What's your favourite result in mathematics?

[Sherlock]

It can be a theorem, a definition, a proof, an identity, a trick/technique/method etc.

 

[Plato]

It can be a theorem, a definition, a proof, an identity, a trick/technique/method etc.

That is an easy one for me: the Russel/Whitehead proof that $1+1=2$.

 

[Chris L T521]

That is an easy one for me: the Russel/Whitehead proof that $1+1=2$.

All 379 pages of it. XD

 

[ThePerfectHacker]

It can be a theorem, a definition, a proof, an identity, a trick/technique/method etc.

Without even spending one second thinking about this, I immediately thought the "pigeonhole principle".
It is amazing how such a simple observation can be so powerful.

 

[CaptainBlack]

It can be a theorem, a definition, a proof, an identity, a trick/technique/method etc.

Cantor's diagonal slash

CB

 

[ThePerfectHacker]

Cantor's diagonal slash

That is a very good choice. I am surprised I did not think of it.

It is quite amazing how often this argument comes up. For example, if $X_n$ are compact topological spaces then $ \prod X_n $ is also a compact topological space, this is the Tychonoff theorem. However, we can keep things more elementary by assuming that $X_n$ are compact metric spaces. Then there is a way to define a metric on $\prod X_n$. The proof of Tychonoff theorem is not so simple. However, the proof of this special case involving countable metric spaces is much simpler. I bring this fact up because the proof of this simpler statement uses a Cantorian kind of an argument. So it is pretty cool that the diagnol process is buried into this proof as well. The funny thing about what you mentioned is that I think Cantor's argument is more of a tool in analysis now than in set theory.

 

[sbhatnagar]

It can be a theorem, a definition, a proof, an identity, a trick/technique/method etc.

My favorite result in mathematics is the Basel Sum.

\( \displaystyle \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots= \sum_{n=1}^{\infty}\frac{1}{n^2}=\zeta(2)=\frac{ \pi^2}{6}\)

 

[AlexYoucis]

The uniformization theorem still blows my mind.

 

[Sherlock]

I think mine would be the Dedekind construction of the real numbers.

 

[Fernando Revilla]

It can be a theorem, a definition, a proof, an identity, a trick/technique/method etc.

Mine has not come out yet.

 

[soroban]

$$\begin{array}{c}\text{What is the next equation?} \\ 3^2 + 4^2 \:=\:5^2 \\ 3^3 + 4^3 + 5^3 \:=\:6^3 \\ \vdots \end{array}$$

 

[Sherlock]

Mine has not come out yet.

This quite simply cannot be true!

 

[Random Variable]

$\displaystyle e^{i \pi} = - 1 $

 

[Ackbach]

$\displaystyle e^{i \pi} = - 1 $

Even better is $e^{i\pi}+1=0$, because then you've also got the additive identity involved in the equation.

 

[mvCristi]

Goedel's work on the relation of consitency-completeness.
Russel's paradox

 

[Also sprach Zarathustra]

[h=1]Parallel postulate-Euclid's fifth axiom.[/h]

 

[Random Variable]

Parallel postulate-Euclid's fifth axiom.

You must really hate hyperbolic and elliptical geometries.

 

[agentmulder]

If $ax^2 = bx + c$ then $x = \frac {b \pm\ \sqrt {b^2 + 4ac}}{2a}$

 

[nimon]

Cauchy's proof of the AM-GM inequality.

 

[Sherlock]

Cauchy's proof of the AM-GM inequality.

Oh, I was reading that just the other day in Linear Analysis: An Introductory Course -Béla Bollobás. It's very clever!

 

[grgrsanjay]

$a^2 + b^2 = c^2$

if a and b are the lengths of the two short sides of a right triangle and c is its long side then this formula holds. Conversely, if the formula holds then a triangle whose sides have length a, b and c is a right triangle.

This formula is about 2,350 years old , that's really brilliant

And this is really good

$ \pi $=3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 60943 70277 05392 17176 29317 67523 84674 81846 76694 05132 00056 81271 45263 56082 77857 71342 75778 96091 73637 17872 14684 40901 22495 34301 46549 58537 10507 92279 68925 89235 42019 95611 21290 21960 86403 44181 59813 62977 47713 09960 51870 72113 49999 99837 29780 49951 05973 17328 16096 31859 50244 59455 34690 83026 42522 30825 33446 85035 26193 11881 71010 00313 78387 52886 58753 32083 81420 61717 76691 47303 59825 34904 28755 46873 11595 62863 88235 37875 93751 95778 18577 80532 17122 68066 13001 92787 66111 95909 21642 01989 38095 25720 10654 85863 27886 59361 53381 82796 82303 01952 03530 18529 68995 77362 25994 13891 24972 17752 83479 13151 55748 57242 45415 06959 50829 53311 68617 27855 88907 50983 81754 63746 49393 19255 06040 09277 01671 13900 98488 24012 85836 16035 63707 66010 47101 81942 95559 61989 46767 83744 94482 55379 77472 68471 04047 53464 62080 46684 25906 94912 93313 67702 89891 52104 75216 20569 66024 05803 81501 93511 25338 24300 35587 64024 74964 73263 91419 92726 04269 92279 67823 54781 63600 93417 21641 21992 45863 15030 28618 29745 55706 74983 85054 94588 58692 69956 90927 21079 75093 02955 32116 53449 87202 75596 02364 80665 49911 98818 34797 75356 63698 07426 54252 78625 51818 41757 46728 90977 77279 38000 81647 06001 61452 49192 17321 72147 72350 14144 19735 68548 16136 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[Sherlock]

This blows my mind: Buffon's needle. Specially the fact that you can get an estimate for $\pi$ by dropping needles!

 

[Deveno]

my favorite is the fundamental isomorphism theorem of universal algebra:

suppose A,B are algebras of the same type, and $f:A \to B$ is an algebra homomorphism.

then:

1) f(A) is a subalgebra of B
2) the relation f(a) = f(b) is a congruence on A
3) A/~ is isomorphic to f(A)

if one understands this on a deep level, it's practically half an undergraduate education right there.

 

[kanderson]

Anything that is elementary, lol. No, I like anything that you can change and have your own explanation to it. That it might be less rigorous in a text book I have in school. Otherwise I like anything from the algebraic field and mathematical finances. I find myself becoming more engrossed in algebra's and mathematical finances due to a large amount of the courses being available at a local community college. I have Business Calculus and Business Statistics. I think the best way at my level to experience something in mathematics before college is to have an application. I also am on the track for an accounting/clerical job and looking into a stock broker license so I can trade stocks for family members.

 

[Deveno]

Anything that is elementary, lol. No, I like anything that you can change and have your own explanation to it. That it might be less rigorous in a text book I have in school. Otherwise I like anything from the algebraic field and mathematical finances. I find myself becoming more engrossed in algebra's and mathematical finances due to a large amount of the courses being available at a local community college. I have Business Calculus and Business Statistics. I think the best way at my level to experience something in mathematics before college is to have an application. I also am on the track for an accounting/clerical job and looking into a stock broker license so I can trade stocks for family members.

one has to be a bit careful about replacing "rigor" with "intuition" (although intuition is often useful for thinking about things we think *may* be true). let me give a simple example:

the derivative is often introduced as "the slope of the tangent line to a curve". but what if the curve has "corners"? for a concrete example of what i mean, think about the function $f(x) = |x|$ which has a corner at $x = 0$. it's not clear what a tangent line there should be...perhaps it's even sensible to say: "it doesn't have one". ok, no problem, we'll just "avoid" that particular point.

but what if our curve looks like the saw-teeth of a sawblade? now we have a lot more corners to "avoid". in fact, it's not hard to imagine a curve with "nothing but corners" (such as the one that crops up in the various $\pi = 4$ "proofs" floating around the internet). so perhaps we should say instead that the derivative only makes sense for "smooth curves". which curves are the smooth ones? if we say: "the differentiable ones", we are stuck in a rather vicious form of circular logic.

so we need some OTHER definition of "derivative" (or "differentiable"). trying to define this without saying: "it is what it is", is one of the motivations behind the concept of LIMIT. but one has to be "careful" with limits, even the pros didn't fully understand them when the idea first came about. the search for functions which "behaved nicely with regards to limits" brought the concept of continuity front-and-center. seen in this light, one realizes that arbitrary functions can act "counter-intuitively", and that the REASON for all those troublesome epsilons and deltas is to keep us HONEST (we don't MEAN to lie, but it's easy to get carried away by what you HOPE is true).

another example from history:

a lot of time and effort down through the ages has been spent on trying to prove or falsify Fermat's Last Theorem:

$x^n + y^n = z^n$ implies $n < 3$.

it turns out that this has a LOT to do with prime numbers. now when complex numbers started being used for lots of different things, a promising avenue of approach seemed to be widening the values for $n$ to gaussian integers:

$n = k + im$ where k and m are integers.

but it turned out that "prime" in the "ordinary" integers does not mean "prime" in the gaussian integers. for example:

$2 = (1+i)(1-i)$ and
$5 = (2+i)(2-i)$

and this turned out to be a serious set-back, although it did help us get a better idea of what "divisibility" really meant. but this shows that what we think is "true" is extremely "context-sensitive" which is another reason for "rigor": to make sure you've got the context correct.