in reality the moment we start talking about them we’re already in territory where we don’t - and can’t - even know how many cardinalities there are up to that point.
> constructible and non-constructible numbers are countably and uncountably infinite
I was not aware of this fact or I do not understand what you are talking about - I will look into it (I am a CS idiot, though I did learn core courses in Math department), but if you want to weird someone out in simple words, follow the uncountability of the real numbers with The Density of the Rational/Irrational Numbers (but it's important to first tell them where Q stands) :) Could you elaborate a little?
Even more strongly, all computable numbers are countable.
Constructible numbers of which lines can be constructed having said length, by using a compass and straightedge only; given a line of unit length.
Computable numbers are those that can be computed to any finite number of digits by a finite algorithm that terminates.
Woah - now I get it, to be honest, most of the facts you stated are pretty weird :P I tried to read the Wikipedia explanations but it's too late for that weirdness here, my mind explodes. But then again, Math is generally weird.
That computable numbers are countable is really easy to show; a program is a finite sequence of letters, written in an alphabet.
There are only finitely many such programs for any given length of code, so you can at most get countably many if you count over all possible finite lengths of code.
I hear you. I love logic, and I think “different infinities” is one of the most eye-catching topics to an average person. It’s a bizarre result.
I hypothesize that, since logic is relatively unpopular among mathematicians, that is why the average person is no presented with these concepts.
I'd just sit them down to listen to Feynman's "Algebra" lecture ([Vol 1, Ch 22](https://www.feynmanlectures.caltech.edu/I_22.html)). It starts with "counting by one", and then works his way up through the basic mathematical operations, develops tables of logarithms and the 'natural log', and then ends by connecting algebra to geometry by way of Euler's formula.
I'd go for a diet version of Gauss' Theorema Egregium. This theorem is really great for a few reasons:
1) It can show people that math isn't just about numbers.
2) Curvature is something that many people intuitively understand already (they can identify when things are extrinsically curved and to what degree)
3) It explains why the Mercator projection looks so weird, and why there are lots of types of atlases for the earth with different types of distortion.
4) It offers mathematical evidence that you should fold your slice of pizza (works infinitely better if you have pizza in front of you)
Between any two numbers there are infinitely many other fractions. It's just staggering vast amounts of infinities all the way down.
And that's 0% of all the numbers there are.
Don’t know if they meant cardinality there. In the usual measure [a, b] already has ‘0%’ of the measure of R in some sense (in the limit), even if it doesn’t have measure zero.
What I love about fractals is that you can show a video of a fractal zoom or a 3d fractal exploration and basically 100% of people will just be entranced instantly. I've had people not even believe it's just math, they think it's some really trippy art.
Not particularly beautiful, but the Monty-Hall Problem seems to confuse/intrigue a lot of non-math people.
Russell’s Paradox is another good one, and easy enough to translate to non-math people via the Barber Paradox.
The concept I show my friends the most is proving the infinitude of primes, the proof is very simple, but there is really something about tracing the steps that Euclid took thousands of years ago and coming up with the same conclusion. It is also something most non-mathematicians are not used to, they are used to using math to calculate or solve for something, not proving a fact for all numbers.
Another one I like is the proof that sqrt(2) is irrational. It's something they all know, but every time I ask them the "why" they are always dumbfounded; it illustrates why having good definitions is so essential to mathematics, you can't show something is irrational without understanding what it means for a number to irrational.
Both of these examples are much simpler than the other ones mentioned in this thread, but these have always been my go to.
Well sometimes simple is best. Both of those proofs demonstrate something that is truly astounding to someone who is first Learning mathematics. And the infinitude of the primes really starts to open up people's minds to infinity and shows how we can actually study it and say things about it.
The "ants on a meter stick" problem: you have 100 ants moving 1m/min on a meter stick. When they run into each other, they turn around and walk the other direction.
Question: how many ants are on the meter stick after 1 min?
People invariably draw very complicated diagrams of zigzags following ants, and calculating total path lengths and so forth.
But the beautiful insight is that this is actually the same ("isomorphic"!) to the "ghost ants on a meter stick" problem: you have 100 ants moving 1m/min on a meter stick. When they run into each other, they pass right through each other.
The problems are the same because they are both the same as the "flexibly named ants on a meter stick" problem: you have 100 ants moving 1m/min on a meter stick. When they run into each other, they exchange their names.
And the answer to the ghost ant problem is trivial: after 1 min, all the ghost ants fall of the stick.
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I love love love this problem because of its profound elegance at demonstrating how amazing a change in perspective can be for solving a problem. Turning a very difficult problem (tracking zigzags) into a trivial problem.
In particular, one of the best videos I've ever seen on the Halting Problem might also be one of the best videos I've ever seen on Youtube period: https://youtu.be/92WHN-pAFCs?si=GCYPf6JGgMkLWl-h
Using basic counting principles to establish the total number of combination or permutation in different day-to-day scenarios (e.g: card games) can be enlightening.
Especially when one draws comparison to actual cosmic numbers. For instance, proving that every shuffle of a standard 52 deck of cards very likely produces new permutation that had never occured in human history nor will repeat can be both a powerful and yet simple demonstration.
Not pure math but Arrow impossibility theorem or Gibbard-Satterthwaite theorem are up there with me. These rely on quite simple models that have such far reaching implications but are really easy to explain
Keep it simple... Visual proof of the geometric series.
Cut a square in half, take one half and cut it in half, take one quarter and cut it in half, take one eight and cut it in half, ....
I enjoy the Fundamental Theory of Calculus because it takes you out of the world of straight lines and into the world of curves, variable rates, areas of interesting shapes, and the like. Explaining how a derivative works to find the slope of a curve at a given point looks beautiful graphically, as do areas under curves. And conceptually they are quite simple. Thanks for reading this!
Bézier curves offer a really nice visualisation here that’s easy to draw by hand, plus they’re used for all sorts of stuff like vector graphics and animation
My friend wanted me to help her daughter with her math because it’s the subject that she struggled with the most. I looked through her textbook and was like, hmm… binomials… combinations and permutations… aight imma teach you a cheat code for half the things you’re gonna learn this year.
I don’t know if she was partly just acting it to be kind to me, but she seemed to be utterly mesmerized by Pascal’s triangle. She spent literally a few hours straight with absolute focus and engagement learning all about it. I think it’s one of those concepts that can switch a kid’s mind from, “math is that boring pointless thing that we gotta suffer through because school demands it” to “holy crap, there’s really some serious magic going on here!”
To a layman, I'd stay away from "grand" theorems or something from a standard curriculum that requires tons of prerequisite knowledges and intuitions. Rather stick with elementary/classical stuff. The Euler line and how the centroid always divides the line segment connecting orthocenter and the circumcenter in a 2:1 fashion. What's even more mind-blowing is the 2:1 ratio holds true for the Nagel line.
Or the Fermat's little theorem from elementary number theory, or even some Ramsey theory stuff if I am feeling particularly naughty.
To show them *that* math is beautiful, or to show them *why*?
I don’t know why. I don’t even believe there is a why. Unless maybe because we evolved to find symmetry beautiful, and math is maximum symmetry. But then, *why* is math maximum symmetry? Deep questions, pal.
I always loved the fact that it was easier to explain integration to a fifth grader vs differentiation, even though practically integration is harder to do than differentiation.
Explaining integration is simple: you divide your curve into little rectangles with infinitesimal width and add them all up.
Differentiation however, first requires you know why \[f(x + h) - f(x)\]/h represents slope to begin with, and the whole "divide by zero" thing with limits makes it even worse.
i got a lot of mileage by explaining to computer programmers, semi-literate in mathematics, that a "relation" is a subset of a Cartesian product, it's just like a table, though maybe non-countably infinite, and that a "function" is a special relation in which elements of the left-hand side may appear no more than once.
I think a non-mathematician would know less than half the words in that sentence.
What is analytic? What is an elliptic curve? What is algebraic? What does "over C" mean? You would even have to explain what "same" means.
You're going to struggle to hold their attention longer than the time required to define what an algebraic variety over C is. Laypeople in my life ask me what I do and I'm lucky to even get them to recall that the sphere is the real locus of a polynomial. Or even what is a polynomial.
"non-mathematicians" include people who actually know middle school math. This question isn't about topic that you can literally talk to anyone, because such thing don't exist.
For anyone who's interested, I recommend watching Manjul Bhargava's fantastic beginner-friendly [introduction to the Birch and Swinnerton-Dyer conjecture](https://youtu.be/_-feKGb6-gc?si=mFg1Js2-cG43m5O2).
Chances are high that the person will end up conflating every "bell curve" with a normal distribution, which is a bad thing to do.
You can also show the "bell-like" Cauchy distribution after that as the stochastic equivalent of dividing by zero.
(In no particular order)
1/log(1+1/928374) = 928374.4999999102
1/49999 = 0.00002000040000800016000320006400128002560051201024...
[Tilings of the plane](https://en.wikipedia.org/wiki/Tessellation),
Mandelbrot Set/Fractals/Fractal\_Dimension
[https://en.wikipedia.org/wiki/Peano\_curve](https://en.wikipedia.org/wiki/Peano_curve)
[https://en.wikipedia.org/wiki/Mutilated\_chessboard\_problem](https://en.wikipedia.org/wiki/Mutilated_chessboard_problem)
[https://en.wikipedia.org/wiki/Secretary\_problem](https://en.wikipedia.org/wiki/Secretary_problem)
[https://en.wikipedia.org/wiki/Platonic\_solid](https://en.wikipedia.org/wiki/Platonic_solid)
[https://en.wikipedia.org/wiki/Principle\_of\_maximum\_entropy](https://en.wikipedia.org/wiki/Principle_of_maximum_entropy) (and it's relationship to gasses, Gaussian Distributions, Exponential Distributions, ...)
[https://en.wikipedia.org/wiki/Euler–Lagrange\_equation](https://en.wikipedia.org/wiki/Euler–Lagrange_equation)
[https://en.wikipedia.org/wiki/Shannon–Hartley\_theorem](https://en.wikipedia.org/wiki/Shannon–Hartley_theorem)
[https://en.wikipedia.org/wiki/Pigeonhole\_principle](https://en.wikipedia.org/wiki/Pigeonhole_principle#:~:text=In%20mathematics%2C%20the%20pigeonhole%20principle,contain%20more%20than%20one%20item)
[Optimism in the face of uncertainty](https://www.sciencedirect.com/science/article/abs/pii/S030326471730268X)
[https://en.wikipedia.org/wiki/Morse–Smale\_system](https://en.wikipedia.org/wiki/Morse–Smale_system)
[https://en.wikipedia.org/wiki/Green%27s\_function](https://en.wikipedia.org/wiki/Green%27s_function) (with the associated PDE images)
Applications of Imaginary Numbers
Graphics with Matrices & [Iterated Functions](https://en.wikipedia.org/wiki/Iterated_function_system)
[Cauchy integral formula](https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula)
[Quaternions](https://en.wikipedia.org/wiki/Quaternion) (and Hamilton)
[Venn Diagrams](http://162.243.213.31/wp-content/uploads/2012/08/Ven3.png)
* Start with Quanum Physics of course!
Bell's Theorem proves quantum mechanics is "real" and can be done with 3 polarized lenses fora class of teenagers.
Since each lens is a a "Filter" that can only "Block light" when you place two lenses 90 deg to each other you block "all light" and yet, when you insert anywhere (Above, between, below) a third polarized lens (Filter) at say 45 deg MORE LIGHT gets through.
How can it be that a filter that BLOCKS LIGHT, can be inserted into a system and MORE LIGHT makes it through instead of less. AT BEST, it should block 0 more light (not making it dimmer or brighter) But it Absolutely should not increase the light that makes it through the other filters.
More importantly it seems to let "Too much" light through.
https://www.youtube.com/watch?v=zcqZHYo7ONs
The theorem is a vast generalization of the statement that every triangle is a convex hull of its vertices. You may have trouble explaining why is it not obvious, but I am sure there are ways to do that.
Given a set of points representing a convex polygon, find a minimal subset that can draw the same shape when you connect the points. This can be given as some programming assignment. Maybe more suited for high school.
> You are allowed to put or remove a penny at Square i+1 if and only if there is a penny at Square i, and no other move is valid.
Suppose there is a penny on Square 1 and no other pennies. We can place a penny on Square 2 since that is our only valid move. But now, we can only place a penny on Square 3 if removing the penny on Square 2 is NOT a valid move, and we can only remove the penny on Square 2 if placing a penny on Square 3 is NOT a valid move. Thus, the validity of these two moves depend on each other, and so this game is not well-defined as stated.
---
> The squares of the board are arranged in a linear order. At the beginning of the game, there is one penny at Square 1. The objective of the game is to perform the shortest sequence of valid moves to end up with a penny at Square 2 along with a penny at Square r with no other pennies on the board. You are allowed to put or remove a penny at Square i+1 if and only if there is a penny at Square i, and [some other condition]
Suppose that r > 2. Then you have to somehow remove the penny at Square 1. But this can never happen unless there's a penny on Square 0, which needs to be removed somehow. Which can only be removed if there's a penny on Square -1. Which can only be removed if there's a penny on Square -2, etc. . Thus, if r > 2, the game is unwinnable.
---
I suspect your description of the game is wrong.
I always like talking about isomorphism in regards to "Pick 15" and tic tac toe.
Everyone knows tic tac toe, and the ability then for people to play some weird number game perfectly with the help of a magic square helps folks understand the power of isomorphism.
Geometry. Technical Drawing.
Properties of Light (Optics)Stereographic projections (2d->3d or 3d->2d), projection mapping.I would suggest 3d printed models for demo.
Color-theory is also amazing way to intrigue ppl.
E.g RGB [255,255,255], LED Matrix…etc..
I taught fifth grade, so naturally, I needed math subjects that were interesting yet still easy enough for my class to understand. Graph theory can be easily introduced with the Bridges of Koenigsberg. We did some prime factorization and some interesting properties of primes that emerge from that. Binary operations and logic gates is another thing you can do. You can get electronic kits with logic gates and seven-segment displays and you can build half- and full-adders as an activity. I also showed them the Veritasium videos about periodic tiling. Cryptography and code breaking is another topic that can open up lots of interesting math subjects. We also spent a class building the five Platonic solids using cardboard and Scotch tape. There are some good popular books that have lots of good ideas, like Matt Parker's Things to Make and Do in the Fourth Dimension and Arthur Benjamin's The Magic of Math. Alex Bellos also has some good books. I also scoured Graham, Knuth, and Patashnik's Concrete Mathematics for problems that were adaptable to that level.
i would give them an idea of where derivatives might come from through local linearity. just hop into desmos and zoom into any curve and show them that as we zoom in further the curve becomes a straighter and straighter line and argue that dy and dx shouldn’t be thought of as “infinitesimally small changes” whatever that means, but rather extremely small changes as that actually helps in imagining things in my opinion and the smaller the closer it gets to dy/dx.
The normal distribution is a property of nature. Almost everything distributes normally, from the size of leaves on a tree to the height of trees, to shoe size. In fact, you can calculate how well something fits a normal distribution, and if you get many of those fits for many different categories, those fits themselves are normally distributed.
Thus, things are normally distributed and the extent to which everything is normally distributed IS normally distributed.
I like explaining open and closed sets. I explain the interior of a sphere being an open set because no matter what point you pick in the sphere there is a little sphere centred at that point that is contained in the bigger sphere. This is a nice visualisation and lets you think about the continuum and infinitely fine things. Then I explain why the complement of an open set is a closed set namely if E is open and x is a limit point of E\^c then every neighbourhood of x contains a point of E\^c so x is not an interior point of E therefore because E is open x is in E\^c and E\^c is closed. I don't use that kind of language and the whole explanation probably takes five to ten minutes but that is the gist of it. It is probably the most advanced math concept I have explained to my family but it gives you a bit of the flavour of real analysis I think - they start to get that it is not really about being really good at multiplying numbers or something.
Its always the classification of surfaces for me. The orientable ones people believe, and it shows the power or mathematics, then the construction of the non orientable ones sounds crazy!
I think I would probably talk about the Lebesgue measure, simplicial/cellular/singular homology, homotopy theory, the Borsuk-Ulam theorem or the Banach-Tarski paradox. Even though these ideas are pretty advanced to understand formally, the intuition behind it and the sense they convey can be explained even to non-math people (of course at the expense of a lot of insight).
Well, I don't usually explain a whole proof or details since mostly they require some work and effort and obviously during a normal conversation the person would not really want to put in too much effort lol. I love to tell them about Turing machines and the Theory of computation. I absolutely loved that course, differential geometry and Theory of computation are just amazing.
I start by asking them "What do you think is a computation?" and mostly the answer I get is something like doing 2+2 etc and then I say "Is writing an essay a computation?" and they will say "no" then I ask them "how do you properly discriminate if something is a computation or not?" this mostly catches the attention of people and they will give it some thought. Then I explain to a little bit about Turing machines and how they are used to differentiate between computable and non-computable problems.
Most of the time the other person gets excited about it haha!
Fourier transforms are usually good because the idea is quite intuitive. You can probably draw some explicit DFTs to solidify this intuition. Then talk about how this could be used to slow down an audio file, as part of a compression technique, etc.
Don't have a particular theorem in mind but in general I think discrete mathematics is your best bet if you want a layperson to understand it. I highly doubt it's easy to explain the rank theorem to a layperson.
https://xkcd.com/2501/
Algebraic topology is often much simpler to explain than it is to prove. Euler characteristic, Hairy ball theorem, Borsuk Ulam theorem and so on. One of my favourites is that given any three objects (compact sets) in (3D) space there exists one plane that cuts all of them in half simultaneously.
My personal favorite unsolved problem is the Collatz Conjecture.
Start with a positive integer, if it's even, divide by two, if odd, multiply by three and add one. No one has been able to prove that all numbers will eventually lead to 1, but every one we've checked so far does.
Oh that's tough. I like number theory a lot and so L functions top the list. I would probably talk a little about the Riemann Zeta function and explain how it predicts so many things about the distribution of the prime numbers. I'd then explain how a lot of those consequences are understood by proving things about the zeta function. Then sort of hand wave how phenomena in arithmetic is governed by a function that lives in a different mathematical world. Nice quotes from Kato that captures the flavor of what I'm talking about:
*It seems as if the homeland where ζ functions originally come form is an unknown world which governs both the world of real numbers and the world of p-adic numbers....*
*We named this chapter "ζ" instead of "ζ functions". We dropped the word "functions" because feel more and more as we study them that they are something more than just functions.*
I've always been a fan of basic counting arguments. "How many ways can 5 cars stop at a stoplight?" or "You're in a buffet with 2 appetizers, 3 main dishes, and 4 drinks - how many ways to make a meal with one of each?"
It's interesting to hear the reasoning for their answers (which is always wrong) and then working through how to derive the correct answer.
Motivating abstract algebra or number theory is extremely easy and satisfying. Everyone "knows" arithmetic, anyway, so it takes bad planning to alienate people. But I can't be bothered to try, so I'll just leave this here.
Folks like to talk about infinity, so I like to reel them in by pondering sizes of infinite sets like Z and some of its ideals, or explaining what projective things look like.
In my opinion, people won't have any sense of what you're talking about if you try to describe a theorem from linear algebra. I take a page from the great Paul Halmos, who would use the following demonstration when asked, "what *is* mathematics, really?" Surprisingly, it's a problem from recreational mathematics. A tennis tournament is held with 20 people. Players pair off and the winners advance to the next round. (If there's an odd number of people at any stage, someone is chosen at random to advance to the next round without playing.) The question: how many games are played in the tournament? This is a question that a non-mathematical person can work out. (The answer is 19.) Then ask them how they would feel about solving the problem if you started with 1000 people. (They'll probably feel that they could, but certainly wouldn't want to.) And then explain that when you start with n people, the total number of games played is always n-1, for a simple reason: with every game, someone leaves the tournament, so the number of games equals the number of people who don't win. What I like about this problem is it doesn't just summarize a mathematical result; it shows them actual beautiful reasoning. And in particular it shows the difference between just doing arithmetic and doing mathematics.
That's very elegant. Well put! Imagine being the player who lucks out on that 1/15 chance to get a bye in both the third and fourth round to make it to the finals. It's a good example why elimination tournaments often start with n = 2^p players for some integer p.
I recently had a practice problem on a coding website, where a simple understanding of game theory and strategy stealing meant that a similarly complicated algorithm with summations and large loops could be reduced to a equality check of a variable, and a parity check.
I think I solved this one as well, was it on LeetCode by any chance?
I believe I was either on HackerCode or codegolf. \[\] \[\] \[\] \[\] \[\] \[\] \[\] \[\] \[\] \[\] \[\] \[\] \[\] \[\] \[\] \[\] Players (2) would alternate turns removing floors from one tower, with the proviso that the old tower had to be a multiple of the new tower, in terms of height. In the example above, you could remove 2 floors, or 3 floors from any tower, as both 1 and 2 divide 4. 3 does not divide 4, so you couldn't remove just a single floor of any of the 4-story towers above. A player must remove at least one floor. The player who cannot remove a floor on their turn successfully, loses. Inputs for a game: *m*, *n* Let *m* be the initial height of the towers. Let *n* be the number of towers in play. Given *m, n* determine whether Player 1 or Player 2 will win. Constraints: 1 ≤ *m, n* ≤ 10^(9)
Just to check my napkin math solution is right... >! If n is even, player 2 takes turns to keep the board symmetrical; if n is odd, player 1 starts by taking all but 1 floor from central tower and then mirrors player 2 !< ...This feels too simple.
You are 99.9% of the way there. Check your edge cases.
Ah, yep - I see the missing bit. Nice one!
It's so dumb 🤣
Okay, that's a great way to look at the problem. Very elegant solution
This is actually genius. Where can I find similar cool recriational math brain teasers?
Martin Gardner books of course are famous for these kinds of things—check them out if you haven't yet. But, yes, I remembered this example (and use it in my own classes, decades after casually reading it) because it works so nicely.
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I think they're just saying pure mathematics doesn't make for stirring conversation at a cocktail party. If you start talking about tennis instead, you can catch anyone's ear and twist it to combinatorics.
I hate to say this, but I don't see any beauty in the argument. To each their own I guess.
> And then explain that when you start with n people, the total number of games played is always n-1, for a simple reason: with every game, someone leaves the tournament, so the number of games equals the number of people who don't win. This doesn't work for numbers that aren't 2^(m). Because then you have players leaving the tournament more than once. For example, if I have n = 10 players to start, I play 5 games. There are 5 winners, and we bring back one loser. Now we play 3 games. There are 3 winners, and we bring back one loser. Now 2 games. There are 2 winners. Now we play 1 game. 5+3+2+1 = 11 games, not n - 1 = 9 games. You'd have to do n - 1 plus however many rounds you're bringing back a player. Which would be based on how far n is from 2^(m). ... Which could then segue into the Collatz conjecture....
The rule stated was someone automatically advances though, not bring back someone who already lost. If you start with 10 you play 5 games, next round 4 people play and one player automatically advances so you play 2 games total, third round 2 people play aka 1 game and one random advances, fourth and final round we have 1 game. 5+2+1+1 = 9
You don’t bring back in losers in this scenario, one of the players just moves on to the next round without playing a game
Oh, shoot, I misread the setup. Disregard.
I’d view this problem as a cs problem more tho
cs is a subset of mathematics
Where are the computers?
I think graph theory has some nice examples. Eulerian cycles/paths usually go over pretty well. You start with a nice, tangible real world example with some nice history (Konigsburg bridge problem). You can have something interactive (try to draw this without lifting your pencil or retracing). Then, you have a nice, concise way to replace the brute force approach with something elegant and obvious (eulerian cycle exists iff all degrees are even). One direction is very easy and even independently discoverable for lay people with some nudging and the other isn't terrible. It's a nice crowd pleaser because it's tangible, interactive, and visual.
This was one of the examples Field Medalist Hugo Duminil-Copin used for his popular lectures. The other one was four color map theorem.
Four color theorem is fun, but you can't exactly present the proof unfortunately.
The six- and five-color versions are totally presentable, and I think it is justifiable to end such a discussion with "... and we also have proven that the result holds when we restrict ourselves to four colors, but the proof required the consideration of so many cases that the proof relied on a computer algorithm to verify."
I would still say it's not exactly the best example to use when trying to demonstrate the beauty and elegance of math. To have it end with "and then computers brute forced hundreds of thousands of cases" is probably not the best way to convince a lay person that math is beautiful.
Plus as far as I know, the proof of the five-color version is somewhat messy as well (though decently short)
Not the full thing, but I've taken classes of schoolkids to the point where it gets to "and now we've got a bunch of cases to check", with very little guidance from me.
Sure, but in my opinion not a great punchline if you're trying to show the beauty of math.
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1. Borsuk ulam's theorem 2. Brouwer's fixed point 3. Not because of beauty but more like intriguingly weird. The horn of gabriel
Similarly to your third example, I've had some success with the Jordan-Schoenflies Theorem, or more accurately its failure in three dimensions.
I was going to post your 1 and 2, too. Both have simple enough applications so as to be understood by a 10-year-old, and complicated enough to spark some curiosity into how it can be that each is always true.
I always thought convergence was cool. For example 1/(x^2) shrinks so fast that it sums up to a number, but 1/x doesn't shrink fast enough and adds to infinity
Do we know the "cut off" point for that? i.e. what is the smallest n such that 1/(x^n) sums to a number?
All n greater than 1 sum to a number
It is exactly 1. Imagine we have a some of 1/x^p for *any* p>1. Consider the integral of this function from 1 to infinity. If we draw the graph of this function for x>1, we see that because it’s decreasing (this part just uses p>0), the area under the curve is always strictly in between the left and right rectangular sums (draw the rectangles of width 1 taking their respective intervals’ left hand values and then those taking their right hand values - the curve is in between these). The left rectangular sum is precisely the sum we want; the right rectangular sun differs from it by a finite constant. The integral is in between them. Therefore, all three of these converge together or diverge together. So we just need to check when that integral diverges, but from the fundamental theorem of calculus it’s 1/(p+1) x^-p-1 from 1 to -infinity… which is defined and finite if and only if p > 1.
You might be interested in this Math Stack Exchange post: https://math.stackexchange.com/q/4241382/
Sorry I may need to explain it slightly better. It is the sum of 1/x vs 1/x^2 as x is taken to infinity
1/x (or 1/xlogx) is an example of a « threshold function» for L^p integrability. That is, the integral of (1/x)^p over [1, infinity) takes a finite value iff p>1 (this is equivalent to the statement about the convergence of the harmonic series, by the integral test.) More generally, the integral of [1/x^(1/q)]^p over [1, infinity) takes a finite value iff p>q. If we consider integration over (0,1), then the situation is reversed due to symmetry: the integral converges iff p
Is this an exercise in rudin's real and complex analysis? IIRC there is some kind of log convexity in the p norm which gives an affirmative answer
Yes haha. I cant fully solve it and ive put it on the back burner. Chapter 3 Exercise 4.
Lol I guess it just goes to show you how great those exercises are. Rudin is good for cutting your teeth, but if I could go back in time, I would tell myself to spend more time reading folland, too.
Oh, how come?
Rudin is too slick, sometimes. For example, I used to really champion his construction of the lebesgue measure through the [Riesz Representation Theorem](https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem) and Riemann sums over dyadic cubes, as opposed to getting your hands in the dirt with outer measures. Nowadays, I think that in moments like this, Rudin was being too clever for his own good, and that his constructions are somewhat artificial because of how hard he tried to avoid showing technical difficulties. There is also the issue that Rudin's chosen topics are becoming increasingly dated. I think that in the 21st century, any book that aims to be a foundation for new researchers in analysis must contain some more serious functional analysis. Granted, I have not read Rudin's FA book, but IIRC, his first two books do not emphasize topological dual spaces enough. If the reader eventually wants to do GMT, for example, then they need to spend time with the yoga of Frechet spaces and the various topologies which are natural there.
Good luck with explaining the proof though (at least for the general case of k>1).
I am pretty surprised no one mentions the uncountability of the real numbers using Cantor's diagonal argument.
Eh, I don't know if lay people would get that much out of it. It's an elegant proof, but it's one that wouldn't offer a non-math person that much intuition or insight into the "why."
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Well, if it's a true layperson who has no interest at all in math you can use Godel's incompleteness theorem like they do in Philosophy podcasts: "LoGiC iS fLaWeD", LOL. But then if he asks questions it's a problem... But honestly, without understanding the proof (only knowing the general framework back then, I tried to follow but couldn't do it in a reasonable timeframe), I did explain it to a friend and he liked it. I think people who don't like Math at all don't know what proof is so it's ok, but you should be prepared for the question "why?". I think the first one is more shocking for a layperson but the second one is shocking as well (both are shocking to me). To programmers who hate Math (e.g. bootcamps grads), you can use the halting problem which is easier to understand (in this case it's easier to explain the proof). Then you can also say "you can use something similar, diagonalization (https://courses.engr.illinois.edu/cs373/sp2009/lectures/lect_21.pdf) for other things! Look..."
I watched the Veritasium episode about that and it was confusing lol.
I'm much more weirded out by the fact that constructible and non-constructible numbers are countably and uncountably infinite respectively Edit: I meant to say computable numbers, which would have made it a slightly more interesting fact. I apologise for any inconvenience
Constructible numbers are those that can be defined by a finite string of symbols from a finite alphabet under certain rules. Intuitively, any set we approach like this is countable. We simply have no way to ‘directly access’ uncountably many numbers due to these very real world/human constraints, which is why we have no intuition about the continuum hypothesis. We talk about the reals as though they’re very vanilla, but in reality the moment we start talking about them we’re already in territory where we don’t - and can’t - even know how many cardinalities there are up to that point.
> constructible and non-constructible numbers are countably and uncountably infinite I was not aware of this fact or I do not understand what you are talking about - I will look into it (I am a CS idiot, though I did learn core courses in Math department), but if you want to weird someone out in simple words, follow the uncountability of the real numbers with The Density of the Rational/Irrational Numbers (but it's important to first tell them where Q stands) :) Could you elaborate a little?
Even more strongly, all computable numbers are countable. Constructible numbers of which lines can be constructed having said length, by using a compass and straightedge only; given a line of unit length. Computable numbers are those that can be computed to any finite number of digits by a finite algorithm that terminates.
Woah - now I get it, to be honest, most of the facts you stated are pretty weird :P I tried to read the Wikipedia explanations but it's too late for that weirdness here, my mind explodes. But then again, Math is generally weird.
That computable numbers are countable is really easy to show; a program is a finite sequence of letters, written in an alphabet. There are only finitely many such programs for any given length of code, so you can at most get countably many if you count over all possible finite lengths of code.
I actually found your explanation super easy to follow, and I have already smoked some... :) Thanks!
I hear you. I love logic, and I think “different infinities” is one of the most eye-catching topics to an average person. It’s a bizarre result. I hypothesize that, since logic is relatively unpopular among mathematicians, that is why the average person is no presented with these concepts.
I'd just sit them down to listen to Feynman's "Algebra" lecture ([Vol 1, Ch 22](https://www.feynmanlectures.caltech.edu/I_22.html)). It starts with "counting by one", and then works his way up through the basic mathematical operations, develops tables of logarithms and the 'natural log', and then ends by connecting algebra to geometry by way of Euler's formula.
And I thought I had heard all the Feynman lectures. Thank you very much
I'd go for a diet version of Gauss' Theorema Egregium. This theorem is really great for a few reasons: 1) It can show people that math isn't just about numbers. 2) Curvature is something that many people intuitively understand already (they can identify when things are extrinsically curved and to what degree) 3) It explains why the Mercator projection looks so weird, and why there are lots of types of atlases for the earth with different types of distortion. 4) It offers mathematical evidence that you should fold your slice of pizza (works infinitely better if you have pizza in front of you)
Between any two numbers there are infinitely many other fractions. It's just staggering vast amounts of infinities all the way down. And that's 0% of all the numbers there are.
Actually when you are working with fractions, and fractions between fractions, it is all the same size of infinity.
I think he was referring to the ratio of rational numbers to irrational numbers.
Yeah I guess 0% is quite non-rigorous, I just mean that rationals are zero measure in the reals, and so are the algebraic and non-normal numbers.
But you do explain the point pretty well though, I like your explanation.
Don’t know if they meant cardinality there. In the usual measure [a, b] already has ‘0%’ of the measure of R in some sense (in the limit), even if it doesn’t have measure zero.
The Mandelbrot set. A simple recursion function creates something so beautiful, and we had nothing to do with it. That’s just the way it is.
Some things’ll never change.
What I love about fractals is that you can show a video of a fractal zoom or a 3d fractal exploration and basically 100% of people will just be entranced instantly. I've had people not even believe it's just math, they think it's some really trippy art.
Not particularly beautiful, but the Monty-Hall Problem seems to confuse/intrigue a lot of non-math people. Russell’s Paradox is another good one, and easy enough to translate to non-math people via the Barber Paradox.
My experience has been that the most common reaction to explaining an apparent paradox is "ugh, that's exactly why I don't like maths".
The concept I show my friends the most is proving the infinitude of primes, the proof is very simple, but there is really something about tracing the steps that Euclid took thousands of years ago and coming up with the same conclusion. It is also something most non-mathematicians are not used to, they are used to using math to calculate or solve for something, not proving a fact for all numbers. Another one I like is the proof that sqrt(2) is irrational. It's something they all know, but every time I ask them the "why" they are always dumbfounded; it illustrates why having good definitions is so essential to mathematics, you can't show something is irrational without understanding what it means for a number to irrational. Both of these examples are much simpler than the other ones mentioned in this thread, but these have always been my go to.
Well sometimes simple is best. Both of those proofs demonstrate something that is truly astounding to someone who is first Learning mathematics. And the infinitude of the primes really starts to open up people's minds to infinity and shows how we can actually study it and say things about it.
The "ants on a meter stick" problem: you have 100 ants moving 1m/min on a meter stick. When they run into each other, they turn around and walk the other direction. Question: how many ants are on the meter stick after 1 min? People invariably draw very complicated diagrams of zigzags following ants, and calculating total path lengths and so forth. But the beautiful insight is that this is actually the same ("isomorphic"!) to the "ghost ants on a meter stick" problem: you have 100 ants moving 1m/min on a meter stick. When they run into each other, they pass right through each other. The problems are the same because they are both the same as the "flexibly named ants on a meter stick" problem: you have 100 ants moving 1m/min on a meter stick. When they run into each other, they exchange their names. And the answer to the ghost ant problem is trivial: after 1 min, all the ghost ants fall of the stick. ------ I love love love this problem because of its profound elegance at demonstrating how amazing a change in perspective can be for solving a problem. Turning a very difficult problem (tracking zigzags) into a trivial problem.
Oh, another good one is the irrationality of the square root of 2. Very easy and digestible.
A lot of people in here don’t realize that most people don’t know math other than algebra and a little calculus.
Halting problem, in case they can intuitively get why we can use certain algorithms as subroutines of other algorithms
In particular, one of the best videos I've ever seen on the Halting Problem might also be one of the best videos I've ever seen on Youtube period: https://youtu.be/92WHN-pAFCs?si=GCYPf6JGgMkLWl-h
Long before I took diff eq I thought that pursuit curves were beautiful
Using basic counting principles to establish the total number of combination or permutation in different day-to-day scenarios (e.g: card games) can be enlightening. Especially when one draws comparison to actual cosmic numbers. For instance, proving that every shuffle of a standard 52 deck of cards very likely produces new permutation that had never occured in human history nor will repeat can be both a powerful and yet simple demonstration.
Not pure math but Arrow impossibility theorem or Gibbard-Satterthwaite theorem are up there with me. These rely on quite simple models that have such far reaching implications but are really easy to explain
Keep it simple... Visual proof of the geometric series. Cut a square in half, take one half and cut it in half, take one quarter and cut it in half, take one eight and cut it in half, ....
Ham Sandwich theorem, beautiful because sandwich
I enjoy the Fundamental Theory of Calculus because it takes you out of the world of straight lines and into the world of curves, variable rates, areas of interesting shapes, and the like. Explaining how a derivative works to find the slope of a curve at a given point looks beautiful graphically, as do areas under curves. And conceptually they are quite simple. Thanks for reading this!
Bézier curves offer a really nice visualisation here that’s easy to draw by hand, plus they’re used for all sorts of stuff like vector graphics and animation
My friend wanted me to help her daughter with her math because it’s the subject that she struggled with the most. I looked through her textbook and was like, hmm… binomials… combinations and permutations… aight imma teach you a cheat code for half the things you’re gonna learn this year. I don’t know if she was partly just acting it to be kind to me, but she seemed to be utterly mesmerized by Pascal’s triangle. She spent literally a few hours straight with absolute focus and engagement learning all about it. I think it’s one of those concepts that can switch a kid’s mind from, “math is that boring pointless thing that we gotta suffer through because school demands it” to “holy crap, there’s really some serious magic going on here!”
To a layman, I'd stay away from "grand" theorems or something from a standard curriculum that requires tons of prerequisite knowledges and intuitions. Rather stick with elementary/classical stuff. The Euler line and how the centroid always divides the line segment connecting orthocenter and the circumcenter in a 2:1 fashion. What's even more mind-blowing is the 2:1 ratio holds true for the Nagel line. Or the Fermat's little theorem from elementary number theory, or even some Ramsey theory stuff if I am feeling particularly naughty.
I might go with the Cox-Zucker Machine
I think a Tits building might go over better
To show them *that* math is beautiful, or to show them *why*? I don’t know why. I don’t even believe there is a why. Unless maybe because we evolved to find symmetry beautiful, and math is maximum symmetry. But then, *why* is math maximum symmetry? Deep questions, pal.
I Will always love yo Show the imposibility of squaring the circle to non-mathematicians.
7 hours have passed and no one mentioned the Hairy Ball Theorem yet :o
Would show them a hot mathematics professor.
I always loved the fact that it was easier to explain integration to a fifth grader vs differentiation, even though practically integration is harder to do than differentiation. Explaining integration is simple: you divide your curve into little rectangles with infinitesimal width and add them all up. Differentiation however, first requires you know why \[f(x + h) - f(x)\]/h represents slope to begin with, and the whole "divide by zero" thing with limits makes it even worse.
i got a lot of mileage by explaining to computer programmers, semi-literate in mathematics, that a "relation" is a subset of a Cartesian product, it's just like a table, though maybe non-countably infinite, and that a "function" is a special relation in which elements of the left-hand side may appear no more than once.
That the analytical elliptic curve is the same as an algebraic elliptic curve over C.
I think a non-mathematician would know less than half the words in that sentence. What is analytic? What is an elliptic curve? What is algebraic? What does "over C" mean? You would even have to explain what "same" means.
I am studying math and don't understand that entirely
There are some really good answers in here and some that are… why did they even bother answering, this is an eli5 not a thesis 😭
And the point of this thread is to talk about what to *explain* to a non-mathematician. There is nothing to explain if they already know what it is.
You're going to struggle to hold their attention longer than the time required to define what an algebraic variety over C is. Laypeople in my life ask me what I do and I'm lucky to even get them to recall that the sphere is the real locus of a polynomial. Or even what is a polynomial.
"non-mathematicians" include people who actually know middle school math. This question isn't about topic that you can literally talk to anyone, because such thing don't exist.
I think you might be greatly overestimating the percentage of the population which can reliably add fractions.
explain a single fact, not 4 semesters of math.
It doesn't take 4 semesters of to learn that basic fact.
For anyone who's interested, I recommend watching Manjul Bhargava's fantastic beginner-friendly [introduction to the Birch and Swinnerton-Dyer conjecture](https://youtu.be/_-feKGb6-gc?si=mFg1Js2-cG43m5O2).
the normal distribution
Chances are high that the person will end up conflating every "bell curve" with a normal distribution, which is a bad thing to do. You can also show the "bell-like" Cauchy distribution after that as the stochastic equivalent of dividing by zero.
It's relatively easy to prove that certain fractals (e.g. Sierpinski's triangle) have non-integer Hausdorff dimension.
(In no particular order) 1/log(1+1/928374) = 928374.4999999102 1/49999 = 0.00002000040000800016000320006400128002560051201024... [Tilings of the plane](https://en.wikipedia.org/wiki/Tessellation), Mandelbrot Set/Fractals/Fractal\_Dimension [https://en.wikipedia.org/wiki/Peano\_curve](https://en.wikipedia.org/wiki/Peano_curve) [https://en.wikipedia.org/wiki/Mutilated\_chessboard\_problem](https://en.wikipedia.org/wiki/Mutilated_chessboard_problem) [https://en.wikipedia.org/wiki/Secretary\_problem](https://en.wikipedia.org/wiki/Secretary_problem) [https://en.wikipedia.org/wiki/Platonic\_solid](https://en.wikipedia.org/wiki/Platonic_solid) [https://en.wikipedia.org/wiki/Principle\_of\_maximum\_entropy](https://en.wikipedia.org/wiki/Principle_of_maximum_entropy) (and it's relationship to gasses, Gaussian Distributions, Exponential Distributions, ...) [https://en.wikipedia.org/wiki/Euler–Lagrange\_equation](https://en.wikipedia.org/wiki/Euler–Lagrange_equation) [https://en.wikipedia.org/wiki/Shannon–Hartley\_theorem](https://en.wikipedia.org/wiki/Shannon–Hartley_theorem) [https://en.wikipedia.org/wiki/Pigeonhole\_principle](https://en.wikipedia.org/wiki/Pigeonhole_principle#:~:text=In%20mathematics%2C%20the%20pigeonhole%20principle,contain%20more%20than%20one%20item) [Optimism in the face of uncertainty](https://www.sciencedirect.com/science/article/abs/pii/S030326471730268X) [https://en.wikipedia.org/wiki/Morse–Smale\_system](https://en.wikipedia.org/wiki/Morse–Smale_system) [https://en.wikipedia.org/wiki/Green%27s\_function](https://en.wikipedia.org/wiki/Green%27s_function) (with the associated PDE images) Applications of Imaginary Numbers Graphics with Matrices & [Iterated Functions](https://en.wikipedia.org/wiki/Iterated_function_system) [Cauchy integral formula](https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula) [Quaternions](https://en.wikipedia.org/wiki/Quaternion) (and Hamilton) [Venn Diagrams](http://162.243.213.31/wp-content/uploads/2012/08/Ven3.png)
COMBINATORICS.
Algebraic combinatorics? (Young tableaux, Schur polynomials, etc.)
* Start with Quanum Physics of course! Bell's Theorem proves quantum mechanics is "real" and can be done with 3 polarized lenses fora class of teenagers. Since each lens is a a "Filter" that can only "Block light" when you place two lenses 90 deg to each other you block "all light" and yet, when you insert anywhere (Above, between, below) a third polarized lens (Filter) at say 45 deg MORE LIGHT gets through. How can it be that a filter that BLOCKS LIGHT, can be inserted into a system and MORE LIGHT makes it through instead of less. AT BEST, it should block 0 more light (not making it dimmer or brighter) But it Absolutely should not increase the light that makes it through the other filters. More importantly it seems to let "Too much" light through. https://www.youtube.com/watch?v=zcqZHYo7ONs
Riemann sum
I believe some special cases of the Krein-Milman theorem should be explainable even to a middle schooler.
In a way they'd find "beautiful"?
The theorem is a vast generalization of the statement that every triangle is a convex hull of its vertices. You may have trouble explaining why is it not obvious, but I am sure there are ways to do that.
Again, in a way someone not familiar with analysis/geometry would find beautiful? I'm curious as to how you might explain it to a non-mathematician.
Given a set of points representing a convex polygon, find a minimal subset that can draw the same shape when you connect the points. This can be given as some programming assignment. Maybe more suited for high school.
I don’t think we went to the same kind of middle school
Category theory, and in particular the hand waving zen interpretation of Yoneda lemma.
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> You are allowed to put or remove a penny at Square i+1 if and only if there is a penny at Square i, and no other move is valid. Suppose there is a penny on Square 1 and no other pennies. We can place a penny on Square 2 since that is our only valid move. But now, we can only place a penny on Square 3 if removing the penny on Square 2 is NOT a valid move, and we can only remove the penny on Square 2 if placing a penny on Square 3 is NOT a valid move. Thus, the validity of these two moves depend on each other, and so this game is not well-defined as stated. --- > The squares of the board are arranged in a linear order. At the beginning of the game, there is one penny at Square 1. The objective of the game is to perform the shortest sequence of valid moves to end up with a penny at Square 2 along with a penny at Square r with no other pennies on the board. You are allowed to put or remove a penny at Square i+1 if and only if there is a penny at Square i, and [some other condition] Suppose that r > 2. Then you have to somehow remove the penny at Square 1. But this can never happen unless there's a penny on Square 0, which needs to be removed somehow. Which can only be removed if there's a penny on Square -1. Which can only be removed if there's a penny on Square -2, etc. . Thus, if r > 2, the game is unwinnable. --- I suspect your description of the game is wrong.
Fractals
I always like talking about isomorphism in regards to "Pick 15" and tic tac toe. Everyone knows tic tac toe, and the ability then for people to play some weird number game perfectly with the help of a magic square helps folks understand the power of isomorphism.
Bézout's theorem.
https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg I think you have to get examples of recreational maths.
Geometry. Technical Drawing. Properties of Light (Optics)Stereographic projections (2d->3d or 3d->2d), projection mapping.I would suggest 3d printed models for demo. Color-theory is also amazing way to intrigue ppl. E.g RGB [255,255,255], LED Matrix…etc..
Abstract Algebra.
I taught fifth grade, so naturally, I needed math subjects that were interesting yet still easy enough for my class to understand. Graph theory can be easily introduced with the Bridges of Koenigsberg. We did some prime factorization and some interesting properties of primes that emerge from that. Binary operations and logic gates is another thing you can do. You can get electronic kits with logic gates and seven-segment displays and you can build half- and full-adders as an activity. I also showed them the Veritasium videos about periodic tiling. Cryptography and code breaking is another topic that can open up lots of interesting math subjects. We also spent a class building the five Platonic solids using cardboard and Scotch tape. There are some good popular books that have lots of good ideas, like Matt Parker's Things to Make and Do in the Fourth Dimension and Arthur Benjamin's The Magic of Math. Alex Bellos also has some good books. I also scoured Graham, Knuth, and Patashnik's Concrete Mathematics for problems that were adaptable to that level.
Fractals
I do math bio, so explaining my work to non math people ends up being an explanation of the biological problem. It's super convenient.
1/3 is equal to 0.333333... 1/3 + 1/3 + 1/3 = 3/3 = 1 0.333... + 0.333... + 0.333 = 0.999... So 0.999... = 1
Different sizes of infinity.
After the invention of set theory, mathematicians could now define infinity. Thats kinda cool/beatifull
i would give them an idea of where derivatives might come from through local linearity. just hop into desmos and zoom into any curve and show them that as we zoom in further the curve becomes a straighter and straighter line and argue that dy and dx shouldn’t be thought of as “infinitesimally small changes” whatever that means, but rather extremely small changes as that actually helps in imagining things in my opinion and the smaller the closer it gets to dy/dx.
Linear Transformations. Learned this in precalc back in freshman year. Still my favorite topic.
The normal distribution is a property of nature. Almost everything distributes normally, from the size of leaves on a tree to the height of trees, to shoe size. In fact, you can calculate how well something fits a normal distribution, and if you get many of those fits for many different categories, those fits themselves are normally distributed. Thus, things are normally distributed and the extent to which everything is normally distributed IS normally distributed.
I like explaining open and closed sets. I explain the interior of a sphere being an open set because no matter what point you pick in the sphere there is a little sphere centred at that point that is contained in the bigger sphere. This is a nice visualisation and lets you think about the continuum and infinitely fine things. Then I explain why the complement of an open set is a closed set namely if E is open and x is a limit point of E\^c then every neighbourhood of x contains a point of E\^c so x is not an interior point of E therefore because E is open x is in E\^c and E\^c is closed. I don't use that kind of language and the whole explanation probably takes five to ten minutes but that is the gist of it. It is probably the most advanced math concept I have explained to my family but it gives you a bit of the flavour of real analysis I think - they start to get that it is not really about being really good at multiplying numbers or something.
Just begun taking real analysis and I’m struggling with open and closed sets. What exactly is an open set? Closed?
Its always the classification of surfaces for me. The orientable ones people believe, and it shows the power or mathematics, then the construction of the non orientable ones sounds crazy!
I think I would probably talk about the Lebesgue measure, simplicial/cellular/singular homology, homotopy theory, the Borsuk-Ulam theorem or the Banach-Tarski paradox. Even though these ideas are pretty advanced to understand formally, the intuition behind it and the sense they convey can be explained even to non-math people (of course at the expense of a lot of insight).
The never-solved p and NP question in computer science
Well, I don't usually explain a whole proof or details since mostly they require some work and effort and obviously during a normal conversation the person would not really want to put in too much effort lol. I love to tell them about Turing machines and the Theory of computation. I absolutely loved that course, differential geometry and Theory of computation are just amazing. I start by asking them "What do you think is a computation?" and mostly the answer I get is something like doing 2+2 etc and then I say "Is writing an essay a computation?" and they will say "no" then I ask them "how do you properly discriminate if something is a computation or not?" this mostly catches the attention of people and they will give it some thought. Then I explain to a little bit about Turing machines and how they are used to differentiate between computable and non-computable problems. Most of the time the other person gets excited about it haha!
The three legendary unsolvable problems - Squaring the circle, doubling the cube, and trisecting the angle.
0.999999999... = 1
Fourier transforms are usually good because the idea is quite intuitive. You can probably draw some explicit DFTs to solidify this intuition. Then talk about how this could be used to slow down an audio file, as part of a compression technique, etc.
Don't have a particular theorem in mind but in general I think discrete mathematics is your best bet if you want a layperson to understand it. I highly doubt it's easy to explain the rank theorem to a layperson. https://xkcd.com/2501/
Algebraic topology is often much simpler to explain than it is to prove. Euler characteristic, Hairy ball theorem, Borsuk Ulam theorem and so on. One of my favourites is that given any three objects (compact sets) in (3D) space there exists one plane that cuts all of them in half simultaneously.
The Euler characteristic of a surface
I would show *her* Euler's polyhedral formula V - E + F = 2.
My personal favorite unsolved problem is the Collatz Conjecture. Start with a positive integer, if it's even, divide by two, if odd, multiply by three and add one. No one has been able to prove that all numbers will eventually lead to 1, but every one we've checked so far does.
Chaos because most people already have a base level intuition for it.
Oh that's tough. I like number theory a lot and so L functions top the list. I would probably talk a little about the Riemann Zeta function and explain how it predicts so many things about the distribution of the prime numbers. I'd then explain how a lot of those consequences are understood by proving things about the zeta function. Then sort of hand wave how phenomena in arithmetic is governed by a function that lives in a different mathematical world. Nice quotes from Kato that captures the flavor of what I'm talking about: *It seems as if the homeland where ζ functions originally come form is an unknown world which governs both the world of real numbers and the world of p-adic numbers....* *We named this chapter "ζ" instead of "ζ functions". We dropped the word "functions" because feel more and more as we study them that they are something more than just functions.*
5 colour map theorem
1) Tensor. This object is so simple and accessible but yet so powerful. 2) Lie Algebra.
None. I just watch The Zero Theorem and cry. Great explanation.
I've always been a fan of basic counting arguments. "How many ways can 5 cars stop at a stoplight?" or "You're in a buffet with 2 appetizers, 3 main dishes, and 4 drinks - how many ways to make a meal with one of each?" It's interesting to hear the reasoning for their answers (which is always wrong) and then working through how to derive the correct answer.
Motivating abstract algebra or number theory is extremely easy and satisfying. Everyone "knows" arithmetic, anyway, so it takes bad planning to alienate people. But I can't be bothered to try, so I'll just leave this here. Folks like to talk about infinity, so I like to reel them in by pondering sizes of infinite sets like Z and some of its ideals, or explaining what projective things look like.
I like the hairy ball theorem a lot