## Favorite Mathematical Equation

For the discussion of math. Duh.

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### Re: Favorite Mathematical Equation

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.

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### Re: Favorite Mathematical Equation

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.
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### Re: Favorite Mathematical Equation

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.

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### Re: Favorite Mathematical Equation

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.
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### Re: Re:

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!"

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### Re: Favorite Mathematical Equation

e as the midpoint between 1 and infinity:

lim, as n goes to infinity, of (1+1/n)^n.
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### Re: Favorite Mathematical Equation

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?
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jestingrabbit
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### Re: Favorite Mathematical Equation

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.
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### Re: Favorite Mathematical Equation

m = cd mod n
3.14159265... wrote:What about quantization? we DO live in a integer world?

crp wrote:oh, i thought you meant the entire funtion was f(n) = (-1)^n
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### Re: Favorite Mathematical Equation

I think it's right this time...
It's child's play in calculus, however I figured it out independantly and that is cool,
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### Re: Favorite Mathematical Equation

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.

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### Re: Favorite Mathematical Equation

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.
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### Re: Favorite Mathematical Equation

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.

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### Re: Favorite Mathematical Equation

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.
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### Re: Favorite Mathematical Equation

It's just the arc length of half a circle, but it's still pretty interesting the way everything comes out as pi.

LoopQuantumGravity
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### Re: Favorite Mathematical Equation

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).
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monkeykoder
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### Re: Favorite Mathematical Equation

I like the proof that 2^(1/2) is irrational

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### Re: Favorite Mathematical Equation

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.
3.14159265... wrote:What about quantization? we DO live in a integer world?

crp wrote:oh, i thought you meant the entire funtion was f(n) = (-1)^n
i's like girls u crazy

LoopQuantumGravity
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### Re: Favorite Mathematical Equation

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.
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Nab
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### Re: Favorite Mathematical Equation

adlaiff6 wrote:m = cd mod n

c = me mod n

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### Re: Favorite Mathematical Equation

I'd say a^φ(n)=1(mod n), but that's just because its the coolest thing I proved by myself.
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### Re: Favorite Mathematical Equation

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] !
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### Re: Favorite Mathematical Equation

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.
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### Re: Favorite Mathematical Equation

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
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### Re: Favorite Mathematical Equation

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).
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### Re: Favorite Mathematical Equation

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.
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### Re: Favorite Mathematical Equation

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.

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### Re: Favorite Mathematical Equation

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
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### Re: Favorite Mathematical Equation

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.)

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### Re: Favorite Mathematical Equation

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.
3.14159265... wrote:What about quantization? we DO live in a integer world?

crp wrote:oh, i thought you meant the entire funtion was f(n) = (-1)^n
i's like girls u crazy

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### Re: Favorite Mathematical Equation

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. . .

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### Re: Favorite Mathematical Equation

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.

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### Re: Favorite Mathematical Equation

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!

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### Re: Favorite Mathematical Equation

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.

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### Re: Favorite Mathematical Equation

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

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### Re: Favorite Mathematical Equation

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.
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### Re: Favorite Mathematical Equation

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
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### Re: Favorite Mathematical Equation

$\int_{0}^{t}ln^e(x)dx$
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### Re: Favorite Mathematical Equation

I don't think that's an equation.

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### Re: Favorite Mathematical Equation

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.
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