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I'm a teacher in mathematics at a high school. Math has fascinated me for almost my entire life, so I would like to bring that enthusiasm to my students with beautiful yet easy to understand proofs or demonstrations. It's meant for students who are in their last grade of high school and will be going to university next year.

So what are simple proofs or visual examples that made you love math? The more examples the better! Answers with pictures would be even better!

Thanks in advance!

P.S. Things that I did already teach my students the basics of are: complex numbers, probability theorem, prime numbers, vectors, functions of more variables, a little bit about group theory, set theory. These are all things that I tried to mix with the things they should actually know for their exams. It's meant to give them an idea of what math is really about, not just repeating formulas.

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closed as too broad by BlueRaja - Danny Pflughoeft, Grigory M, muaddib, user223391, Chris Brooks Jun 29 '15 at 1:00

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Assuming that this is done after them finishing calculus, looking over the Weirstrass function and proving that it's continuous everywhere but differentiable nowhere was very interesting to me. en.wikipedia.org/wiki/Weierstrass_function $\endgroup$ – NoseKnowsAll Jun 24 '15 at 15:31
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    $\begingroup$ I like combinatorial ideas with pictures: Sum of first $n$ (positive) odd integers is $n^2$; the connection between binomial coefficients, Pascal's triangle, and path-counting; those sorts of things. $\endgroup$ – pjs36 Jun 24 '15 at 15:35
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    $\begingroup$ It's a really small thing, but it marked me when I was in higschool, because it showed that sometimes, having another formalism simplify everything : proving that $\lbrace a^2+b^2 \; | \; a,b \in \mathbb{Z} \rbrace$ is stable by mulitplication with complex numbers. $\endgroup$ – Sylvain L. Jun 24 '15 at 15:36
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    $\begingroup$ Euclids proof of infinitude of primes is nice. $\endgroup$ – mathreadler Jun 24 '15 at 15:50
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    $\begingroup$ Is it better on matheducators.stackexchange.com ? $\endgroup$ – Vi0 Jun 24 '15 at 16:59

34 Answers 34

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The proof of Thales' Theorem (that any triangle constructed using the diameter chord of a circle and a third point on the circle that doesn't coincide with the endpoints of the diameter is a right triangle) is a pretty nice one:

enter image description here

The triangle $\triangle OAB$ is an isoceles triangle because $OA$ and $OB$ are both radii of the circle and thus by definition must have equal length. That means that $\angle OAB = \angle OBA$, the measure of which we identify as $\alpha$. The same is true for $\triangle OBC$ and for the same reasons; the congruent angles are $\angle OBC = \angle OCB = \beta$.

Now, we know that the sum of all interior angles of a triangle is 180 degrees. The proof of that is also pretty intuitive in visual form:

enter image description here

We also can see by inspection that $\angle ACB = \alpha + \beta$. Therefore, the sum of the interior angles is $2\alpha + 2\beta = 180$, and when simplified by dividing everything by 2, $\alpha + \beta = 90$, and thus the third angle of a triangle of this construction is always 90 degrees, making the triangle a right triangle.

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  • $\begingroup$ Much easier proved by the inscribed angle theorem (the angle subtans a arc of 180 deg), os its measure is 180/2=90 deg. $\endgroup$ – Cyclohexanol. Jun 28 '15 at 18:08
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The fact that we can give finite result even to divergent sums like for examples the sum of all natural numbers:

$$\sum_{n\in \Bbb N}=1+2+3+4+\dots =-\frac 1{12}$$

This fact really astonished me and continues to have a sort of fashion on me :)

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    $\begingroup$ Oh yes I know that. I think my students will call me crazy if I try to tell them that XD $\endgroup$ – Peter Jun 25 '15 at 13:19
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    $\begingroup$ I dislike this because it's always presented as, "If $\sum_{k} (-1)^{k + 1} k = s$, then $\sum_{k} k = - \frac{1}{3} s$." But in reality, these types of arguments are hand-waving and almost assuming the conclusion. In calculus, $\sum_{k} a_{k} = \lim_{N \to \infty} a_{1} + \cdots a_{N}$, while $\sigma_{k} k$ is an entirely different thing. It's fun, but honest presentation requires explaining that we're not "really" summing. Banach limits might be cool, though. Almost-convergent sequences are interesting when you're explicit about what you're assuming, and let you solve $\sum_{k} (-1)^{k}$. $\endgroup$ – AJY Jun 25 '15 at 18:40
  • $\begingroup$ How did you reach the conclusion that $\sum_{n\in \Bbb N}=1+2+3+4+\dots$ equals $-1/12$? (I think it requires some explanation) Let's say that it does, what would that knowledge be useful for? $\endgroup$ – HelloGoodbye Jun 26 '15 at 8:14
  • $\begingroup$ @HelloGoodbye It comes from the analytic continuation of the Riemann Zeta function, this and other similar result can be obtained with summation method as Cesaro's or Ramanujan' s. This has application in number theory and, surprisingly in theoretical physic. $\endgroup$ – Renato Faraone Jun 26 '15 at 8:22
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    $\begingroup$ The question was about simple proofs or visual examples. Now which one of those is this?? $\endgroup$ – Marc van Leeuwen Jun 26 '15 at 10:05
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guess the proof

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

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How about Galileo's paradox?

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

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  • $\begingroup$ Hi I've noticed a few down votes which I'm a bit confused about. If someone could explain errors in the answer or why the answer is completely wrong for the question I would very much appreciate the advice. Thanks :) $\endgroup$ – J-S Jun 28 '15 at 21:09

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