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### Re: Favorite Mathematical Equation

Posted: Sun Sep 30, 2007 4:54 pm UTC
Lagrange Interpolating Polynomial, mainly because the problem it solves seems to be tricky, yet if you think about it for a short while, Langrange's solution seems mind-numbingly simple.

### Re: Favorite Mathematical Equation

Posted: Mon Oct 01, 2007 6:38 am UTC
My favourite is probably Zeta(-1) = -1/12, where Zeta(s) is the Riemann-Zeta function at s. I like it because you can write Zeta(-1) = 1 + 2 + 3 + 4 + 5 + ..... There is an anecdote of Ramanujan giving Hardy and Littlewood a paper in which he declares 1 + 2 + 3 + ..... = -1/12. Whether the story is true or not, I like it.

### Re: Favorite Mathematical Equation

Posted: Tue Oct 02, 2007 9:49 am UTC
d(e^x)/dx=e^x

I know, it's really simple. But this little identity is just so fracking useful.

It's just.. ah.. when I have a problem that include e^x, my eyes light up. The best part is, in physics, you see e^x quite frequently.

### Re: Favorite Mathematical Equation

Posted: Tue Oct 02, 2007 11:48 am UTC
roundedge wrote:d(e^x)/dx=e^x

I know, it's really simple. But this little identity is just so fracking useful.

It's just.. ah.. when I have a problem that include e^x, my eyes light up. The best part is, in physics, you see e^x quite frequently.

It's just an eigenvalue.

I've always liked the integral forms of the gamma function,
Gamma(x) = (x-1)! = Integral_0^infinity t^(x-1) e^-t dx

You can also write it in a nifty log form with appropriate substitution of variables. And it's the unique function that's equal to factorial given certain conditions (IIRC, increasing, concavity, and continuity). One of the MAA's website's "how Euler did it" articles has a neat derivation.

Torn Apart By Dingos wrote:Cauchy's integral formula is rad:

Is also a favorite.

### Re: Re:

Posted: Tue Oct 02, 2007 4:54 pm UTC
jeff_attack wrote:
brodieboy255 wrote:dunno if it counts as a formula, seeing as its more of a property, but i love that d/dx(ex) = ex

Euler's formula is also cool. I love the way it merges trig, e, and imaginary numbers.

Isn't e^x the greatest? It was like Christmas when you got an e^x question on a test. Just knowing that its integral and its derivative are the same function, it's kinda strangely beautiful when you think about it. I hate myself for using those words, but I don't know how else to describe it.

I remember very clearly when I learned that. "Really, that's weird. A pretty big coincidence, isn't it? . . . oh. I get it. That's where "e" came from. Why didn't you tell me that before!"

### Re: Favorite Mathematical Equation

Posted: Tue Oct 02, 2007 5:08 pm UTC
e as the midpoint between 1 and infinity:

lim, as n goes to infinity, of (1+1/n)^n.

### Re: Favorite Mathematical Equation

Posted: Tue Oct 02, 2007 5:12 pm UTC
Yakk wrote:e as the midpoint between 1 and infinity:

lim, as n goes to infinity, of (1+1/n)^n.

How does that show that e is the midpoint of 1 and infinity?

### Re: Favorite Mathematical Equation

Posted: Wed Oct 03, 2007 2:00 am UTC
The Hurewicz ergodic theorem. It works for any nonsingular T (ie T such that mu ~ muT).

Via http://www.mathbin.net - the best site in the whole internets.
14476_0.png

### Re: Favorite Mathematical Equation

Posted: Wed Oct 03, 2007 4:52 am UTC
m = cd mod n

### Re: Favorite Mathematical Equation

Posted: Wed Oct 03, 2007 5:19 am UTC

I think it's right this time...
It's child's play in calculus, however I figured it out independantly and that is cool,

### Re: Favorite Mathematical Equation

Posted: Thu Oct 04, 2007 2:37 am UTC
I saw a proof for euler's phi that used only inclusion / exclusion. I thought that was an extremely sexy method.

I am a big fan of working with x*sin(1/x). it fully convinced me that continuous functions are quite strange beasts.

### Re: Favorite Mathematical Equation

Posted: Thu Oct 04, 2007 2:44 am UTC
Sygnon wrote:I saw a proof for euler's phi that used only inclusion / exclusion. I thought that was an extremely sexy method.

I am a big fan of working with x*sin(1/x). it fully convinced me that continuous functions are quite strange beasts.

You know there are everywhere continuous nowhere differentiable functions, right?
When you think about the topological definition of continuity, though, it doesn't seem quite so strange that completely insane things could exist.

### Re: Favorite Mathematical Equation

Posted: Thu Oct 04, 2007 7:52 pm UTC
LoopQuantumGravity wrote:
Sygnon wrote:I saw a proof for euler's phi that used only inclusion / exclusion. I thought that was an extremely sexy method.

I am a big fan of working with x*sin(1/x). it fully convinced me that continuous functions are quite strange beasts.

You know there are everywhere continuous nowhere differentiable functions, right?
When you think about the topological definition of continuity, though, it doesn't seem quite so strange that completely insane things could exist.

That's one of my favorite continuous but not differential funcitons.

### Re: Favorite Mathematical Equation

Posted: Fri Oct 05, 2007 6:29 pm UTC

It's not that fast converging, but I like it since about six months ago, I saw Euler's series for arctangent, and used it knowing that 4 * arctan(1) = pi, so I simplified it and got that. The thing is I'm 15, and learned about series in wikipedia.

### Re: Favorite Mathematical Equation

Posted: Fri Oct 05, 2007 7:49 pm UTC
It's just the arc length of half a circle, but it's still pretty interesting the way everything comes out as pi.

### Re: Favorite Mathematical Equation

Posted: Fri Oct 05, 2007 8:52 pm UTC
Ooooh, how I could I forget the Gauss-Bonet theorem?

K is the gaussian curvature of a surface (or in general, manifold) M. In general, this can be some crazy complicated function, since it's easy to imagine a crazy complicated surface. X(M) is an integer that's related only to how many holes a surface has.

So, the double integral of a crazy complicated function is always an integer*pi! That's totally crazy.

Conversely, this also means you can calculate a really complicated integral by counting holes (as long as what you're integrating is the curvature of something).

### Re: Favorite Mathematical Equation

Posted: Sat Oct 06, 2007 4:13 am UTC
I like the proof that 2^(1/2) is irrational

### Re: Favorite Mathematical Equation

Posted: Sat Oct 06, 2007 6:27 am UTC
monkeykoder wrote:I like the proof that 2^(1/2) is irrational

I like the proof that there exist irrational n and m such that nm is rational.

### Re: Favorite Mathematical Equation

Posted: Sat Oct 06, 2007 7:49 am UTC
monkeykoder wrote:I like the proof that 2^(1/2) is irrational

I like the proof that there exist irrational n and m such that nm is rational.

That's fairly trivial, e.g., e^(log(2)) = 2. It's more interesting in light of the following theorem:
Gelfond's Theorem wrote:If,
1. a =/= 0,1 is algebraic
2. b is irrational and algebraic
then,
ab is transcendental.

More general cases aren't known, though, AFAIK.

### Re: Favorite Mathematical Equation

Posted: Sun Oct 07, 2007 7:52 am UTC
adlaiff6 wrote:m = cd mod n

c = me mod n

### Re: Favorite Mathematical Equation

Posted: Sun Oct 07, 2007 3:21 pm UTC
I'd say a^φ(n)=1(mod n), but that's just because its the coolest thing I proved by myself.

### Re: Favorite Mathematical Equation

Posted: Sun Oct 07, 2007 4:12 pm UTC
LoopQuantumGravity wrote:
Gelfond's Theorem wrote:If,
1. a =/= 0,1 is algebraic
2. b is irrational and algebraic
then,
ab is transcendental.

...that is so awesome.

[transcendental] (-1)i [/transcendental] !

### Re: Favorite Mathematical Equation

Posted: Sun Oct 07, 2007 4:37 pm UTC
Ended wrote:
LoopQuantumGravity wrote:
Gelfond's Theorem wrote:If,
1. a =/= 0,1 is algebraic
2. b is irrational and algebraic
then,
ab is transcendental.

...that is so awesome.

[transcendental] (-1)i [/transcendental] !

I'd say it's more interesting that (-1)i is real, but that's just me.

### Re: Favorite Mathematical Equation

Posted: Sun Oct 07, 2007 4:55 pm UTC
I'm now wishing I knew slightly more about working out complex powers, at the moment the best I can come up with from (-1)i is domain error

### Re: Favorite Mathematical Equation

Posted: Sun Oct 07, 2007 5:37 pm UTC
SimonM wrote:I'm now wishing I knew slightly more about working out complex powers, at the moment the best I can come up with from (-1)i is domain error

Me too. For now, it's just unbelievable to think that -1^i = -1 (or at least that's what MATLAB says).

### Re: Favorite Mathematical Equation

Posted: Sun Oct 07, 2007 5:39 pm UTC
Govalant wrote:
SimonM wrote:I'm now wishing I knew slightly more about working out complex powers, at the moment the best I can come up with from (-1)i is domain error

Me too. For now, it's just unbelievable to think that -1^i = -1 (or at least that's what MATLAB says).

(-1)i = ei*log(-1)
= ei*i*π
= e
≈ 0.043

Though, actually, there is more than one answer, given that the complex log function is multivalued. Any number of the form e(2k+1)π, where k is an integer, is a possible solution.

### Re: Favorite Mathematical Equation

Posted: Sun Oct 07, 2007 6:00 pm UTC
Govalant wrote:
SimonM wrote:I'm now wishing I knew slightly more about working out complex powers, at the moment the best I can come up with from (-1)i is domain error

Me too. For now, it's just unbelievable to think that -1^i = -1 (or at least that's what MATLAB says).

Don't forget your parentheses. -1^i = -(1^i) = -(1) = -1, which is not very unbelievable at all.

### Re: Favorite Mathematical Equation

Posted: Sun Oct 07, 2007 11:44 pm UTC
Heh, my favourate equation is the one for ii, because we had to work that out for one of my courses in the first year. I won't post it, because not only am I not sure of the answer right now, and too lazy to work it out, but also it is fun to work it out.

Spoiler:
use Euler's identity

### Re: Favorite Mathematical Equation

Posted: Mon Oct 08, 2007 7:55 am UTC
LoopQuantumGravity wrote:
monkeykoder wrote:I like the proof that 2^(1/2) is irrational

I like the proof that there exist irrational n and m such that nm is rational.

That's fairly trivial, e.g., e^(log(2)) = 2.

But using that as your example requires proving that e is irrational which is not really all that trivial. The proof that adlaiff6 was almost certainly thinking of could be understood by anyone who can understand the proof that \sqrt{2} is irrational. (A group which is much larger than those who even know what e is.)

### Re: Favorite Mathematical Equation

Posted: Mon Oct 08, 2007 11:32 pm UTC
bray wrote:But using that as your example requires proving that e is irrational which is not really all that trivial. The proof that adlaiff6 was almost certainly thinking of could be understood by anyone who can understand the proof that \sqrt{2} is irrational. (A group which is much larger than those who even know what e is.)

You seem to indicate that I was referencing the "well, one of these must be true..." proof, in which case you'd be correct.

### Re: Favorite Mathematical Equation

Posted: Wed Oct 10, 2007 8:07 am UTC
ii=eln(i^i)=ei*ln(i)=ei*ln((-1)^1/2)=ei*ln(-1)/2.

By taking the ln of both sides of Euler's Identity, we get ln(-1)=i*pi, so continuing:

=ei^2*pi/2=cos(pi*i/2)+i*sin(pi*i/2), which is probably about as far as you can go. . .

### Re: Favorite Mathematical Equation

Posted: Wed Oct 10, 2007 2:56 pm UTC
quintopia wrote:ii=eln(i^i)=ei*ln(i)=ei*ln((-1)^1/2)=ei*ln(-1)/2.

By taking the ln of both sides of Euler's Identity, we get ln(-1)=i*pi, so continuing:

=ei^2*pi/2=

=e-1*pi/2, which is real.

### Re: Favorite Mathematical Equation

Posted: Wed Oct 10, 2007 6:45 pm UTC
I like the Meredith axiom.

((((p->q) -> (~r->~s)) -> r) -> t) -> ((t->p) -> (s->p))

You can derive any true formula with this axiom, substitution and modus ponens (well, at least in the context of classical propositional logic). And it just might be the shortest way to do it!

### Re: Favorite Mathematical Equation

Posted: Wed Oct 10, 2007 9:05 pm UTC
GBog wrote:=e-1*pi/2, which is real.

Shows what you get for doing derivations at 3 in the morning, after working on graph theory proofs for hours on end. God, I'm dead.

### Re: Favorite Mathematical Equation

Posted: Thu Oct 11, 2007 12:25 am UTC
e^{i\pi}+1=0

This question is obvious to anyone with even the smallest sense of beauty

It has:
- exponentiation
- multiplication
- equality
- e
- pi
- zero
- one

... which is more than enough to do any true mathematics

### Re: Favorite Mathematical Equation

Posted: Sat Dec 15, 2018 3:39 am UTC
Mine is cos(pi/5) = phi/2 because it can be used to explain the difference between pi and phi (in my country some people can't even distinguish them, writing pi as phi because they think it looks more foreign) without involving the more complicated e.

### Re: Favorite Mathematical Equation

Posted: Mon Dec 17, 2018 2:17 pm UTC
BioTronic wrote:Integral z squared dz
from one to the cube root of three

(Doesn't rhyme, in English English. )

From physics:
P1V1/T1 = P2V2/T2

From QI:
y=ln(x/m-as)/r²
If I back-formed it again correctly without being an idiot, can be re-resolved to: merry=x-mas

### Re: Favorite Mathematical Equation

Posted: Mon Dec 17, 2018 4:36 pm UTC
$\int_{0}^{t}ln^e(x)dx$

### Re: Favorite Mathematical Equation

Posted: Sat Dec 22, 2018 10:38 am UTC
I don't think that's an equation.

### Re: Favorite Mathematical Equation

Posted: Sat Dec 22, 2018 7:06 pm UTC
Carmeister wrote:I don't think that's an equation.

It's equal to something, just left as an exercise to the reader. Which is everyone's favorite, really.