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A large part of my fascination in mathematics is because of some very surprising results that I have seen there.

I remember one I found very hard to swallow when I first encountered it, was what is known as the Banach Tarski Paradox. It states that you can separate a ball $x^2+y^2+z^2 \le 1$ into finitely many disjoint parts, rotate and translate them and rejoin (by taking disjoint union), and you end up with exactly two complete balls of the same radius!

So I ask you which are your most surprising moments in maths?

  • Chances are you will have more than one. May I request post multiple answers in that case, so the voting system will bring the ones most people think as surprising up. Thanks!
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big-list usually means community wiki. For this question it applies. –  Aryabhata Aug 21 '10 at 19:01
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And maybe also mathoverflow.net/questions/18100/… . –  Qiaochu Yuan Aug 21 '10 at 21:21
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I'm getting tired of this question being bumped every once in a while. It seems to have served its purpose and there's no need to accumulate more than 100 answers. Therefore I voted to close it. –  t.b. Sep 5 '11 at 22:09
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closed as too localized by t.b., Zev Chonoles Sep 5 '11 at 22:18

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91 Answers

Rather basic, but it was surprising for me:

For any matrix, column rank = row rank.

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Computational instability of the Quadratic Formula. Who would have thought?

Due to this computational stability an alternative formula is also employed. Here is the relevant quote from the Wikipedia article:

“The alternative formula can reduce loss of precision in the numerical evaluation of the roots, which may be a problem if one of the roots is much smaller than the other in absolute magnitude.”

And here is the link to the full article:

http://en.wikipedia.org/wiki/Quadratic_equation

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Erdős's Probabilistic Method because it is so elegant.

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If I recall correctly, it wasn't Erdos who first came up with it. –  Aryabhata Nov 18 '10 at 23:27
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How about the Anti-Calculus Proposition (Erdős): Suppose $f$ is analytic throughout a closed disc and assumes its maximum modulus at the boundary point $a$. Then $f^\prime(a)$ is not equal to $0$ unless $f$ is constant. (Source: Bak & Newman, Complex Analysis 2nd ed.)

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Henry Ernest Dudeney's Spider and Fly Problem: With a cuboid $30\times12\times12$, what is the minimum surface distance from a point which is on a $12\times 12$ face and in $1$ from the mid-point of an edge to the opposite point across the centre of the cuboid?

The surprise is that the minimum distance requires a route using five of the six faces of the cuboid. enter image description here

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The Feit-Thompson theorem, although I don't know yet how to prove it. It was impressive for me that only from the condition of having an odd order a group would be solvable.

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The fact that surface area is simply the derivative of volume!

Bear in mind that in mathematics, “volume” is a generic term (i.e., a convenient handle, no pun intended) for a “container” of arbitrary dimension. So, this holds not only for dimension 3 (i.e., a sphere), but also for dimension 2 (i.e., a circle, a circle being a 2-sphere, and perimeter of a 2-dimensional object corresponding to the surface area of a 3-dimensional object), and for dimension 1 (i.e., an interval, an interval being a 1-sphere, and the set of endpoints of a 1-dimensional object corresponding to the surface of a 3-dimensional object). So, we have:

Dimension 3: d(4/3)πr^3/dr = 4πr^2

Dimension 2: dπr^2/dr = 2πr

Dimension 1: d2r/dr = 2

Beautiful!

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$r/{\mathrm{d}r}=1$ does not make sense to me. –  Rasmus Jun 10 '11 at 9:11
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I was very surprised when I discovered that $$0.\overline{9} = 1$$

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Why would you write $0.\overline{99}$ instead of $0.\overline{9}$? –  Rasmus Jun 10 '11 at 9:09
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The extension principle of fuzzy subset theory. More generally, the extension of several parts of classical mathematics to an infinite-valued context.

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(Mazur) If $E$ is an elliptic curve over $\mathbf{Q}$ then the torsion subgroup of $E(\mathbf{Q})$ is one of

$\mathbf{Z}/N\mathbf{Z}$ for N=1,2,3,4,5,6,7,8,9,10,12

or

$\mathbf{Z}/2\mathbf{Z} \times \mathbf{Z}/N\mathbf{Z}$ for N=1,2,3,4

I find it very surprising that there are so few possibilities for the rational torsion on an elliptic curve. It's also strange to see every number from 1 through 12 except 11 in that first list.

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Kind of simple, but I find it really counterintuitive:

A strictly increasing function can have zero derivative.

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A strictly increasing function can have zero derivative almost everywhere. –  Jonas Meyer Sep 5 '11 at 20:25
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The divergence of the Harmonic Series.

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Almost everything I've seen so far, but especially that the area under Gaussian curve converges to the square root of the ratio of circumference of the circle to its diameter. This result is old and well-known, but these two things seem so unrelated that I still find it amazing!

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The fact that any known first order property of $\mathbb{C}$ in the language of rings is valid in any algebraically closed field.

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Well, of characteristic 0... –  Harry Altman Jun 10 '11 at 8:29
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This one goes hand in hand with the enumerability of $\mathbb{Q}$

The fact that though most of the real numbers are transcendental, it is extremely difficult to find one. (excluding some slight modification of the already known ones)

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I wonder what is the exact meaning of it is extremely difficult to find one (excluding some slight modification of the already known ones). –  Did Jan 7 '13 at 10:34
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Wedderburn’s Theorem: Every finite division ring is a field.

Why should finiteness imply commutativity???

(Background: The only way a division ring can fail to be a field is if its multiplication is not commutative.)

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Mike: you have indeed many "one result that surprised you the most" :) –  t.b. May 21 '11 at 1:12
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At different times:) - Remember the one about being able to count on and on and on:) –  Mike Jones May 21 '11 at 7:33
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The Chinese Magic Square:

816

357

492

It adds up to 15 in every direction! Awesome! And the Chinese evidently thought so too, since they incorporated it into their religious writings.

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This is not a very specific answer but I was struck with awe when I saw the entanglement between partial differential equations and stochastic processes.

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The fact that the sum of the first n odd numbers is n squared. Also, not just this fact, but the nice visual “wrapping” proof of it (as opposed to the induction proof).

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Skolem’s Paradox:

From the Wikipedia article: “Skolem’s paradox is that every countable axiomatisatin of set theory in first-order logic, if it is consistent, has a model that is countable.”

Here’s the link:

http://en.wikipedia.org/wiki/Skolem%27s_paradox

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The fact the every set can be well-ordered (given the Axiom of Choice, of course).

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One surprise for me -- What is the optimal way to cover an equilateral triangle with two squares?

It wasn't solved correctly until 2009. http://www2.stetson.edu/~efriedma/squcotri/

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It seems weirdly arbitrary to me that you can comb a hairy n-sphere if n is odd, but that this is impossible when n is even.

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But seems to be similar to the fact that a polynomial with real coefficients alwasys has a real root if the degree of the polynomial is odd, but there is no such guarantee if the degree is even:-) –  Mike Jones May 21 '11 at 0:57
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Morley’s Miracle: The three points of intersection of the adjacent trisectors of the angles of any triangle always form an equilateral triangle.

This is a stunning gem that slipped through the fingers of the ancients.

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I absolutely was shocked when I learned about the exact formula for the number of partitions of an arbitrary natural number. This formula is amazing for so many reasons, including not only the simple fact that it exists at all, but also that it is so intimidatingly complicated, in the typical style of a result of Ramanujan's.

$p(n)=\frac{1}{\pi \sqrt{2}} \sum_{k=1}^\infty \sqrt{k}\, A_k(n)\, \frac{d}{dn} \left( \frac {1} {\sqrt{n-\frac{1}{24}}} \sinh \left[ \frac{\pi}{k} \sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}\right] \right) $

where

$A_k(n) = \sum_{0 \,\le\, m \,<\, k; \; (m,\, k) \,=\, 1} e^{ \pi i \left[ s(m,\, k) \;-\; \frac{1}{k} 2 nm \right] }.$

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The formula you have provided is actually an improvement of Ramanujan's work, not a result from Ramanujan himself. "In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan's results by providing a convergent series expression for p(n). It is[. . .]" Ramanujan's result (with the help of Hardy) was the asymptotic expression: $$p(n) \sim \frac {1} {4n\sqrt{3}} e^{\pi \sqrt {\frac{2n}{3}}} \mbox { as } n\rightarrow \infty.$$ –  000 Jan 15 '12 at 23:33
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How the "inverse" of area under a curve is the slope.

(I mean: $\frac{d}{dx} \int x {\mathrm dx} = x$)

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The fact that the curve of fastest descent (i.e., the brachistochrone) dips beneath its target!

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The infinitude of primes! – and the simplicity of its proof!

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If it weren't for me stumbling at this result while browsing wikipedia, I doubt that I would even have math.SE account right now. This result and its proof showed me that there is a world of elegance and beauty in mathematics, contrary to what I gathered from HS mathematics classes. –  user5501 May 21 '11 at 15:24
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That one can count on and on without end.

(Of course, this surprise was a while ago.)

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