# Tag Info

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I would recommend the numberphile youtube channel: https://www.youtube.com/user/numberphile He gives short, really good introductions to cool areas of math, some of which stand on their own and others that can be used as a starting point for a class.

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I do not really get what you are asking exactly, but still will try to answer your question. Correct me if I am wrong. First and utmost, "the methods of theoretical physics" is Mathematics. There are a lot of results in physics obtained by means of "mathematical" work, i.e., with mathematical models, pencils and paper, but there are not that many results in ...

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By far the coolest thing about complex numbers is that nature embraces them. Throw a die. Since Newton we know how non-linear dynamics laws are supposed to govern the probabilistic outcomes. Now we know better. We understand nature at a more fundamental level. And at that level we work with complex amplitudes, the 2-norm of which corresponds to observed ...

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Complex numbers can be used to give a very simple and short proof that the sum of the angles of a triangle is 180 degrees. Construct a triangle in the plane whose vertices are complex numbers z1.z2 and z3 in the plane. Recall that the angle $\theta$ at a vertex like $ABC$ is the angular part $\theta$ of $(C-B)/(A-B) = r\exp(i\theta)$. It is written ...

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Even though I have only a single line of hearsay as an application of such, the derivation of primitive/all Pythagorean triples via complex factorisation (see here or here; they might not look brief but really are) always did strike me as both natural and really neat. Can't factor a sum of squares? Surprise, it's a difference after all!

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An Introduction to Abstract Algebra (Pinter) is rigorous but should be accessible to bright high school students who can handle proofs.

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If you're unaware of a substitution, which usually happens when you have'nt attempted a problem of that particular type, the substitution usually seems non-obvious(happened to me a lot when beginning to learn integration), some of the examples which I liked the most are: From here:$$\int \frac{\mathrm{dx}}{x^4[x(x^5-1)]^{1/3}}$$ This one's by ...

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Sets for Mathematics by Lawvere and Rosebrugh is a serious introduction to category theory, intended for high school students.

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The Knot Book, by Colin C Adams is a good one.

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An Introduction to Mathematical Cryptography (Hoffstein, Pipher, Silverman) could fit the bill.

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A circle is two-dimensional. More generally, the least $n$ so that a space can be embedded in $\mathbb{R}^n$ is a reasonable definition for the "dimension" of that space.

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Answers by Christian Blatter often contain A little bit of Physics. This one is a wonderful example: How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$

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Here's a really cool application: Complex Step Differentiation The basic idea is that you normally compute a derivative as: $$f'(x) \approx \frac{f(x+h)-f(x)}{h}$$ This requires evaluating $f$ twice. But what if we use complex numbers? $$f'(x) \approx \frac{f(x+ih)-f(x)}{ih}\approx \frac{\Im{f(x+ih)}}{ih}$$ Now we only need to evaluate $f$ once!* ...

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"There exists, unless I am mistaken, an entire world consisting of the totality of mathematical truths, which is accessible to us only through our intelligence, just as there exists the world of physical realities; each one is independent of us, both of them divinely created, and appear different only because of the weakness of our mind; but, for a more ...

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The Big Bang theory - but we get a good explanation for many observables, e.g. the Hubble constant. Gravitation in general since we don't know fundamentally how it works. We have good models to explain what happens due to gravity, however, e.g. General Relativity. The Mind - We say our perceptions are handled by neural networks. We know how neurons ...

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The other answers nicely cover specific examples of alternating current and wave equations. Basically, wherever you encounter an oscillatory phenomenon of any type, complex numbers are a natural tool to describe them easily and efficiently. I'd like to add a related point here. The relationship between the exponential function and trigonometric functions is ...

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I use the following in my calculus class to find limits of various quotients when the variable approach zero. They are: $\begin{eqnarray}(BIG + small)^r &=& BIG^r + r\ BIG^{r-1} small + \cdots\\ \sin(small) &=& small + \cdots\\ \cos(small) &=& 1 - {1 \over 2}small^2 + \cdots\\ \tan(small) &=& small + \cdots\\ e^{small} ... 2 Since you mentioned "real world". The "real world" consists of miniscule particles: protons, electrons, etc. Which are not exactly particles: quantum mechanics says each of them looks like a wave. Normal waves have some "value" or "displacement" or "magnitude" in each point of space. Magnitude (amplitude) of waves in quantum mechanics are complex! Just ... 0 A nice text explaining a lot of modern physics from the viewpoint of a professional mathematician is Albeverio, Sergio; Høegh-Krohn, Raphael; Fenstad, Jens Erik; Lindstrøm, Tom. Nonstandard methods in stochastic analysis and mathematical physics. Pure and Applied Mathematics, 122. Academic Press, Inc., Orlando, FL, 1986. 3 There is a splitting of the Banach-Tarski theorem into a constructive and a nonconstructive part. The first, which as von Neumann showed is the principle behind many similar problems, is the existence of a noncommutative free group$F$inside the group of congruences of 3-dimensional space. The proof that such a group exists does not use the axiom of ... 0 Whatever you intend by "condensed", the standard mathematical physics book is Reed/Simon: Methods of modern mathematical physics. It is condensed in the sense that it is written concisely, without a lot of fuss, very elegantly written. There are four volumes, in increasing difficulty. It is written for mathematicians and physicists alike, in any case the ... 1 Have a look at The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose. Penrose as you probably know a great mathematician (e.g., Penrose tilings) and a mathematical physicist (e.g., major contributions to relativity). This book is a synthesis of his worldview of the physical world. It uses a great deal of mathematics. So if as a ... 0 Voronoi summation formula:$\sum \limits_{n=1}^{\infty}d(n)(\frac{x}{n})^{1/2}\{Y_1(4\pi \sqrt{nx})+\frac{2}{\pi}K_1(4\pi \sqrt{nx})\}+x \log x +(2 \gamma-1)x +\frac{1}{4}=\sum \limits _{n\leq x}'d(n)4 Complex numbers are very useful for 2D geometric operations. You can rotate a point by multiplying, move it around by addition and subtraction. It gets really interesting when you can do things like project a line segment onto another one by dividing.. 1 Complex numbers can be though of as an algebraic modelisation of groups of geometrical transformations: every direct similarity of the plane can be represented by a function: \begin{align*} f: &\mathbb{C}\rightarrow \mathbb{C} \\ &z\mapsto az+b \end{align*} for somea,b\in\mathbb{C}$. This allows to calculate the centers of composed similarities, ... 19 Compare solving an electrical AC circuit with capacitors and inductors with complex quantities with any other method. Generalizing resistance to impedance makes the whole computation just shuffling around a few numbers (which may or may not have any intuitive meaning), compared to doing essentially the same thing by, e.g., graphical methods. 4 I used to love using complex numbers to derive tangential- and normal components of acceleration. The position function is$s(t) = re^{i \theta}$Differentiating once gives velocity:$v(t) = \dot{r}e^{i\theta} + i \dot{\theta}re^{i\theta} = (\dot{r} + i\dot{\theta}r)e^{i\theta} $Differentiating once more gives acceleration, broken down into tangential ... 3 Since it does not appear that Fractals are mentioned, I would mention those. The pictures on wikipedia are absolutely gorgeous: http://en.wikipedia.org/wiki/Mandelbrot_set Fractals such as the Mandelbrot set are built using the boundedness of an iterative complex function. The explanation of how the set takes on its shape is beyond me as a mechanical ... 20 The solutions to$x^n=1$are$n$evenly spaced points around the unit circle. For example, all 5 values of$x$that satisfy$x^5=1$are found at the points (Image from Wikipedia.) It is results such as this that motivate and justify the 2-dimensional representation of complex numbers by the complex plane. 4 If you look at the progression of number systems, each step was taken because of operations that exposed a fundamental incompleteness in the number system, eg negative numbers were introduced to compensate for incompleteness of the natural numbers under subtraction, rational numbers compensate for lack of divisibility of any integer by another... Complex ... 18 If your high school student has seen Taylor series, the following might work. The Taylor series of$\frac{1}{1-x}$about$x=0$has radius of convergence$1$. This can be explicitly computed, but it also makes sense because of the pole at$x=1$. It may come as a surprise that the$C^\infty$function$\frac{1}{1+x^2}$also has radius of convergence$1$. ... 14 Your problem: factor$x^4 - 2$over the real numbers. Easy solution: Find its complex roots: they are$\sqrt[4]{2}$,$\mathbf{i}\sqrt[4]{2}$,$-\sqrt[4]{2}$,$-\mathbf{i}\sqrt[4]{2}$. Its factorization over the complexes is thus $$(x-\sqrt[4]{2})(x-\mathbf{i}\sqrt[4]{2})(x+\sqrt[4]{2})(x+\mathbf{i}\sqrt[4]{2})$$ Group them into real factors and complex ... 3 Most integrals that cannot be solved with real methods can be solved with complex contours. For example, the integral $$\int_{-\infty}^{\infty}\dfrac{1}{(x^2+1)^2}dx=\frac\pi 2$$ by using contour integration. 18 Every polynomial with real (or complex) coefficients has a complex root. This is the Fundamental Theorem of Algebra. The important fact here is that if we restrict to real numbers the root may not exist; complex numbers are required in general. 5 De Moivre's formula can be used to compute identities like $$\sum_{k=0}^{n-1}\sin(k\cdot\theta) = \Im\left(\frac{\cos(n\cdot\theta)+i\sin(n\cdot\theta) - 1}{\cos(\theta)+i\sin(\theta)-1}\right)$$ etc. I remember using this one in geometry when proving that the sum of the squares of all the diagonals and sides of a regular n-sided polygon inscribed in the ... 0 If the question is “What's behind the Banach-Tarski paradox?” as in the title, then the answer is: The Axiom of Choice. To me the moral of the story is that the Axiom of Choice is not as obvious or intuitive as it seems. 2 Functiones et Approximatio, Commentarii Mathematici is off most of the published lists of journals. Nevertheless it is a serious journal for refereed papers in mathematical research, with a 42-year history of continuous publication. Based in Poznan, Poland, it publishes, in English, papers in analysis and number theory. Among the well-known authors whose ... 29 Complex numbers are necessary for Cardano's formulae to work in all cases of the third degree equation:$x^3+px+q=0$. Setting$\Delta=4p^3+27q^2$, the root(s), if any, are given by $$x=\sqrt[3]{\frac{1}{2}\Biggl(-q -\sqrt{\frac{\Delta}{27}}\Biggr)}+ \sqrt[3]{\frac{1}{2}\Biggl(-q +\sqrt{\frac{\Delta}{27}}\Biggr)}.$$ Now this formula works fine when there ... 1 If you happen to know that$\sin '=\cos$, then that limit is by definition$\sin'(0).$6 Complex numbers are extremely useful in physics, but that may require some more preparation than high school math provides. The complex numbers though are cool for many reasons. For one, it's cool to be able to solve$x^2=-1$, which if you only know of the reals, then you are probably indoctrinated to think no such solution can meaningfully exist. Another ... 63 Using$e^{i\theta} = \cos \theta + i \sin \theta$it is very easy to find (and remember) many trigonometric identities. For example,$e^{i(\alpha+\beta)} = e^{i\alpha}e^{i\beta}gives the sine-of-sums and cosine-of-sums formulas. \begin{align} e^{i(\alpha+\beta)} &= e^{i\alpha}e^{i\beta} \\ \cos(\alpha+\beta) + i \sin(\alpha+\beta) &= (\cos ... 23 Mandelbrot set http://en.wikipedia.org/wiki/Mandelbrot_set should be cool, however rather not easy to describe in a formal way. But if it can be done informally... 6 In my point of view complex analysis is just cool stuff, but perhaps that is somewhat broad. If he knows some analysis: If a complex function is (complex-) differentiable once, it is differentiable infinitely many times! This blew my mind. Or if you know a complex differentiable function on the boundary of a certain area, you can calculate any point of that ... 13 How about the fact thate^{i\pi} +1 = 0?$That's one of the coolest equations around, because: It contains all elementary operations (exponentiation, multiplication and addition) It contains five important constants of mathematics ($e,\pi, i, 1$and$0$) It's short and easy to memorise 4 I would suggest that the axiom at fault is the axiom of infinity. The physical world deals only with finite sets, which is why we never encounter these paradoxical decompositions in practice; they come up automatically any time infinite sets are involved. Almost by definition, an infinite set is one which is equal in size to a proper subset. Geometrically ... 12 There is an implicit axiom that people assume sets of points in the plane should be meaningful geometrically This, of course, isn't an axiom of ZFC or of real analysis; this is an implicit expectation of people who want to apply set theory to geometric problems. Measure theory, one of the more general and powerful tools for defining contexts where we ... 1 Given a closed topological manifold$M$, we can ask whether$M$admits a smooth structure and whether that structure is unique. It's easy to show that it's the case in dimensions$1$and$2$, and it's nontrivial but also true in dimension$3$. It fails in dimension$4$, but there has been a lot of progress on trying to determine which$4\$-manifolds are ...

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It's not particularly famous, but it should be; something almost all mathematics students encounter without realizing it, the definite integration problem. The problem of whether a given indefinite integral has a closed form antiderivitive expressible in elementary functions is solved, in the form of a semi-algorithm, the Risch algorithm. There is no similar ...

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If anything, the non-reality of the Banach-Tarski decomposition is because the sets the theorem speaks about lack continuity that reality does have. Even though the theorem divides the sphere into a finite number of "pieces", those pieces cannot exist as physical objects; the points inside and outside of the set are too finely intermingled to make it ...

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