Greatest Equations Ever

sgant writes "What is your favorite equation? This was the question asked by Physics World in a recent poll. This is also covered in a New York Times article about the same poll. Some of the equations mentioned were the simplistic 1+1=2 and Euler's equation, eⁱ + 1 = 0. What are some of your favorite equations?"

Comments

sum of cubes

[themusicgod1]

a^3+b^3 = (a+b)(a^2-ab+b^2)
first proof, that i'd seen at least, of the existance of negative numbers.

Re:sum of cubes

[themusicgod1]

I mean, you're right it does. But man, was I skeptical.

I say, that until I saw the sum of cubes I internally denied the existance of negative numbers. I mean I could work with them and all, I just didn't believe in them. If you deny the existance of negative numbers, you cannot have an expression 0-1, because -1 is meaningless, so therefor the result is meaningless. It's circular reasoning, and this is why[according to my youthful very non-standard way of thinking of things]:

there is a number -1
there is a number 0
if you have two numbers, there is a third number which represents their sum. :.
there is a number -1 + 0

if there is a number -1 + 0 there must be a class of numbers known as negative numbers
[the direction you were going in?]
but if you cannot prove there is a number -1 + 0, you cannot even get that far.

a^3+b^3 = (a+b)(a^2 - ab + b^2 ), on the other hand, shows quite clearly that no matter what numbers a and b you pick, you end up, in your equation, with a negative number.

V=IR

[oddbudman]

Gotta Love V=IR. Works pretty well, I use it daily, well that and P=VI.

Re:V=IR

[djdead]

When I was taking physics for the first time in high school, a EE from MIT taught me the following:

twinkle twinkle little star
power equals I squared R

I remembered it.

Geometry and Algebra

[metlin]

In my opinion, the most important equations are those that brought together Algebric representation of Geometry -- that has been the single most fundamental basis for today's advancement in mathematics and physics.

Dirac's equation of 1/2 spin:

[Jesrad]

ih/2Pi dPhi/dt = hc/2iPi (A1 dPhi/dx1 + A2 dPhi/dx2 + A3 dPhi/dx3) + A4 mc(squared)Phi

Said by Hotson to be the Equation of Everything. First part, second part. Worth a read IMO.

Re:correction

[SamSim]

...Which is in turn not to be confused with Euler's equation, which is V+F=E+2.

Euler has a ridiculous amount of stuff named after him.

The importance of notation

[tootlemonde]

Over at the Historia-Matematica discussion list, the members debated a similar question:

As you know, notation has helped the progress of mathematics. Consider, for example, the limitations of the Roman number system, the importance of the invention of a symbol for zero, etc.

Which were, in your opinion, the notations that have permitted the greatest advances in mathematics?

Apropos to the current discussion was this response:

the interest of the question:

> Which were, in your opinion, the notations that have permitted the
> greatest advances in mathematics?

(which is very different from any question concerning the history of math. notations) is very close to the interest of the question: who has been the greatest mathematician in the history, e.g. near zero.

Re:Take a guess....

[bigdreamer]

"The difficulty of formal logic was demonstrated in the monumental Principia Mathematica (1925) of Whitehead and Russell's, in which hundreds of pages of symbols were required before the statement 1 + 1 = 2 could be deduced."

http://mathworld.wolfram.com/Logic.html

Re:Submitter and Parent are stupid

[sgant]

I submitted it the equation to /. wrong...(thanks for calling me stupid btw, very helpfull)

But the equation IS e^(i*pi)+1 = 0

That's Eurler's equation. That's it. You're simply writing it in a different way.

Hell you can even plug in e^(i*pi)+1 into Google and it will spit out zero. Go ahead, give it a try.

Also, I won't call you stupid for making this mistake....I'll let it slide.

At the moment... generalized Fourier series

[dysprosia]

I would have to say, at the moment, my favorite equation would have to be the one giving the coefficients of the generalized Fourier series involving a set of eigenfunctions {p_n}, ie., c_n = <f, p_n>/||p_n||^2.

Simple stuff, but incredibly cool, considering that Fourier series don't always have to involve just sines and cosines, and you get similar sorts of behaviour.

Comment removed

[account_deleted]

Comment removed based on user account deletion

Re:Actually...

[rsidd]

i is the square root of -1, which is about as abstract a concept as you'll ever come up with - it certainly doesn't correspond to any physical quantity (unless you define a physical system using complex coordinates for the sake of convenience).

Quantum mechanical wavefunctions are complex. You could define them as two real wavefunctions and work out the appropriate algebra, but it's exactly complex algebra. So i could correspond to the phase difference of two wavefunctions, which would be observable via interference effects.

Not disagreeing with what you're saying though -- the equation is fundamental mathematics, independent of the physical universe, it doesn't make sense to imagine an "alternative universe" where it doesn't apply.