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2

Symbols carry meaning contextually. Sure, they have a literal "English" counterpart, but the sentence "plums deify" is pretty much nonsense, unless you're readying a fictional work in which plums are intelligent and have organized some kind of religion. In order to separate these kinds of nonsense sentences from the ones that really do make sense, you have ...


0

Here is where I used to go to learn new symbols. Also, if $f : X \rightarrow Y$ and $g : Y \rightarrow Z$ are functions, we write $g \circ f$ for the unique function $X \rightarrow Z$ specified by the following constraint. $$\forall x \in X : (g \circ f)(x) = g(f(x))$$ This is called composition of functions. Make sure you also learn the terms "domain" ...


-1

Juast read and learn math and you'll see that there is nothing to learn ;)


0

$\bullet$ Patience! $\bullet\bullet$ Persistence. $\bullet\bullet$ Work hard. $\bullet\bullet\bullet$ Learn things very well. (in detail) $\bullet\bullet\bullet\bullet$ Ask yourself lots of questions, even stupid ones! (when does this lemma work? when it doesn't? is there a generalization of it? is there a similar lemma about ...) ...


0

Give a good answer to this question may be also a way to give a rigorous definition of "perpendicular" and of what is the "measure" (in radians) of an angle (a not simple task at elementary level) without using angles measured in degrees. I sketch the reasoning in steps: 1) Let's $O$ the intersection point of two straight lines $r$ and $s$ and take an ...


0

Let $$R: \mathbb{N} \rightarrow {0, 1, 2, 3, 4, 5}$$ be the function which returns the remainder when you divide $n$ by 6. Then $$\forall n, m \in \mathbb{N}: R(n \cdot m) = R(R(n) \cdot R(m))$$ and especially \begin{align} R(3x) &= R(R(3) \cdot R(x))\\ &= R(3 \cdot R(x)) \end{align} Now go through all possibilities: \begin{align} R(x) = 0 ...


3

Note that if $$5-3x=6t\implies 5=3\cdot(x+2t)$$, but 5 is a prime number with a prime factor 3 and this is a contradiction


1

A proof that needs five seconds rather than five minutes, but I find instructive nonetheless: Suppose you have a line in $\mathbb{R}^2$ together with two points $A$ and $B$ on the same side of the line. Determine the point where the distance travelled by an object moving from A to B being reflected at the line is minimal. Too lazy to draw up a diagram. Show ...


1

You can go to various website, such as US News & World Reports here, that drills down rankings by department, but I never saw any ranking that drills down to one particular subject in algebra. Hope this helps.


8

In this case it's exactly the same: \begin{align} \left|\frac{3+\sqrt{9-x^2}}{x}\right|^{\!-1} &= \left|\frac{x}{3+\sqrt{9-x^2}}\right|\\[2ex] &= \left|\frac{x}{3+\sqrt{9-x^2}}\frac{3-\sqrt{9-x^2}}{3-\sqrt{9-x^2}}\right|\\[2ex] &=\left|\frac{x(3-\sqrt{9-x^2})}{9-9+x^2}\right|\\[2ex] &=\left|\frac{3-\sqrt{9-x^2}}{x}\right|\\[2ex] ...


-2

My favourite 'immediately proven' theorem is that all natural numbers are interesting. Suppose the contrary, that some natural numbers are not interesting. Then by natural ordering there would exist the smallest uninteresting natural number ...and there would be only one such number, which would be very interesting! So, there is no uninteresting natural ...


0

A standard differential geometry class would certainly contain this material, but it would probably not spend significant time on them. This is more the flavor of a multivariable calculus/analysis class; if this wasn't covered to your satisfaction in yours, you might want to study these on your own.


5

Problem: A red ribbon is tied tightly around the earth at the equator (assume the earth is a perfect sphere). How much more ribbon would you need if you raised the ribbon 1 ft above the equator everywhere? Answer: Only a tad bit more than 6 ft! Solution: Let $r$ be the radius of the earth in feet. Then the circumference (length of the ribbon) is $2\pi r$. ...


3

Disclaimer: All I know of Dhuruva numbers comes from the paper which you linked. That being said, the numbers listed, $53955$ and $59994$, are the first entries in OEIS-A099010. This sequence has to do with the Kaprekar Routine, which the extremely vague definition given in your referenced paper could be referring to.


1

How about Marilyn vos Savant's explanation of which door to choose in a game show? http://marilynvossavant.com/game-show-problem/ The problem asked of her was: Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the ...


1

I really wondering why nobody has already mentioned this classic books: Classical Mathematical Physics - W. Thirring Quantum Mathematical Physics - W. Thirring If someone already knows a lot about Topology and Differential Geometry, Chapters 1, 12 and 13 from Geometry, Topology and Physics - M. Nakahara will give some nice introduction to the ...


0

One of the first proofs I was shown was in a discrete structures class. I don't think this would survive the rigors of formal proof, but I really enjoyed it. Definitions: $N$ is the set of numbers starting with 0, 1, 2, $\dots $ A number in our set $N$ is odd if and only if there exists another number in our set $N$ that when multiplied by two and added ...


1

"Evaluating the solution along a path" assumes meaning only in a complex environment. We then have a differential equation $$w'(z)=f(z,w)\ ,$$ where both the independent and the dependent variable are assumed to be complex, and furthermore the right hand side is defined in some domain $\Omega\times{\mathbb C}$ and is analytic in both variables. A simple ...


1

Note that $t$ is considered a complex variable now. The solution to the equation $x'(t) = \frac{n x(t)}{t}$ is a function proportional to $t^n$. This function, defined on $\mathbb{C}\backslash\{0\}$ is in fact multivalued. The way it is defined is $$t^n = |t|^n e^{i n \arg t}$$ The problem is with the argument: it is not uniquely defined, as a function ...


1

First, I am assuming the project is about or heavily involves mathematics. To come up with a good project you should find a problem that needs to be addressed (problem statement), decide on how your project will address that problem (purpose statement), and be able to explain why what you are doing matters. Here are some slides that go over this process. ...


0

I have spent many decades studying why so many highly intelligent people are so mystified by mathematics. Lockhart's view is very serious and cannot be negated by the personal experiences of mathematically inclined people. My study has clearly shown that the best advice is to be simple and sensible. For example, our place number system is an ingenious ...


4

I really like proofs using the pigeonhole principle I give two examples I think most people should know should know. Example 1: In a party with $n$ persons there are always two persons who have shaken hands with the same number of people. Proof: clearly in parties people don't shake hands with the same person twice (for sufficiently low alcohol levels). ...


3

While possibly a bit silly, I find that the (utterly trivial) proof of the uniqueness of identity elements very nicely illustrates how "abstract" mathematical proofs "work" and how, at least not totally trivial, questions can get very simple answers if posed correctly. While the proof itself obviously does not require 5 minutes to present one would probably ...


3

There's no way to tune a piano in perfect harmony. There are twelve half-steps in the chromatic scale, twelve notes in each octave of the keyboard. Start at middle "C", and ascend a perfect fifth to "G". That's seven half steps up, with a frequency ratio of 3/2. Drop an octave to the lower "g" -- that's twelve half steps down, and a frequency ratio of ...


2

Whittaker and Watson's A Course of Modern Analysis is a standard source for these types of problems.


1

Proof that $\sqrt 2$ is irrational: Any non-integer fraction multiplied by itself cannot be an integer. (So a full length proof along these lines would first have to show that the prime factorization of integers is unique, and this turns out to be rather hard. But any young kid who has learned about prime factorization will accept this without proof.)


1

I always loved the explanation of the Hilton-Eckmann argument, given by J. Baez in This week's finds in mathematical physics 100. The Hilton-Eckmann argument itself is a rather easy result (which one could certainly present in an undergrad abstract algebra class as an exercise), and the "visual proof" (using "higher-dimensional reasoning") which is hinted ...


0

I read this book from Lara Alcock and I think it answers your question: "How to Study as a Mathematics Major". I warmly suggest it! http://ukcatalogue.oup.com/product/9780199661312.do


2

The discipline of mathematics requires the following from you in order to excel in it: 1) Your love of the subject. 2) Your desire to do well in it. 3) Intellectual curiosity. 4) Willingness to work very hard. To excel in ANY discipline, you must work very hard. 5) Mental aptitude. This is the "luck component" of life. You have what you're born with as far ...


0

Definition 1 is not sufficient to characterize all parabolas in a coordinate plane. It characterizes some--namely, those whose axes of symmetry are coincident with one of the coordinate axes, but for parabolas whose axis of symmetry is not parallel to either coordinate axis, definition 1 is not sufficient. Consequently, Definition 2 is preferable. There ...


39

The fact that $$1 = \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ ...$$ by drawing the following picture of a square with a side length of $1$:


15

Here's a variation on $$\sum_{n=1}^{R} n =\frac{R(R+1)}{2}$$ I stumbled onto this a while back. shows $$1+2+3+4+5+6+7 = \frac{1}{2}\cdot b\cdot h + \frac{1}{2}\cdot b$$ $$1+2+3+4+5+6+7 = \frac{1}{2}\cdot 7\cdot 7 + \frac{1}{2}\cdot 7$$ then $$1+2+3+4+5+6+7 = \frac{1}{2}\left( 7\cdot 7 + 7\right)$$ thus $$1+2+3+4+5+6+7 = \frac{7(7+1)}{2}$$


3

One of my favorite proofs given constraints such as these is: Theorem. $n^2 - n$ is even for all natural numbers $n$. The proof can be carried out in many different ways depending on your "general" audience. I have written up a sketch of the entire talk on MO and re-mentioned it on MESE. See also the couple of different generalizations around the ...


3

Taking $\epsilon = 5$ minutes here, we presented a lecture to psychologists, engineers, philosophers and also mathematicians over Game Theory, and how to take better decisions with examples, and different problems. So we took about $10$ minutes to explain Nash Equilibrium to them, using the most famous Prisoner's Dilemma. I believe everyone on the ...


34

The Mutilated chessboard problem: Suppose a standard 8x8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2x1 so as to cover all of these squares? has an easy parity-based solution: The puzzle is impossible to complete. A domino placed on the chessboard will always cover one white ...


43

The harmonic series diverges because otherwise there exists a finite number \begin{align*} S &= 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\dotsb \\ &= \left(1+\frac{1}{2}\right)+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}\right)+\dotsb \\ &> ...


12

For an audience with a bit of geometrical intuition about the sphere (earth), and with a markable sphere at hand to draw on, I like the proof that on a sphere, the area $\Delta$ of a spherical triangle that has angles $A$, $B$, and $C$ has area $\dfrac{(A+B+C-180°)}{720°}\odot$, where $\odot$ is the area of the sphere. (The audience doesn’t even need to know ...


7

I would explain the Pigeon Hole principle, or one of its many guises. In fact, I remember explaining the PHP to a non math student while playing bridge. I told him that one gets at least $4$ cards of some suit- which is really the PHP. You can come up with many other interesting "real life" examples, and it's fun.


15

Every rational number has a repeating decimal. Basically, once you know how to do long division to generate digits, all you need is a very naive pigeonhole principle. Basically, when doing the long division for $\frac{p}{q}$, the remainder is always in $0,1,\dots,q-1$. And thus, you must eventually get the same remainder to the right of the decimal, at ...


59

I'm a bit reluctant to throw another answer on the pile, especially because I think there are other lists on this website which serve a pretty similar purpose. However, I think an excellent 5 minute blurb could be given to a general audience on the trick, attributed to von Neumann, for performing a fair coin toss when only a biased coin with unknown bias is ...


2

I love the proof that a finite group of even order has an element that is its own inverse. I realize that groups are not nearly as well understood as they should be to a general audience, but that is the fault of our education priorities, as groups are everywhere in math and nature. Anyway, the proof. Let $G$ be a group of finite even order. We want to show ...


7

In high school one learns that $\left(-a\right)\left(-b\right)=ab$, but in my experience the proof is never given, or a clear reason. The proof is easy using the number systems that people in America are comfortable with, i.e $\mathbb{R}$, and of course the notion that $a\left(0\right)=0$. Proof: $\left(-a\right)\left(0\right)=0$ $\implies ...


-2

I would like to talk about Pictorial Presentation of Phythagoras Theorem has always fascinated me . Another thing i would like to explain is about Gauss did some of numbers in his school 1 + 2 + 3 + 4 + … + 98 + 99 + 100 Gauss noticed that if he was to split the numbers into two groups (1 to 50 and 51 to 100), he could add them together vertically to get a ...


27

Plenty of people are dumbfounded when you suggest to them that $0.\overline{999}$ might be equal to $1$. Regardless of whether or not the following is rigorous, I've found it to be a wonderful way of demonstrating the property that only requires an understanding of simple algebra. \begin{align} \textbf{let } x &= 0.\overline{999}\\ 10 \cdot x &= 10 ...


18

Problem: An old car has to travel a 2-mile route, uphill and down. Because it is so old, the car can climb the first mile, the ascent, no faster than an average speed of 15 mi/hr. How fast does the car have to travel the second mile, on the descent it can go faster, of course, in order to achieve an average speed of 30 mi/hr for the trip? Solution/Proof: ...


35

I am reading some great responses, thank you all, keep them coming please, I love these. I'd also contribute with another of my favorite ones. Not so much a proof but a 'reasonable argument" for the area of a circle. Most 'general audience' members know $$a=\pi r^2$$ but I am often surprised at how few of them have ever seen a proof for this, similarly but ...


9

Geometric series. \begin{align} S&=\quad\;\,1+\frac1x+\frac1{x^2}+\frac1{x^3}+\dotsb\\ Sx&=x+1+\frac1x+\frac1{x^2}+\frac1{x^3}+\dotsb\\ Sx-S&=x\\ S(x-1)&=x\\ S&=\frac{x}{x-1} \end{align} And, if time permits, a variant: \begin{align} S&=\quad\;\,\quad\;\,\frac1x+\frac2{x^2}+\frac3{x^3}+\dotsb\\ ...


19

It might be too short, but my favourite mathematical proof of a not-too-mathematical concept is the ancient Chinese proof (actually1, not apocryphally) of Pythagoras' theorem. First you can explain what Pythagoras' theorem means. Then, you introduce this image: Now you're ready to tell them the proof. 1. Behold! (QED)


14

I like the proof of the handshaking lemma in graph theory. I've taught it to a few students from ages 12-18 and they all seem to understand. Lemma: There is an even number of vertices in a (finite) graph of odd degree. The proof revolves around the fact that odd$\times$odd$=$odd, and even$+$odd$=$odd and similar facts like that. It involves another simple ...


10

The Stable Marriage Theorem of Gale & Shapley. There is hardly anything in the statement or proof that even looks like mathematics to a "general audience". I haven't tried telling it to a general audience myself, but I'm sure a skilled expositor could get it across in $5+\varepsilon$ minutes, for some sufficiently large value of $\varepsilon$. ...



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