# What's your favourite result in mathematics?

#### Sherlock

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

#### Plato

##### Well-known member
MHB Math Helper
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

##### Well-known member
Staff member
That is an easy one for me: the Russel/Whitehead proof that $1+1=2$.
All 379 pages of it. XD

#### ThePerfectHacker

##### Well-known member
It can be a theorem, a definition, a proof, an identity, a trick/technique/method etc.
It is amazing how such a simple observation can be so powerful.

#### CaptainBlack

##### Well-known member
It can be a theorem, a definition, a proof, an identity, a trick/technique/method etc.
Cantor's diagonal slash

CB

#### ThePerfectHacker

##### Well-known member
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.

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#### sbhatnagar

##### Active member
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

##### New member
The uniformization theorem still blows my mind.

#### Sherlock

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

#### Fernando Revilla

##### Well-known member
MHB Math Helper
It can be a theorem, a definition, a proof, an identity, a trick/technique/method etc.
Mine has not come out yet.

#### soroban

##### Well-known member

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

#### Random Variable

##### Well-known member
MHB Math Helper
$\displaystyle e^{i \pi} = - 1$

#### Ackbach

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

#### mvCristi

##### New member
Goedel's work on the relation of consitency-completeness.

#### Also sprach Zarathustra

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

#### Random Variable

##### Well-known member
MHB Math Helper
Parallel postulate-Euclid's fifth axiom.

You must really hate hyperbolic and elliptical geometries.

#### agentmulder

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

#### nimon

##### New member
Cauchy's proof of the AM-GM inequality.

#### Sherlock

##### Member
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

##### New member
$a^2 + b^2 = c^2$

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 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#### Sherlock

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

#### Deveno

##### Well-known member
MHB Math Scholar
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

##### Member
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

##### Well-known member
MHB Math Scholar
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.