Greatest Equations Ever 1017
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?"
sum of cubes (Score:5, Interesting)
first proof, that i'd seen at least, of the existance of negative numbers.
V=IR (Score:5, Interesting)
Geometry and Algebra (Score:5, Interesting)
Dirac's equation of 1/2 spin: (Score:5, Interesting)
Said by Hotson to be the Equation of Everything. First part [zeitlin.net], second part [zeitlin.net]. Worth a read IMO.
Paper on this (Score:3, Interesting)
Some beautiful equations of mathematical physics [arxiv.org]
Re:sum of cubes (Score:5, Interesting)
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.
Re:Take a guess.... (Score:3, Interesting)
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?
khinchin's constant (Score:3, Interesting)
That's my favorite.
I used to even use "exp(pi^2/12ln2)" as my name in Quakeworld.
'e' is for "cool formula" (Score:1, Interesting)
Re:correction (Score:5, Interesting)
...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.
Point nine recurring equals one (Score:3, Interesting)
This one from string theory (Score:1, Interesting)
In fact it's not a joke. It's called a zeta function regularization.
Re:1+1=10 (Score:2, Interesting)
1/3 + 1/3 + 1/3 (Score:1, Interesting)
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.
Re:Impressions of math equitations. (Score:3, Interesting)
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!
The importance of notation (Score:5, Interesting)
Apropos to the current discussion was this response [mathforum.org]:
Re:Take a guess.... (Score:5, Interesting)
http://mathworld.wolfram.com/Logic.html [wolfram.com]
Re:Submitter and Parent are stupid (Score:3, Interesting)
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.
Navier-Stokes Equation (Score:2, Interesting)
As a former Aerospace student, I just had to pitch for good-old N-S
Re:V=IR (Score:5, Interesting)
twinkle twinkle little star
power equals I squared R
I remembered it.
Re:Take a guess.... (Score:3, Interesting)
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.
At the moment... generalized Fourier series (Score:5, Interesting)
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 axioms of set theory (Score:1, Interesting)
Re:sum of cubes (Score:3, Interesting)
a-a=0, is not totally far off. Comutativity even less so(a-a+b+a=a+b+a-a etc). I think it is weaker, because it assumes that additive inverses/negative numbers exist. if -a doesn't exist, then a-a does not equal zero, since by definition there is no a-a.
The suggestion that this is similar to proving god's existance via god's existance *in the bible* isn't really appropriate, as I do pull from two positive numbers, a negative number. Perhaps a similar argument involving god may turn out to be descartes';
nothing can be created by something less perfect than it is. something [known as i] exists therefor something perfect exists, [by induction]. you start the argument with not-god, and you end with god. sure there's probably plenty wrong with that argument, but it doesn't fail in the kind of way that a-a=0 does; that you can simply define a-a=0 and be done with it; but you cannot define away the sum of two cubes, at least without doing a lot of damage.
ie
1^3 + 2^3
1 + 8
9
(3)(3)
(1+2)(3)
(1+2)(1-2+4)
OK so nothing really spectacular happens on the last step here. But I think that negative numbers, in this view, become something of a property of regular, positive numbers. That they only exist insofar as relationships not between positive numbers and zero[ie, the standard a-a=0 view], but between different collections of items. There is a ratio which is *always* upheld, whether or not negative numbers exist. but if -ab is not negative, (ie, it is some ab instead) then the numbers a and b must have been subtracted. etc.
where I think this thread will actually get interesting is here:
god exists in the bible.
therefor god exists.
It is the nature of god in which my opinion may differ with others; Whereas some believe him to be, well, whatever they believe him to be, I will suggest that god is the collection of ideas, motivations and actions of those who believe in it, much the same as I am the sum of my actions [moonside.org], God is practically everywhere, and the effects of god can be long reaching on an almost unimaginable scale...at least for my imagination. How can you argue against this? if you believe that god has X nature, I too believe that god has X nature, in the amount that your opinion matters. if your opinion matters greatly, for some reason, X is relatively greatly true. I say.
damn am I ever hungry. I'm going for some KD.
Comment removed (Score:5, Interesting)
what about (Score:3, Interesting)
Re:Everything = 42 (Score:3, Interesting)
Anyways, it knows this constant too:
"the answer to life the universe and everything"
Made me chuckle the first time I saw it...
Fractals and Iteration (Score:1, Interesting)
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?
Getting the Ideal Gas Law Right (Score:3, Interesting)
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.
The basis of asymmetric keys (Score:2, Interesting)
Re:V=IR (Score:3, Interesting)
delta G = delta H - T(delta S)
This equation balances the contributions of entropy (S) and enthalpy (H) and tells you if a given reaction is energetically favorable. delta H is the total energy in a reaction, while T(delta S) is the energy unavailable for work. A quick rearrangment shows that delta G is the energy available for doing work.
I'm also fond of Shannons juggling theorem. [bc.edu]
Re:one of the more famous misquotes there (Score:2, Interesting)
Re:Actually... (Score:5, Interesting)
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.
Re:Actually... (Score:2, Interesting)
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....
Re:Women = Evil (Score:3, Interesting)
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.
Exp[ i*Cir/2] + 1 = 0 (Score:3, Interesting)
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.
Re:Actually... (Score:2, Interesting)
Another view, that I find interesting, and am tempted to subscribe to, is:
Newton and Navier-Stokes (Score:2, Interesting)
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)
Re:correction (Score:3, Interesting)
Re:The axioms of set theory (Score:3, Interesting)
Well, strictly speaking the axioms are represented as well-formed formulas (wffs) that aren't displayed in the form of equations on the pages I linked to. But using what is called "class notation" in set theory it is always possible to rewrite a wff with an equivalent expression that has form of an equation. For example: "P(x) imples Q(x)" can be expressed as "{x:P(x)} union {x:Q(x)} = {x:Q(x)}" where "{x:P(x)}" means "the class of sets x such that P(x) is true". Or more generally, any statement P(x) that is true of all sets x (such as any of the axioms) can be rewritten "{x:P(x)} = V" where V stands for {x:x=x} i.e. the universe of all sets. Class notation is an extremely powerful device. Classes need not exist as sets, by the way; the class V above is not a set but is called a "proper class". Does this answer your question?
An anagram equation (Score:3, Interesting)
11 + 2 = 12 + 1
ELEVEN + TWO = TWELVE + ONE