# Favorite equation

1. Feb 9, 2007

### rock4christ

list your favorite equation and why

mine is -b + or - the square root of b2 -4ac all over 2a

the quadratic formula. I found it easier than factoring for finding my x's

2. Feb 10, 2007

### FunkyDwarf

e^(i.pi) = -1

just because its so Goddamn freaky

3. Feb 10, 2007

### murshid_islam

without a doubt: $$e^{i\pi} = -1$$

i think it will be the favourite equation of many ppl around here.

4. Feb 10, 2007

### rdt2

5. Feb 10, 2007

### Gib Z

Please, if it is your favourite, give Euler's Identity the respect it deserves:

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

I probably wouldn't be able to state my favorite, but heres one I found VERY useful: $$\int^b_a f(x) dx = F(b) - F(a)$$ where F'(x)= f(x)

EDIT: Favorite equation of mine, here it is. Im a mathematician, but this is really beautiful.
$$F^{\rightarrow} = \frac{dp^{\rightarrow}}{dt}$$. Simple, effective, and has withstood the test of time. It's still correct to this date.

Last edited: Feb 10, 2007
6. Feb 10, 2007

### neutrino

The Euler's identity, of course, because I didn't realise a thing about it when I first saw it; it didn't hit me a like a rock in the stomach, not like a lightning out of the blue, and so on. And I still don't understand what the hype is all about! People see it like some sort of Hollywood movie featuring all the top stars, that's all. :tongue2:

7. Feb 10, 2007

### murshid_islam

is that the way Euler originally wrote it? i think i read somewhere that Euler origially wrote it as $$e^{i\pi} = -1$$ and not in the more beautiful way you or other ppl nowadays writes it.

isn't it $$\overrightarrow{F} \propto \frac{d\overrightarrow{p}}{dt}$$
and is it true for velocities close to the velocity of light?

Last edited: Feb 10, 2007
8. Feb 10, 2007

### uart

Although I think it's definitely a bit too geeky to have a "favorite" equation I have to say that the first time I saw the following it equation it really impressed me.

$$\prod_{n=1}^\infty \frac{1}{1-(1/p_n)^a} = \sum_{n=1}^\infty 1/n^a$$

where p_n is the n_th prime and a>1.

Last edited: Feb 10, 2007
9. Feb 10, 2007

### Hootenanny

Staff Emeritus
No, its as Gib Z written it;

$$\vec{F} = \frac{d\vec{p}}{dt}$$

And yes, if applied correctly, is valid in Special Relativity. However, note that F=ma is not valid when v>0.01c

10. Feb 10, 2007

### Curious3141

Definitely Euler's identity for me. When I first saw it (I think when I was around 15 or so), my mind was blown because the equation immediately suggested a way to define the logarithms of negative reals. And that's really, really cool.

11. Feb 10, 2007

### ranger

I'll go simple. My favs are Kirchoff's voltage and current law. I use 'em practically everyday.

12. Feb 10, 2007

### 3trQN

Hmm, i have a few favourites, in order of their cognitive bias from greatest to least:

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

$$e^{ix} = cos(x) + i sin(x)$$

$$F(t) = \frac{1}{\sqrt{(2\pi)}}\int^{\infty}_{-\infty} e^{itx}f(x)dx$$

and finally Maxwell's equations, which i don't fully remember or understand but when i do they will be next in the list....

Last edited: Feb 10, 2007
13. Feb 10, 2007

### Gib Z

In fact, F=ma is incorrect for any velocity larger than zero :P as small as the error is.

BTW uart, its not too geeky for have a favorite equation :) and thats a nice relation you've chosen, perhaps you could prove the related Riemann hypothesis for me? :P

EDIT: Forgot to address this. Euler Originally wrote it as Murshid_islam states it, but on later realisation of the equations profound consequences, changed it to the new form. He was inclined to do so by many collegues and espically number theorists, who found beauty in equations that were equated to zero.

Last edited: Feb 10, 2007
14. Feb 10, 2007

### quasar987

Written as such, this is nothing. The cool-looking thing is that

$$f(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty}f(\chi)e^{-i\omega\chi}d\chi \right)e^{i\omega t}d\omega$$

15. Feb 10, 2007

### rock4christ

I also kinda like an + bn =/= cn

Fermat's theorem. was a pain to prove

16. Feb 10, 2007

### MeJennifer

Forgive my ignorance but why do some here consider Euler's identity so special?

To me it seems that Euler's identity is a trivial instance of Euler's formula.
Furthermore the presence of $\pi$ is not significant, it is only there if you decide to measure angles in radians.

Last edited: Feb 10, 2007
17. Feb 10, 2007

### Curious3141

I don't know about others, but speaking for myself :

1) It's a breathtakingly simple looking result that beautifully ties up four mathematical constants (e, pi, 0 and 1) in a single equation (at least when you write it with the RHS equal to zero).

2) It allows one to define the logarithm of a negative number as a complex number, as I've already alluded to.

3) It is an important result that allows a quick proof of pi's transcendence via the Lindemann-Weierstrass theorem (actually the exp(2*pi*i) = 1 variant is the one used most often here).

Maybe trivial to derive (from Euler's formula, which in itself is a beautiful tie-up between exponentiation and trig and leads to the shortest possible proof of De Moivre's theorem), but hardly trivial in its significance.

Pi is a mathematical constant, it need have nothing to do with measuring any angle as far as (the "four constant") Euler's identity goes. As for Euler's original formula, well, it's understood that the trig ratios have to be evaluated with arguments in radians. There is nothing arbitrary in this, radian measure is also considered by most to be far more fundamental than any other commonly used unit of angle measure (degrees, grad, etc.) It's the same sort of "natural bias" (pun not intended) when one compares a natural log with a common one.

And if you want to treat trig functions as abstractions without any immediate reference to angles or triangles (as is often the case in analysis), then you should leave everything in radians, far more elegant that way.

Last edited: Feb 10, 2007
18. Feb 10, 2007

### DieCommie

Well, it contains unique numbers... The square root of negative one, pi, e, 1 and 0. All these are very special numbers and they can be equated. Who would have thought that raising e (crazy number with elegant properties) to the power of i (an imaginary number that doesnt quite make sense) times pi (the ratio of circumference/diameter) and add one (an obvious important number) and it all equals zero (a number humanity struggled to come to grips with)? wtf? <--thats what I think. Like it was said before, kind of creepy.

Last edited: Feb 10, 2007
19. Feb 10, 2007