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As a high school student, I would say that I would find the following the most appealing: Ramsey Theory: You can start with the "Happy Ending Problem" and then introduce the idea of different pigeon hole principles and end with VDW theorem for some simple cases. Surreal Numbers: Even though they are practically useless, they are neat when you involve ...


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A few suggestions: $1.$ Hyperbolic Geometry. This gives you the opportunity to explain what an axiomatic system is (something rarely discussed in high school geometry), and then explain some of the interesting results that arise when Euclid's fifth postulate is false. For example, it is very simple to prove that Euclid's fifth postulate is equivalent to the ...


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I gave such a talk once and focussed at the applied end of mathematics, introducing them to polar coordinates and elementary ODEs. In 2-D and 3-D polar coordinates the mystery as to why the surface area $4\pi r^2$ is the derivative of volume $\frac{4}{3}\pi r^3$ and circumference $2\pi r$ of area $\pi r^2$ is solved. This is something that immediately set ...


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Try presenting Goldbach's conjecture, and let them have a go at it. :) In all seriousness, they should get an introduction to these types of open problems, and your class may be it. An apparently easy problem can actually turn out extremely difficult, as is evident in such a conjecture. Others could include the Collatz conjecture, which is also interesting, ...


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I suggest graph theory: an one hour lecture can for example start with the bridges on Koningsberg and move up to proving that a graph contains an Eulerian cycle if and only if every vertex has even degree.


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You may want to check this site for over 46.000 free ebooks. Copyrights of the books contained in this site are expired so you can freely download books. You can download the books in PDF, EPUB or LATEX. main site : https://www.gutenberg.org/wiki/Main_Page math bookshelf : https://www.gutenberg.org/wiki/Mathematics_%28Bookshelf%29


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In the field of optimal control, there is a set of lecture notes by Hector Sussmann with source code located here.


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Rob Beezer's "A First Course in Linear Algebra" represents the future of OER textbooks for math, imho. His MathBook XML production flow allows a single source input (written in xml) to output in multiple formats (right now pdf-via-LaTeX and html, but the future could include more). To compete with commercial textbooks, both a quality book and a quality ...


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On what to take It depends where you are. If there's a rigorous course for freshman who expect to be math majors, I would recommend it even if you've already seen the material. This has many benefits: 1) You admit you struggled through some of the college material. Someone advanced enough to be taking college math courses in high school should not be ...


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First of all you will have to judge where you are in the grand scheme of things. I'd say get yourself steady with algebra and calculus first, you could do some probability on the side as well if you want. These 3 things will give you a pretty good foundation to take maths further. There's a book called Thomas Calculus its often recommended for people ...


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There is a course on Coursera, called Analysis of a Complex Kind. Here is the link: https://www.coursera.org/course/complexanalysis


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The India Institute of Technology has a series which I have found helpful. There are a lot of videos and the presentation is straightforward lecture. Here is one. I believe there are about 40 total. The accent is pleasant to my ear and makes these easier to sit through than they might otherwise be. I have only listened to a few so far because my ...


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MIT video lectures part 1 There are 5 videos covering complex numbers, functions, conformal mappings, sequence and series, and integration. The speaker is professor Herbert Gross There is also the Kahn Academy's basic complex analysis videos


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The Stacks Project and the CRing project are good for learning algebraic geometry and commutative algebra, respectively: Stacks Project: http://stacks.math.columbia.edu/ CRing Project: http://people.fas.harvard.edu/~amathew/cr.html


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To add to jameselmore's answer, you also need some background in the following courses: Survival Models - Survival analysis attempts to answer questions such as: what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into ...


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You might look at the special case of solving $x^2 = a$, where $x_{n+1}$ is the average of $x_n$ and $a/x_n$. Then you can see that the "error" $a - x_{n+1}^2$ is a constant times the square of the previous error $a - x_n^2$.


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Here are a few I've found over the years. I've included some brief comments on the first four, which I've personally used in my classes. Stitz and Zeager Precalculus materials Outstanding. Includes a lot of ancillaries like answers and youtube videos. CCL Mooculus Very, very good calculus MOOC offered at Ohio State University. Text is freely available ...


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Hints: Perpendicular lines have slopes that are negative reciprocals of each other. There are (infinitely) many possible $y$-intercepts you can use for the fence. By inspection, $2$ should work. An inequality of the form $y \leq mx + b$ means that you shade below the line, while an inequality of the form $y \leq mx + b$ means that you shade above the line. ...


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If you can read German, here is a detailed development of the approach suggested by Gowers (see the link given by Mark Bennet), albeit in terms of binary fractions: http://www.math.ethz.ch/~blatter/Dualbrueche_2.pdf But note the following: Whichever approach you take, the amount of work to be done in order to verify all the details is about the same.


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This site about modern Greek has a guide for handwriting the Greek letters apparently based on how people actually write in modern Greece and Cyprus. It can sometimes be useful to think about how the lowercase letter might have developed from the uppercase letter through people attempting to write it faster, smaller and without lifting the pen off the page. ...


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The course Measure Theory by D.H.Fremlin includes TeX source. Topology Course by Aisling McCluskey and Brian McMaster in HTML. Diverse lecture notes by Conor Houghton. Cryptography homework by Boaz Barak. Digital Image Processing. Abstract Algebra handouts and Number Theory lecture notes.


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Many of the courses in MIT's OCW have such notes: http://ocw.mit.edu/courses/find-by-department/


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No, there is no simpler way to do this than to compute all the digits and add them. What you can find easily is the sum mod 9 (the "digital root").


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This is true because square root is not a Many$\to$1 function. i.e., 2 numbers can't possibly have the same square root. Thus, if square root of 2 numbers is the same, it means that the 2 numbers must be equal.


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I think people should have a background in analysis and algebra before attempting to learn serious set theory and logic. Those subjects tend to be a bit too abstract to be the place where you first learn to reason mathematically. (I'm not talking about the rudiments of set theory that are a prerequisite for analysis and algebra.) I think in your shoes I ...


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While this is doubtless too late for the OP, it may help others studying analysis. Lara Alcock, who does research on how people understand abstract mathematics, has recently written a book, How to Think About Analysis. It addresses the original question by providing helpful advice on how to study introductory real analysis and what the common pitfalls are. ...


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Page 505 of Spivak's Calculus (the very final problem in the chapter on "Construction of the real numbers") contains an exercise that begins "This problem outlines a construction of "the high-school student's real numbers." We define a real number to be a pair $(a, \{b_n\})$ where $a$ is an integer and $\{b_n\}$ is a sequence of natural numbers from $0$ to ...


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Although your question was mainly about irrational exponents, let me first answer for non-integer rational exponents, which you also mentioned, since these are the more important pedagogical problem. I can think of two approaches. The first would be to study some problem with exponential growth, such as bacteria. Assume that at $t = 0$ you have one unit ...


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Can't that be said about any advanced degree in a topic that's not directly vocational? Even physicists after a certain point are just expanding society's collective knowledge about the topic. The only difference between very advanced physics and society's perception of it, versus very advanced mathematics and society's perception of it is that physicists ...


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Why learn abstract mathematics ? What is the point ? $\quad$ A century ago, Hardy argued very passionately about the non-utility of abstract mathematics. Number theory, in particular. He was quite proud of this. Now, guess what branch of knowledge is used today overwhelmingly in telecommunications security ? :-) P.S. : Please don't write a follow up ...


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it says Fifty minutes ago it was four times as many minutes past three o'clock IT means X WAS means = FOUR TIMES means 4 * AS MANY MINS PAST 3 means 130-X so equation becomes X=4*(130-X); X=104 mins How many mins past 3 = 130-X=130-104= 26mins example If a Father's age is 2 times the age of Son and Son's age is 30yrs. then the ...


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At about 10 or 11 I discovered that the area of a circle was half the circumference multiplied by the radius.


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The set $\{ f \colon \mathbb{N} \to \mathbb{N} \colon f $ is constant $\}$ is indeed countable, you have the bijection $n \longmapsto f_n$ where $f_n(x) = n$ for all $x \in \mathbb{N}$


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Define $f: \{All constant functions N to N\}->N$ by f(g)=g(0). Note that $f$ is bijective, therefore the set is countable.


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I read 'Unknown Quantity' by John Derbyshire a while back to get a feel for the History of Mathematics and focus on derivation of principles in Algebra.


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Since $f_a$ is a homomorphism, you only need to show differentiability at $0$, for $$\frac{f_a(x+h) - f_a(x)}{h} = f_a(x)\frac{f_a(h)-1}{h}.$$ Since $f_a$ is convex [you need to show that, of course], you know that $$\frac{f_a(h) - 1}{h}$$ is monotonically increasing in $h\in \mathbb{R}\setminus \{0\}$, hence the one-sided derivatives $$D^+f_a(0) = ...


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I suppose that you find Improved Latin Square based Secret Sharing Scheme and Secret Sharing Schemes Arising From Latin Squares not elementary enough, but some technical jargon is inevitable.


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A true arithmetic progression does follow the rule $U_n=U_1+(n-1)r$, so this is not an arithmetic progression. There is no general rule for finding the general term of a sequence. If the sequence is the list of values of a polynomial, taking the difference of successive terms will result in a constant eventually. In your case:$$2\quad 8 \quad 18 \quad 32 ...


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Let's try to acquire general term By writing sequence recursively we have $$\begin{align}a_1&=2\\ a_2&=a_1+(6+4(1-1))=8\\ a_3&=a_2+(6+4(3-2))=18\\ a_4&=a_3+(6+4(4-2))=32\\ \vdots\\ a_n&=a_{n-1}+(6+4(n-2))\\ a_n&=2+\sum_{k=1}^{n-1}{(6+4(k-2))} \end{align}$$ (By summing all steps) $$a_n=2+2(n-1)+4\frac{n(n-1)}{2}$$ ...


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In Hebrew (& in Israel) you always read equations in LTR. There are no exceptions (not even inline equations, as one might expect). RTL math doesn't exist here, so that's just a no, and it would be just as confusing and odd as it would in any other language or place. So basically I'd say that no one will understand you, certainly won't bother to get ...


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I think the best way to discover "obvious" questions is to act as though you're going to present the material to others. Presuming one of a few pedagogical approaches, then preparing the material in this way will force you to anticipate questions in such a way that you'll tend to see these "obvious" questions materialize.


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If you have successfully studied Widder's Advance Calculus book (and don't fool yourself, work the problems to see if you really know the material!) then you have the chops to take two math courses together. But calc II and calc III are the wrong choices, since one leads into vector calculus and the other develops it. (Of course, at schools like MIT, the ...


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I don't know about the educational system in India, but North American graduate schools would be very unlikely to give much consideration to someone who dropped out of university to study at home. I would suggest that you complete your engineering degree while taking as many physics and mathematics courses as possible within that degree program. Also, try ...


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The binomial coefficient $$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$ is the number of ways of selecting $k$ elements from $n$ elements when order does not matter (the number of subsets with $k$ elements in an $n$ element set). The number $n!$, read "$n$ factorial," is defined recursively as follows: $1! = 1$ $n! = n(n - 1)!$ for $n \geq 1$ If you ...


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it means the number of combinations possible when you pick 2 items out of n, that is n(n-1)/(2*1)


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It sounds like you need to run a web scraping script. Python has a pretty useful library called URL that makes writing one short and easy. The URL library allows you to grab and store an HTML file into a single string. Then you just parse what you need out of the string and store it into whatever structure you want to use (list, dictionary, etc.). ...


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Given the equation y=mx + b, we can draw a triangle ABC with the vertical leg length m and the horizontal leg length 1. Next draw triangle ADE with DA perpendicular to AC. ADE is congruent to ABC since angle DAE=angle CAB We then have slope AC= rise/run = m/1 And slope DA = -1/m


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In advance I apologize for my folly. Wolfram Alpha is puzzled :) You can see results here But i think, we can try more simplify, if it is possible, i hope we will succeed. (sorry for my bad English, i'm Russian student). We can make the change of variable: To expand arctan, we need to get a tan of something, but we cant(i hope clearly - why). I tried to ...



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