Which one result in mathematics has surprised you the most?

Question

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!

Answer 1

If a function of a complex variable is once differentiable, it's infinitely differentiable.

Answer 2

$e^{i\pi} +1 = 0$

This still blows my mind.

Answer 3

Maybe this is too obvious, but the fact that the Rationals are countable blew my mind.

Answer 4

The infinite-dimensional sphere is contractible.

Answer 5

Cauchy's Integral Formula.

The fact that the values of an analytic function on the edge of disk (or a simple closed curve) are enough to determine all the values within the curve was very surprising to me.

Answer 6

The fact that you can turn a sphere inside out differentiably.

Answer 7

Brouwer's fixed point theorem, which has several non intuitive consequences in the real-world such as:

The fact that if you lay a piece of paper on your desk and trace around its outline, then crumble/wad the paper up and put it back inside the lines that there will always be a point on the paper exactly above where it started relative to the desk

And, no matter how you stir your coffee there will always be some point in the liquid that ends in the same place that it was before mixing.

Answer 8

Riemann's rearrangement theorem.

This is responsible for the counter-intuitive results of, for example this and this.

Answer 9

I think one of my favorites would be Gödel's incompleteness theorem, which tells us that a consistent formal system containing basic arithmetic cannot prove its own consistency.

Answer 10

Euler's Polyhedral Formula: $\text{vertices} + \text{faces} - \text{edges} = 2$ for convex (more generally, sphere-like) polyhedra.

Euler discovered this about 1750 though the Greeks might well have discovered this fact. The first proof, however, was given by Legendre, using spherical geometry.

Answer 11

The function $f(x)=\begin{cases}e^{-1/x^2} & \text{ if } x\neq 0\\ 0 & \text{ if } x=0 \end{cases}$ has Maclaurin series equal to $0$.

Answer 12

I found the simplicity of Pick's Theorem pretty surprising when I first stumbled across it.

Answer 13

One of the most surprising results in numerical math is Wilkinson's polynomial. Wilkinson gave an example in which a very tiny change to one coefficient of a polynomial can have a drastic impact on the location of the zeros. The change in the location of the roots is seven orders of magnitude larger than the change in the coefficient.

(This is an exact result. The impact of the coefficient change is not due to numerical precision. The point of the example, however, is that changes such as the perturbation of the coefficient are inevitable in numerical computing.)

Answer 14

The Gauss-Bonnet theorem. The integral of the curvature of a manifold, a totally geometric concept and one that looks dependent on the embedding, is equal to $2\pi$ times its Euler characteristic, an algebraic homotopy invariant.

Answer 15

A beautiful result by Erdős:

In any sequence of distinct $n^2 + 1$ integers, there is always some increasing or decreasing subsequence of length $n + 1$.

Pigeonhole!

Answer 16

Exotic Spheres. Kervaire and Milnor's proof that there exists 27 distinct differentiable $7$-manifolds that are homeomorphic, but not diffeomorphic, to the standard $7$-sphere (giving $28$ differentiable structures for $S^{7}$).

Answer 17

The nine point circle.

Three sets of three points, each of which obviously determines a circle. That these three constructions always give the same circle!?

Answer 18

PRIMES is in P. This was surprising to me both because I knew it as an open problem before it was proved, and because the proof is simple enough that I can follow the outline and understand some of the details. The proof of FLT was not as surprising to me because comprehending it seems to require a lot of background that I don't have.

Answer 19

The one result that puzzled me most is from the ACM's communication ... "Puzzled" section by Peter Winkler: "We are in a large rectangular room with mirrored walls, while elsewhere in the same room is our mortal enemy, armed with a laser gun. Our only defense is our ability to summon graduate students to stand at designated spots in the room blocking all possible shots by the enemy. How many students would we need, assuming for the purposes of the problem that we, our enemy, and the students are all thin enough to be considered points?" The answer is 16. I still didn't do the calculation end-to-end, though a buddy of mine did it and got the result. What i find most puzzling is that the trajectory may turn dense, but still has 16 such points.

Answer 20

In class of number theory, identities of Ramanujan(continued fractions).

For example:

If $\alpha, \beta >0$ with $\alpha\beta=\frac{1}{5}$, then:

$\left\{ \left(\displaystyle\frac{\sqrt{5}+1}{2} \right)^{5}+ \displaystyle\frac{e^{-\frac{2\pi\alpha}{5}}}{1+\displaystyle\frac{e^{-2\pi\alpha}}{1+\displaystyle\frac{e^{-4\pi\alpha}}{1+...}}}\right\}\cdot\left\{ \left(\displaystyle\frac{\sqrt{5}+1}{2} \right)^{5}+ \displaystyle\frac{e^{-\frac{2\pi\beta}{5}}}{1+\displaystyle\frac{e^{-2\pi\beta}}{1+\displaystyle\frac{e^{-4\pi\beta}}{1+...}}}\right\} = 5\sqrt{5}\left(\displaystyle\frac{\sqrt{5}+1}{2} \right)^{5}$

Beautiful result!!!

Answer 21

Picard’s Great Theorem: In every neighborhood of an essential singularity of an analytic function, the function takes on every value, with at most one exception.

Answer 22

Rather basic, but it was surprising for me:

For any matrix, column rank = row rank.

Answer 23

Fermat's "two square theorem".

G.H. Hardy's A Mathematician's Apology is a book everyone should read, but for those who haven't here's something Hardy mentions that is rather surprising:

(If we ignore 2) All primes fit into two classes: those that leave remainder $1$ when divided by $4$ and those that leave remainder $3$.

This much is obvious. The surprising thing is that all of the first class, and none of the second can be expressed as the sum of two integer squares.

That is, for all prime $p$, if $p = 1 \mod 4$ then there exist $x,y$ integers such that $p = x^2 +y^2$ and if $p = 3 \mod 4$ there exists no such $x,y$

Answer 24

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

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

Answer 25

I recall vividly the moment I learnt of Thomae's function, which is continuous at all irrational numbers and discontinuous at all rational numbers.

Answer 26

Oh, I've been surprised a lot of times, but a particularly memorable one for me was learning the maximum modulus principle of complex analysis.

On the numerics front, I still find it amazing that the humble trapezoidal rule is the best one to use for integrating periodic functions over a period, better than Simpson's rule or the other fancier quadrature methods. This can be seen by appealing to Euler-Maclaurin.

Answer 27

There exists a non-reflexive Banach space that is isomorphic to its dual.

Answer 28

1) Exotic structures on $\mathbb{R}^4$ probably puzzles anyone learning about differentiable topology. Even more, the fact that it is only for $n=4$ is quite remarkable.

2)$S^n$ not being a Lie group for all $n$ ...

Answer 29

Erdős's Probabilistic Method because it is so elegant.

Answer 30

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.

Answer 31

Fractals, especially the ones related to simple dynamical processes like the Mandelbrot set or this eerie Burning ship fractal really still inspire me with awe.

---

It's not really a mathematical result, but after seeing all the nice entries here, I thought this lighter one would fit in well:

"Young man, in mathematics you don't understand things. You just get used to them."

When I see all the examples here, this dictum by von Neumann comes to mind. I'm always remembered of how true it is.

Answer 32

The Thom-Pontrjagin theorem: $\Omega_n^{SO} \cong \pi_n(MSO)$. The group of equivalence classes of n-manifolds with respect to oriented cobordism is isomorphic to the n-th homotopy group of the Thom spectrum MSO. This can be generalized to include different cobordisms (unoriented, ...) and different Thom spectra. See for example the minor thesis by T. Weston.

Answer 33

The solution to Hilbert's 10th problem, i.e. the MRDP theorem.

Number theorists were trying to find a general method to solve Diophantine equations. Special cases of the Diophantine equations were/are studied extensively and the theorems are quite nice. Learning the fascinating fact that there is no general method (algorithm) to solve arbitrary Diophantine equations was surprising for me.

Answer 34

One of the most surprising results I have ever seen is the Universality Theorem of Voronin which states that any nonvanishing analytic function can be well -approximated by $\zeta(s)$ somewhere in the critical strip for $0 < Re(s) < 1$.

Answer 35

As an undergraduate, the fact that |P(x)| > |X|. I recall being surprised at how both how short and easy this was to prove, and that it implied there were infinitely many "sizes" of infinity. (The standard diagonalization of decimals proof only showed there were two sizes and took more time.)

Answer 36

That one can count on and on without end.

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

Answer 37

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

Answer 38

When young, that there exist dense sets with zero measure and smaller cardinality, and later that there exist nowhere dense sets with positive measure.

Answer 39

When I started studying elliptic curves and modular forms I was really amazed by the fact that for a normalized eigenform the Fourier coefficients are the Hecke eigenvalues.

Answer 40

Gold's theorem provides pretty convincing mathematical evidence supporting the universal grammar hypothesis in linguistics. This hypothesis is two-fold: (1) children are not presented logically with enough information to actually learn their native language; (2) hence there exists a universal grammar which is encoded somehow in the human brain and which facilitates the logical gap between the positive data given to the child and the data necessary to determine the language's grammar. While the universal grammar hypothesis isn't universally accepted, it has been one of the most important ideas in linguistics so far.

Gold's theorem shows that certain classes of languages are logically not learnable. Of course, it operates in a purely formal setting. I'll provide up this setting now following the definitions and notations of Gabriel Carroll, pg. 41.

Start with a finite alphabet $\Sigma$ and let $\Sigma^*$ designate the set of finite sequences of elements of $\Sigma$. A language $L$ is a subset of $\Sigma^*$. A text of $L$ is an infinite string $w_1, w_2, \dots$ of elements of $L$ such that every element of $L$ occurs at least once. A learner for a class $\mathcal{L}$ of languages is a function $\Lambda : (\Sigma^*)^* \rightarrow \mathcal{L}$ that intuitively takes a sequence of strings of $\Sigma$ and guesses the language in $\mathcal{L}$ in which all these strings are grammatically correct. The learner $\Lambda$ learns the language $L \in \mathcal{L}$ if for every text $w_1,w_2,\dots$ of $L$ there exists a natural number $N$ such that $\Lambda(w_1,w_2,\dots,w_n) = L$ for $n \geq N$. The learner $\Lambda$ learns the class $\mathcal{L}$ if it learns each language in $\mathcal{L}$, and the class $\mathcal{L}$ is learnable if there exists a learner which learns it.

This is Gold's theorem, first proved by Gold in his seminal paper (but my wording is taken from Carroll):

• If the class $\mathcal{L}$ contains all finite languages and at least one infinite language, then $\mathcal{L}$ is not learnable.

In particular, any finite language is regular. Hence the class of regular languages is unlearnable, and it follows at once that every class of the Chomsky Hierarchy is unlearnable.

The proof of Gold's theorem is, as Carroll shows, not very hard, although certainly not intuitive, and it can be reduced to a corollary of the following characterization of learnable classes of languages (Carroll, Lemma 9):

• A countable class $\mathcal{L}$ of nonempty languages is learnable if and only if, for each $L \in \mathcal{L}$, there is a finite ''telltale'' subset $T \subseteq L$ such that $L$ is minimal in $\{L' \in \mathcal{L} : T \subseteq L'\}$.

Answer 41

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.

Answer 42

The fact that the axiom of foundations (also known as regularity in some places) is independent of the rest of the axioms, and not only you can have an infinitely descending chain $x_0 \ni x_1 \ni \ldots$ but you can have $x = \{ x \}$ and even more than that! $P(a) \in a$ is possible as well.

Crazy set theoretic voodoo, that's what this is! (And I'm loving every single bit of it :))

Answer 43

I was very surprised when I discovered that $$0.\overline{9} = 1$$

Answer 44

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.)

Answer 45

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

Answer 46

that properties of recursive function are not always provable. For example, existence of an algorithm which non-terminates and whose non-termination cannot be proved.

Answer 47

I would not rate this example as surprising, but it did provoke in me a little epiphany when I finally understood it. There is a theorem of category theory that characterizes adjunctions as a pair of functors and a pair of natural transformations satisfying a bunch of equations. Now in some sense, this is a pure formality (the proof is easy), but on the other hand, an adjunction encodes a parameterized universal property, with some implicit quantifiers (over potentially proper classes) floating around. Now think of all the adjunctions you have come across that encode huge amounts of information. The characterization theorem says that this is the same as a pair of 2-cells in a 2-category satisfying a pair of equations. Look, Ma, no quantifiers, no isomorphisms, no nothing. Just a bunch of equations in a 2-category. The single most important concept of category theory and what do we end up with? a pair of equations...

Answer 48

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

Answer 49

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.

Answer 50

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

It wasn't solved correctly until 2009. https://erich-friedman.github.io/packing/squcotri/

Answer 51

For me it would be the Green-Tao theorem, which states: For any natural number $k$, there exist $k$-term arithmetic progressions of primes.

Answer 52

Kuratowski's Complement problem, is the one which i came across recently, and i was clearly flabbergasted.

Answer 53

I was very surprised to learn about the Cantor set, and all of its amazing properties. The first one I learnt is that it is uncountable (I would never have told), and that it has measure zero.

I was shown this example as a freshman undergraduate, for an example of a function that is Riemann-integrable but whose set of points of discontinuity is uncountable. (equivalently, that this set has measure zero). This came more as a shock to me, since I had already studied some basic integrals in high school, and we had defined the integral only for continuous functions.

Later, after learning topology and when learning measure theory, I was extremely shocked to see that this set can be modified to a residual set of measure zero! I think the existence of such sets and the disconnectednes of topology and measure still gives me the creeps...

Answer 54

The primitive element theorem is quite surprising.

Theorem: Let $E \supseteq F$ be a finite degree separable extension. Then $E=F[\alpha]$ for some $\alpha \in E$.

Answer 55

There exists $f\colon \mathbb{N}\times\mathbb{N}\to\mathbb{N}$ which is bijective.

Answer 56

While not as surprising as, say, the countability of the rationals, and even fairly obvious to some people, the fundamental theorem of calculus joins two operations (differentiation and integration) which didn't look completely related to each other at first to me if you define them as the rate of change of a curve and the area beneath it.

Answer 57

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.)

Answer 58

The divergence of the Harmonic Series.

Answer 59

(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.

Answer 60

Apart from the above sets of interesting and surprising results, this one should also be mentioned.

Gaussian quadrature rule:

An $n$ point Gaussian quadrature rule, yields an exact result for polynomials of degree $2n − 1$ or less.

Answer 61

The use of compactness to show existence of solutions to differential equations. It was not surprising as in unexpected, but in the sense that it opened so much possibility almost unreal. It felt like I was given an amazing super toy that can never be destructed.

Answer 62

When I was a freshman at the University of Rome I took a course in Geometry in which some projective geometry was covered. I was really amazed by two facts:

1. a Moebius strip sits inside the real projective plane.

2. the group of rotations of the real plane fixes two (complex) lines and all the circumferences pass through the points at infinity of these two lines (the socalled cyclic points). A few years later I realized that I was the only grad student in David Eisenbud's class at Brandeis to know this (actually simple) fact!

Answer 63

The connection between syntax and model theory. For example, you can tell that you can't define "field" (the algebraic structure) by equations because the category of fields doesn't have products. In other words, a property of the models controls the logical connectives you must use to say what it is. There are many results like this.

Answer 64

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

Answer 65

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!

Answer 66

To me the Cayley-Salmon theorem is an example of a result that still strikes me as rather surprising.

Theorem(Cayley-Salmon)

A smooth cubic surface $\mathcal{S} \subset \mathbb{P}^{3}_{k}$ contains exactly 27 lines, where $k$ is an algebraically closed field.

Here is a link to some history about it. There's a really nice treatment of this result in chapter 5 of Klaus Hulek's book Elementary Algebraic Geometry.

Answer 67

Fintushel and Stern's construction of exotic K3's by surgery on torus fibered knots in S^3. If the Alexander polynomial of the knot is not monic then the smooth structure doesn't admit a symplectic structure.

http://arxiv.org/pdf/dg-ga/9612014.pdf

It's also very beautifully explained in the last chapter of Scorpan's "The Wild World of 4-manifolds." The entire construction is available in the Google books preview.

Answer 68

Bounded holomorphic function is constant; integration of a meromorphic function.

Answer 69

I got really struck by duality, when my professor lectured about it the first time. I think that even though the algebraic concept is easy to understand, to think that there exists a space such that all inclusions are switched always had a special place in my mind.

Answer 70

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.

Answer 71

The consequences of busy beaver problem are really suprising. For every conjecture/hypothesis about countable number of cases if we can write a program that can verify whether this conjecture holds for some case then we need to check only FINITE number of cases to prove that this conjecture is true for every case.

Answer 72

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

Answer 73

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.

Answer 74

Aside from some results I found amazing that have already been mentionned, Lagrange's Theorem in group theory is one that amazed me for some time.

For those who don't know about it, it tells us that the order of any subgroup of a group $G$ divides the order of $G$.

Answer 75

I remember a homework question in elementary school. Something like this: Billy and Jane's house is x blocks east and y blocks north of school. Billy walks home by walking east for x blocks and then north for y blocks. Jane decides to take a short cut: she walks alternately a block north and a block east. (There is a picture: Jane's route is a step-like hypotenuse.) Is Jane's route really a short cut?

Of course it is exactly the same distance, but I found this really hard to digest. I knew that in the triangle the sum of the two sides would exceed the hypotenuse.

Answer 76

If $G$ is a (Hausdorff, locally compact) totally disconnected abelian group which is a filtered union of its compact open subgroups (e.g. the additive group of $\mathbb{Q}_p$), then the category of smooth complex representations of $G$ (smooth meaning the action map is continuous when the vector space has the discrete topology) is canonically equivalent to the category of sheaves of complex vector spaces on the Pontryagin dual of $G$.

This is a beautiful example of an algebro-geometric duality in representation theory and quite shocking to me. The situation is sort of analogous to the equivalence, for a commutative ring $R$, of $R$-modules with quasi-coherent sheaves on $\text{Spec} \ R$.

Answer 77

Supersymmetry. From a mathematical standpoint it implies that every bosonic (resp. fermionic) particle in the universe has a fermionic (resp. bosonic) superpartner, i.e., the existence of two physical bijections.

Answer 78

Another example from the numerics front: it's surprising that despite the theoretical fact that Gaussian elimination can be unstable (even with pivoting!), examples that trigger this instability are in fact very rare in practice, and can be handled by a simple fix if they do arise.

Answer 79

Similar to Thomae's function, I was impressed by the Dirichlet function, which is not only discontinuous everywhere, but impossible to plot. The function is defined as:

$f(x)=\begin{cases} 1 \mbox{ if } x\in\mathbb{Q} \\ 0 \mbox{ if }x\notin\mathbb{Q} \end{cases}$

Answer 80

All eigenvalues of a Hermitian matrix A are real. No immediate intuition as to why it must be true. If we think of the Riemann hypothesis "All non-trivial zeros of the Riemann zeta function are of the form $ \frac{1}{2}+zi $ where z is real" and try to think of Hermitian matrices having real eigenvalues in equal awe, the wonderment increases. The two ideas are also related.

Answer 81

Dirichlet’s Theorem: Every proper arithmetic sequence contains a prime.

Answer 82

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!

Answer 83

How the "inverse" of area under a curve is the slope.

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

Answer 84

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).

Answer 85

The extension principle of fuzzy subset theory. More generally, the extension of several parts of classical mathematics to an infinite-valued context.

Answer 86

I have had two results in the last year that surprised me.

One appeals to my intuition in physics, although it may seem more obvious to those more versed mathematics.

The set of pauli matrices (with identity) when multiplied by $i$ form the group of unitary quaternions (under matrix multiplication).

The was surprising to me how you can connect such a physical concept as electron spin to an abstract algebraic concept. Its what led me to dive into group and representation theory as it applies to physics. Now I understand that we can consider spin symmetry as SU(2) , which is injective into SO(3) the group of symmetry of $R^3$ of which the quaternions can be thought of as a representation.

The second result, which I had to prove in an algebra assignment:

For a field $F$ of characteristic $p$ where $p$ is prime $(x+y)^p = x^p + y^p$ for $x,y \in F$. Lovely little result that spits in the face of everything that our grade school teachers taught us in algebra.

Answer 87

Mamikon’s Theorem: The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the original shape of the curve.

This theorem allows you to, among other things, easily obtain results that were obtained before only with difficulty, such as the area under one arch of a cycloid. This theorem is the basis of what has come to be known as Visual Calculus. Here’s the link to Tom Apostol’s account of this awesome insight:

http://eands.caltech.edu/articles/Apostol%20Feature.pdf

Answer 88

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.

Answer 89

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)

Answer 90

Kind of simple, but I find it really counterintuitive:

A strictly increasing function can have zero derivative.

Answer 91

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