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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?"
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.
GENERATION 26: The first time you see this, copy it into your sig on any forum and add 1 to the generation.
Gotta Love V=IR. Works pretty well, I use it daily, well that and P=VI.
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.
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.
Maybe we deserve this world ?
Some beautiful equations of mathematical physics
Banu
I don't get the whole mystery over 1+1=2 and huge proofs.
Let's construct a number system from the very basics. We'll construct an infinite field over addition and multiplication. We have an additive unit which we'll call 0 and a multiplicative unit which we'll call 1. So we can add two multiplicative units to get 1+1. We call this 2. Therefore 1 + 1 = 2 *by definition of 2*.
So what am I missing?
That's my favorite.
I used to even use "exp(pi^2/12ln2)" as my name in Quakeworld.
= 1/0! + x/1! + x^2/2! + x^3/3! + x^4/$! + ...
...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.
qntm.org
Point nine recurring equals one.
qntm.org
1+2+3+4+...=-1/12 !!!
In fact it's not a joke. It's called a zeta function regularization.
There is a department store here in Japan called 0101 (Marui-marui) - when I first got here, I wasn't sure what to call it, and the geek in me asked somebody, what do they sell at 5? (10...) Needless to say, that's only funny to one of the 10 types of people.
1/3 + 1/3 + 1/3 is interesting too. If expanded to decimal form, .3333.... + .3333.... + .3333.... = .999999....
.000....00001 comes from?
where does the last
I like to ask people if they know how to add, and when they answer 'of course I do', I ask them to explain that one to me.
Suppose you have something like this (apologies for loss of indentation)This is ripped off from a web application I once wrote. You should be able to modify the time by typing figures into the box or by using up and down arrows. What happens in practice is that adding 1 actually concatenates "1" on the end of the string, so you find that 1 + 1 = 11, and 11 + 1 = 111.
This is what you get when you borrow one idea from Perl about how the computer should be able to work out from context whether or not something is a number or a string; and one from BASIC about re-using operators obviously out-of-context {strings cannot be added} to mean something different {such as concatenation}. The result, as they say, is a mess. For all practical purposes, JavaScript lets you subtract, multiply and divide numbers; but if you want to add, you'd better subtract a negative number.
I mean, it's not freaking rocket science, is it? = is for telling, == is for asking. + is for adding, . is for joining strings. Sheesh!
Je fume. Tu fumes. Nous fûmes!
Apropos to the current discussion was this response:
"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
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.
"Leo Fender was in a 'state of grace' when he designed the Stratocaster." -- Paul Reed Smith
I haven't got the full form handy, but these're the three dimensional equations for motion of fluids.. very elegant, very complete, and spawns a huge mass of special cases.
:)
As a former Aerospace student, I just had to pitch for good-old N-S
1+1=2 in the most popular formal systems translates to:
S(0)+S(0)=S(S(0))
where S(x) is the successor operation. To prove that, you have to use the addition axioms:
x+1=S(x) // 1 is shortcut for S(0)
S(a+b)=a+S(b)
and of course the Peano axioms (look them up on google, I'm too lazy to retype).
Try to prove 1+1=2 with this simple set of axioms. Note that you don't have x+y=y+x, x+(y+z)=(x+y)+z nor even x+0=x. The proof won't be several pages long, but still quite long.
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.
The answer is simple. The most beautiful equations, hands down, are those from which all of mathematics can be derived. These are the axioms of ZFC set theory. What could possibly be more beautiful or more important than that? And it's a shame so few people know about them. See Zermelo-Fraenkel Axioms and Metamath Proof Explorer.
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x=x+1 ? or does that count as two.
GENERATION 26: The first time you see this, copy it into your sig on any forum and add 1 to the generation.
Reminds me of a fun trick with google. Google's calculator knows all kinds of constants - "c", "pi", "e", etc. (Just put those in the standard search box and hit search and you'll see what I mean. Now you can use them in equations - "2*pi+7" or whatever.)
Anyways, it knows this constant too:
"the answer to life the universe and everything"
Made me chuckle the first time I saw it...
Fractals of course can be produced from many different equations, but the iterative process of many simple equations produces amazing fractals:
x=x^2+1 where the "=" should be a two way arrow.
P.S. is there a keyboard encoding for a two way arrow? How about in UTF-8?
I was amused to see the ideal gas law amongst the contenders, written as PV = nRT where n is in some weird units and R is some weird constant.
A much nicer form is:
P = nkT
where n is the number density of particles and k is Boltzmann's constant.
For some reason chemists persist in using 12 divided by the mass of the proton in grams as the basis for all measurement, and this choice leads to a proliferation of strange constants and units. I know there are historical reasons for this, but one only has to look at the way physics has re-invented its notation and concepts repeatedly over the years to realize that historical reasons are no excuse.
Written in a sensible form, the idea gas law is a very beautiful equation, though not so beautiful as the Dirac equation, which is the only differential equation in physics that I'm aware of that describes reality and only reality.
All the other equations we use have non-physical as well as physical solutions, and we quietly throw out the non-physical solutions. We sometimes even try to maintain that mathematics is "unreasonably successful" as a means of describing reality, when we know perfectly well that half of what our equations describe has no physical counter-part, but is just an ugly artefact of an imperfect description.
Blasphemy is a human right. Blasphemophobia kills.
Surely a serious contender for most elegant equation must be M = PQ and the associated factoring problem when given M and the knowledge that {P, Q} are prime and P Q. This is without doubt one of the simplest bit of maths to explain to anyone. It is also almost as old as multiplication itself and fundementally uncracked by modern number theory. A number theory problem that is expressed by 4 symbols and can be explained to a child of 10 that has confounded mathemeticians for centuries and continues to do so. How elegant and wonderfully fiendish is that?
...and, on the seventh day, God switched off his Mac.
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.
what you have stated so plainly may not be so plain. you are taking the philosophical standpoint that mathematics is nothing more than a model of physical reality.
most platonists would differ. in their view, mathematics has an existence all to it's own, and transcends the physical universe. they claim that their equations have an intrinsic existence of their own, regardless of their expression or discovery.
it is interesting to note that every great civilization that has endured for hundred or thousands of years was mathematically advanced. mathematical knowledge is directly proportional to ones power.
it is also incredible that many mathematical discoveries have preceded the discoveries of physical laws which use those mathematics....
It's such an old joke and I'm such a math teacher that I'm forced to point out that:
let x = -3then x^2 = 9
if you take the square root of both sides you get x = 3.
Technically you should instead write |x| = 3 which covers the actuality that x is in fact -3. I had to find a way to explain the + or - part of the quadratic formula to my Algebra 2's and that's what I did.
What you've really proved is that women are either evil or the opposite of evil.
When the axe came to the forest, the trees said, "Look out - the handle was once one of us."
Richard Feynman once famously remarked that Euler's Identity was the most remarkable equation in mathematics, since it combined all the really important numbers into one formula. Recently while attempting to formulate a technically-oriented conlang, I was considering what numbers really were important and concluded that there was one number of massive significance that was left out, and another was formulated somewhat arbitrarily.
Firstly, 2 is a very important number. 0 is null and the origin, 1 is unity - but 2 is the purest expression of difference and distinction. Dualism is everywhere: 0-1, On-Off, Up-Down, Left-Right, In-Out, Real-Imaginary. Everything has its opposite. Many concepts can only be considered in the context of two objects or states. 2 is the base of the humble but indispensible bit - and consequently the base of the logarithm that yields the number of bits necessary to express a number or code. 2 is indispensible.
Secondly, Pi was chosen somewhat haphazardly. For the unit circle of radius 1, Cir = 2*Pi. Pi is the ratio of a circle's circumference to it's diameter. But from a mathematical standpoint the diameter is not what's important - the radius is. Wouldn't it make just as much sense if not more to use the ratio of the Circumference to the radius (here designated as Cir)? The way things are formulated now, Pi is half a cycle in radians, halfway around the unit circle. Wouldn't a constant that represents a full cycle, Cir, make more sense? Have we grown so used to Pi that we have forgotten the arbitrariness of it's formulation?
Of course if you choose to use Cir, 2 naturally works its way into Euler's equation as well.
Exp[i*Cir/2]+1=0
Mathematicians, please respond.
---If you can't trust a nerd, who can you trust?
Another view, that I find interesting, and am tempted to subscribe to, is:
Great minds think alike; fools seldom differ.
The equation I use the most is definitely "F = m a" in all of its interesting forms. I would give that a number one rating.
But the most intriguing are the Navier-Stokes Equations. It's amazing that just by changing the boundary conditions on these dynamical equations, you can completely change the behaviour of the flow.
For incompressible flows of common fluids, these 3 simple equations make incredibly accurate predictions:
du/dx + dv/dx = 0 (incompressibility eqn)
du/dt + u du/dx + v du/dy + dP/dx = 1/R ( d^2 u/dx^2 + d^2 u/ dy^2 ) (momentum-x eqn)
dv/dt + u dv/dx + v dv/dy + dP/dy = 1/R ( d^2 v/dx^2 + d^2 v/ dy^2 ) (momentum-y eqn)
Actually, the most important result, albeit not as famous, is that taking an irrational number to the power of an irrational imaginary number and adding a rational number gives you zero. For example, sqrt(2)^sqrt(-2) + 7 = 0.
This one I find interesting because the two sides are not only equal, but anagrams of each other when spelled out in English:
11 + 2 = 12 + 1
ELEVEN + TWO = TWELVE + ONE
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