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# Most Beautiful Equation

Dennise
Many feel the most beautiful math equation is Einstein's E=MC^2.

What are some of yours?
silver67
My favorite equations are from physics that combine linear and angular velocity and acceleration.

v=rΩ
a=rα

I don't really know why. Just makes things so easy to convert back and forth.
_AVG_
If you're talking strictly math, I must say, I have never come across an equation as elegant and at the same time, equally useful ... as Leonard Euler's Formula:

e^(i*pi) + 1 = 0

Of course, the general case is: e^(i*x) = cos (x) + i sin (x)
Afaceinthematrix
 _AVG_ wrote: If you're talking strictly math, I must say, I have never come across an equation as elegant and at the same time, equally useful ... as Leonard Euler's Formula: e^(i*pi) + 1 = 0 Of course, the general case is: e^(i*x) = cos (x) + i sin (x)

The thing that's nice about that equation is that it's very easy to prove... Most nice equations are tricky to prove... But that one can easily be proved in several ways. You can prove it by looking at the Taylor series or by solving a simple differential equation..

As far as E=MC^2... That one is overrated in my opinion... Yeah it's cool and interesting, but everyone seems to talk about that one when most people don't even truly understand it... There are millions of cools results in maths (especially in fractal geometry and complex analysis) that are cooler in my opinion... Some are:

Cauchy's integral formula
de Moivre's formula
Cauchy–Riemann differential equations
Etc...
iman
 _AVG_ wrote: If you're talking strictly math, I must say, I have never come across an equation as elegant and at the same time, equally useful ... as Leonard Euler's Formula: e^(i*pi) + 1 = 0 Of course, the general case is: e^(i*x) = cos (x) + i sin (x)

I agree with him. I mean, this equation has e, i, and pi at the same time. It's a beautiful relationship.
Bikerman
Euler has to be right up there - it is breathtakingly lovely.

As a physics fan I'm a little biased, so I also think Maxwell's equations are real stunners

noobcake
Euler's identity is great. We don't know what the identity means, we just know it to be true. My only qualm with it is that it's the generic answer for "What is the most beautiful mathematical equation?" Personally, I find the Pythagorean Theorem pretty nice too. I also like cos^2(x) + sin^2(x) = 1.
Afaceinthematrix
 noobcake wrote: Euler's identity is great. We don't know what the identity means, we just know it to be true. My only qualm with it is that it's the generic answer for "What is the most beautiful mathematical equation?" Personally, I find the Pythagorean Theorem pretty nice too. I also like cos^2(x) + sin^2(x) = 1.

What do you mean "We don't know what the identity means"? I certainly know what it means. And anyone who's studied even a little complex analysis knows what it means. It is both easy to prove and not too difficult to understand when you understand how complex numbers work and the graphs of them work.
iman
[quote="Afaceinthematrix"]
 noobcake wrote: And anyone who's studied even a little complex analysis knows what it means. It is both easy to prove and not too difficult to understand when you understand how complex numbers work and the graphs of them work.

I agree. The proof is actually pretty straightforward
saratdear
The photoelectric equation:

And the equation to calculate impedance in a series LCR circuit:

Z = root of ( R^2 + (WL - 1/WC)^2)

I think I like it just because of the squares and roots.
icool
V=I R
where
V=Voltage
I=Current
R=Resistance

without this equation you cannot solve any electrical circuit. This is the basic equation which would apply in almost every circuit.
_AVG_
 icool wrote: V=I R where V=Voltage I=Current R=Resistance without this equation you cannot solve any electrical circuit. This is the basic equation which would apply in almost every circuit.

Actually, in my opinion, what's even cooler is the general statement of Ohm's law:

J = sigma*E

where J is the Current Density Vector, E is the Electric Field Vector and sigma is the Conductivity of a substance.

Of course, it gets messy when you have to replace sigma with a Conductivity Tensor.
Dennise
Pythagorean Theorem:

C^2 = A^2 + B^2

Newton's 2nd Law:

F = ma

Newton's Gravitational law:

Fg = (G*m1*m2)/r^2

Heisenberg's Uncertainty Principle:

(deltaX)(deltaP) > h/2

Sorry duno how to get the Greek "delta" character on this PC.

Looking all the equations in this post, Euler's equation has to be the gold standard for mathematical beauty in my opinion.
Bikerman
It is ASCII 916 (dec) which you can enter as &#<number>;
Δ
Plank's constant is number 8462

(I won't steal your thunder by re-writing the equation in the more beautiful form - I'll let you do it as practice )
LittleBlackKitten
I suck at math. I don't even understand why there are letters involved in math. (Lol yes seriously). Its been explained to me, but I can't understand it, I just don't get it. I used to google the answers on my phone then throw gibberish in it and screw the teacher up. Like:

(Tx4)+(r-1)=____ {No clue if that's arranged right....}
T=7
r=12
t+r=76

So then my favorite equation comes from a middle school rhyme:

Add the mood, subtract the clothes, divide the legs, and multiply. (lol...)
bukaida
Maxwell, Dirac & Newton gave us some amazing equations.
Bikerman
 LittleBlackKitten wrote: I suck at math. I don't even understand why there are letters involved in math. (Lol yes seriously). Its been explained to me, but I can't understand it, I just don't get it.
Therefore it has been explained badly, or you gave up. You are clearly not stupid and the concept of using letters to represent quantities is not beyond anyone with reasonable intelligence.

Try this:
We can calculate the area of a triangle by measuring the base and the height and multiplying them together and halving.
Thus, for the following triangle:

the area = 1/2 times 15 times 4 = 30 inches squared.

OK so far?
Now, we want a way to write that general method down so that it works for any triangle, so instead of 'base' and 'height' we use letters - b and h - and our general method is then:
area = 1/2 * b * h

Now you are using letters. The letters simply stand-in for some quantity which may have many values - such as base and height.
LittleBlackKitten
Wait, how can numbers represent a letter?...

And how can you multiply a fraction???...

My math IQ was graded at 78....Everything else, around 129.
Bikerman
 LittleBlackKitten wrote: Wait, how can numbers represent a letter?...
Other way around - letters represent a number.
 Quote: And how can you multiply a fraction???...
Easily.
1/2 * 1/4 = 1/8 (multiply the top terms together, multiply the bottom terms together - voila...so 1*1 = 1 (top line) and 2*4 = 8 (bottom line)
Bikerman
Try this:
If you take any number, double it, then take away the same number then you end up where you started - ie with the number again. That is pretty obvious, yes? So specifically, we can say
2 doubled is 4. Take away 2 and you end up back with 2.
OK?
So, using the letter x to represent 'any number' we can write that procedure down as follows:
2x-x=x
Whatever value you set for x the procedure works fine.
That is all algebra is - a way of making a general case rather than being limited to specific examples (such as the specific example of 2*2-2=2).
I would say a Gaussian Integral:

LittleBlackKitten
You totally lost me after "2 doubled is 4. Take away 2 and you end up back with 2. ".

2x....how can we possibly know what x is unless its a number? Then it makes letters redundant if we know THAT...

Brain go sploot.
Bikerman
The point is that x can be any number and it still works. That is the whole reason to use a letter. If you stick a specific number in then you have one statement about that number. If you use a letter then you have a general statement about ALL numbers.
2x-x = x is a general statement that works for ANY number. There is no way to make such a statement using real numbers because you would end-up with an infinite list:
2*1-1=1
2*2-2=2
2*3-3=3
2*4-4=4
etc etc
In one simple statement we can write ALL the possible versions of the above simply with 2x-x=x and we know it works for any value of x - therefore any number. This is much more profound and powerful.
LittleBlackKitten
WAIT....So if a letter can be ANY number, how is it even useful? You can't possibly know the answer with a variable like that...Its like, making a shopping list without the consonants and guessing what you're buying......

XXeaX
XXoXXoXi
XXeeXe

You see? Makes no sense! I still don't get why they use letters if it's that silly....
Nah, it's very useful LBK! Let's say you got this question:

Find the number so that 2 plus that number is 8.
Writing this in an equation:

2 + x = 8

Moving from one side to another is changing the sign of the number/unknown, so if we change 2 from side, it becomes minus two.

x = 8 - (+2)
x = 8 - 2
x = 6

So the number, you were looking for is 6.

Of course this is all very obvious and basic but once you start with functions and all that, it gets quite interesting.

Helios
Ha, we should open a "Teach LBK math" thread Joking, of course

My favorites are:
Newton's second law: F=ma. Brilliant.
Ohm's law: V=IR. Being an EE, how can it not be one of the fav.s ?
How can it be that nobody mentioned Snell's law?
Last one: Euler's Formula. Of course. Contains all the basic elements of math, so to speak
LittleBlackKitten
Well no cause if you add 2 onto x it becomes z....right???? @_@
Helios
 LittleBlackKitten wrote: Well no cause if you add 2 onto x it becomes z....right???? @_@

In that case z is a function of x. Every time you plug a new value into x, z will be equal to that value + 2.
Say you replace x with 3, then z will be 5.
Why is it useful? That's a different question altogether
metalfreek
Being a student of Physics I would say four equation of Maxwell are very elegant. Just four equations unified Electricity and Magnetism. Beautiful.

Bikerman
 LittleBlackKitten wrote: Well no cause if you add 2 onto x it becomes z....right???? @_@
No. Don't try to think of the letter as being important. It is simply a 'token' which stands for any number. You can't manipulate the symbol, only the numbers that the symbol represents. Adding 2 to the symbol x just means that you have x (whether x be 1 or 100000) and another 2 = x+2. x doesn't suddenly become y or z or any other symbol. It can sometimes be cancelled-out by other xs, as in x*y=x
(here we can divide both sides of the equation by x to get rid of it, leaving y=1).
The point is that although x represents any number it keeps that value for the whole equation. So if I plug a value of 20 in to replace x, then I must replace every occurrence of x with that same value - 20.

x is often representing a physical quantity. Thus in Ohm's law we say that
V = I.R (where V is voltage, I is current and R is resistance).
This tells us that for any given voltage we can calculate the current, given the resistance, or the resistance, given the current. It is observed to be true in practice and this use of letters to represent the physical properties allows us to say something powerful about electricity that will be true in all (or nearly all) cases.
LittleBlackKitten
But, if its not important than why use it??

I'm so lost. xD

If x is 20 then just write in 20! Lolllll...
Bikerman
 LittleBlackKitten wrote: But, if its not important than why use it?? I'm so lost. xD If x is 20 then just write in 20! Lolllll...

OK..so here in the UK we have Value Added Tax (VAT) of 20%. How are you going to express this?
You could try to write down every possibility
£1 ticket price, 20%=20p, therefore sale price is £1.20
£2 ticket price, 20%=40p, therefore sale price is £2.40
and so on.
Now we want a till to actually do the sum and work out the tax. The till reads the product via a bar-code reader and it looks up the ticket price. How am I going to tell the program to calculate the VAT?
The way to do it is to enter
sp=tp*1.2 (sp = sale price, tp = ticket price).
I could use sp, and tp, but conventionally we would probably use y and x (just a convention - we don't have to, but it doesn't hurt) so we get y = 1.2*x
There isn't any other way of doing it - we assign a symbol to mean ticket price (call it tp or y or whatever you like), we do the same for sale price and we can then write down a method that will work for any price, very simply, in a single 'sentence', and in a form that can be used by a calculator (whether that is human or machine).
therimalaya
ha ha, I think it is very hard to teach mathematics, if anyone can not imagine (virtual) things. I'm not trying to say anything about numbers and letters, may be i can not but instead, if i continue reading these things I'll certainly get confused what i have learned.

Well, My beautiful equation (Not actually equation it is a theorem), is the Central Limit Theorem (CLT), it simply says that any sequence of random number shows the properties of Normal Distribution if it is large in number.
LittleBlackKitten
But the numbers are useless because you can just go

\$20 (or whatever the price is) +20%
or

20 x 1.20.

No y no x no sp/tp....

Why make it more complex than it needs to be?

Letters just screw me over because you can't add numbers onto letters....
Helios
 LittleBlackKitten wrote: But the numbers are useless because you can just go \$20 (or whatever the price is) +20% or 20 x 1.20. No y no x no sp/tp.... Why make it more complex than it needs to be? Letters just screw me over because you can't add numbers onto letters....

The letters are just a convention. You can say that an area of your room equals the length of the room times the width of the room. But it's much easier to call area by some letter, the width by some letter and the height by some letter.. like a=b*c. When they write E=mc^2, everybody (who studied the theory once) know what E, m and c mean. Much shorter and easier.

Another thing is to make calculations easier. To help us reach certain conclusions based upon the fact that we can say that a certain unknown is actually known.
Eh, WHAT??? Take a simple train problem from school:
 Quote: Train ABC, traveling constantly at 50 miles per hour takes off from station 1 towards station 2 which is 260 miles away. At the same time Train XYZ, traveling 60 mph, leaves station 2 and it's heading towards station one, because of some mistake it's on the same rail as train ABC. When do the two trains explode?

We know the distance between the two stations: 260miles.
We know their speeds: ABC at 50mph, XYZ at 60mph.
We do not know the time they will travel until the collision, but let's say that we do know and we'll mark it as T. This is the key point, and marking the unknown by a letter will later help us fine the numerical value!
So, train ABC will travel 50*T miles (speed multiplied by time) before collision.
Train XYZ will travel 60*T miles. We do not know what T is yet, but we will find out!!!
The sum of both distances equals 260, as we know:
so, 260 miles = 50*T + 60*T = 110*T. Using this equation, T can be found. t= 260 / 110 hours which is about 8509 seconds.

So it is possible to reach the conclusion that the trains will collide after 8509 seconds without using that letter T, there are several ways even, but marking the unknown time by some letter makes things much easier for most people.
And say that we didn't use that letter T at all, but now I say that ABC's speed is 51mph and not 50... so instead of plugging 51 into the equation and pressing some buttons on the calculator, you need to rethink everything.

Also how cool can it be to say that L is the distance between the two stations, V is speed of train ABC, U is speed of train XYZ and T is the time after which they'll collide...
so L=(V+U)*T.
Now I can say whatever I want about the distances, speeds.etc.. you can always plug in values and calculate, no need to think too much because the distance or speeds have now changed.

However, try to solve a problem which is a bit more complex without using any letters, say... train ABC (same as above) now instead accelerates from 0 mph to 50mph (which is more realistic) at a rate of 10mph^2, instead of travelling constantly at the same speed. When will the trains collide now?
Using letters in this case makes things much easier.

If you're not convinced now, I give up
LittleBlackKitten
Well...You just write the numbers in here too because once you have the numbers you don't NEED the box. Like when I have to make a raster graphic, (I know, I know, bad form, not vector), I just think of the numbers, I don't go x by z = 560 pixels squared...It just makes no sense to me, (and I still don't totally get why letter are there)...
Bikerman
Well, you've had the gentle and fairly obvious explanation. If you want to abstract from real numbers you use letters. If you don't want to abstract then unfortunately you remain in the stone-age. If I give you an irregular area to measure, you can write down all the numbers, but you won't get an accurate answer unless you have measuring devices for every irregularity - bumpy ground, odd curves etc.
Moreover you won't be able to generalise your experience and pass it on to others. It is no use telling te apprentice that if someone pays for an 80p card with a pound coin that they need 20p back. What if they pay with a fiver? Are you going to write down every possible quantity for the apprentice? Much better to teach him/her how to perform the general calculation, using algebra - then they can do it for any amount of change.

There is another problem, however, with the 'use the numbers' approach - well, 2 actually.
Firstly, what do you think a '2' is? It is a symbol, just like a letter - no difference. We all agree that a symbol in the shape of a 2 can represent the quantity which arises when you treat objects in groups rather than individually.
Second problem is that numbers only take you so far. If you want to examine more complex things then the numbers don't work. You might be OK with negative numbers - although they would also seem to be against your basic approach - but what about quantities that don't lie on the normal plane of numbers and are, instead, found on the complex plane? (for example, the square root of -1).
These quantities are incredibly useful and no less 'real' than the numbers you normally deal with...
LittleBlackKitten
I'm not against it, I just honestly don't grasp it.
therimalaya
 LittleBlackKitten wrote: I'm not against it, I just honestly don't grasp it.

Think about a general name like Human, this is just like 'a letter' in this discussion, the human can be 'john', 'Jeniffer',... and so on just like 'a number'. When you consider john, that is just a john but when you consider human that can include any name. That is when you talk in general sense. I hope you get the point.
Helios
 LittleBlackKitten wrote: Well...You just write the numbers in here too because once you have the numbers you don't NEED the box. Like when I have to make a raster graphic, (I know, I know, bad form, not vector), I just think of the numbers, I don't go x by z = 560 pixels squared...It just makes no sense to me, (and I still don't totally get why letter are there)...

Of course I write the numbers eventually if I want a specific solution, but there are 2 key points:
1. to reach the numerical solution, in some cases (like I demonstrated), I have to use numbers and letters to help me out. So try to solve those problems yourself without using any letter to represent something. You'll see how hard it is.
2. In the end the purpose isn't really to find a certain numerical value, like 2.32 hours or w/e, but to find an equation which will allow me to plug in any values of speed, distance etc to be able to calculate the time until collision in every possible case in the future.
Why is it useful? Well say a computer program wants to check for possible train collisions, alert the train operators about it and say how much time they have...
The computer needs an equations to be able to take the numerical values, plug them in, and calculate time. A computer can't 'think', therefore it needs that equation. You can 'think', so it's up to you to program the computer.

 LittleBlackKitten wrote: I'm not against it, I just honestly don't grasp it.

Yeah, I believe you and I think I understand what you mean by that.
To make it clear, is it hard for you to understand why do we need equations and functions at all?
And/or is it hard to understand how to describe a certain situation using letters?
IceCreamTruck
I like bringing Einstein's equations down to earth for people. Put your weight into "E=MC^2"... it's not hard... how much you weight goes in the "M" spot (remembering units of measure is important!). This will tell you how much energy is in your body in terms of foot/pounds, but the speed of light must be in English measurement and mass (M) converted to pounds. This will give you a solution that is in foot/pounds which is how they measure the energy in TNT.

Can you do the reverse? ~1,450,800,000,000ft/pounds of energy is in my body... can you tell me how much I weigh? I'll give you a hint: "1,450,800,000,000ft/pounds=MC^2" ... you'll have to work it out from there.
therimalaya
If i put my weight in place of M and 3 times 10^8 in place of C than would the equation give how much energy i would release if i will run in that speed. Am i wrong...?
_AVG_
I recently came across another dazzling and elegant equation ... yes, it's Euler again, only this time, one pertaining to graph theory / 3-dimensional geometry:

F + V = E + 2
for any polyhedron (where F = number of faces of the polyhedron, V = number of vertices of the polyhedron, E = number of edges of the polyhedron)

What an elegant equation! I urge you to try it for any possible polyhedron if you don't believe me ...
IceCreamTruck
 therimalaya wrote: If i put my weight in place of M and 3 times 10^8 in place of C than would the equation give how much energy i would release if i will run in that speed. Am i wrong...?

Well, you are wrong. Think about it this way -- Einstein stated that if you accelerate matter to the speed of light then mass becomes infinite and the matter becomes energy (ignore the fact that as it did so it would basically destroy the universe). He stated that you cannot accelerate to the speed of light, but in a sense you can use that equation to find out how much energy is in any object given that you know it's weight and what the speed of light is. Now you fill in the equation...

I weigh 115 pounds, so "pounds" (lbs) is our unit of measure

The speed of light should now also be considered in English measurement, because it wouldn't make sense to switch to kilometers here... you could argue it does, but just make sure you use Kilos for weight.

The speed of light: 670,616,629 miles per hour (keep all of your units in mind.. lbs/mi/hr)

"3 times 10^8" is your measurement for the speed of light, but honestly it's hard to wrap my mind around any number that doesn't have some kind of units. Is this kilometers/hour?

So put that in the equation and be careful to do the multiplication in order.. As a rule anything inside [brackets] or (parentheses) should be evaluated first, and [brackets] or (parentheses) inside [bra [brackets]ckets] or (pare(parentheses)ntheses) sould be evaluated from the inside out.

oops... almost forgot... miles must be converted to feet. 1 mile = 5280 feet

This means the speed of light converted is = 3,540,855,801,120feet/hour

E = MC^2
E = 115 * (3,540,855,801,120)^2
E = 115 * 12,537,659,804,325,156,993,254,400
E = 1,441,830,877,497,393,054,224,256,000 ft lbs

And this shows me there is a LOT of energy in my body... but we have nothing to compare that to so we can grasp it with our minds.

The energy in TNT is = 10,340,000 joules/pound

1 joule = 0.7376 ft lbs

E = 1063494455242077116795811225.6 joules

divid that by the joules in one pound of TNT, and you find out how much TNT detonated would equal the amount of energy in my body.

Pounds = 1063494455242077116795811225.6 joules / 10,340,000 joules

~102,852,461,822,251,171,837 lbs of TNT

Long story short I'm a HUGE pile of TNT exploding in a very small body... fortunately matter/energy - equivalence determines that my matter, the substance of who I am and all that energy, won't change forms even if you set me on fire, so don't even think about it!

"Matter/energy - equivalence" ... this means even if I burn all the energy stays right where it is, and all the mass stays in the equation too, but just changes places. Even the heat from the fire of my burning body is really just energy released, and was present as energy inside me the whole time too. Crazy stuff, I know, but it's a crazy world we live in.

BTW, I am a true believer in Einstein ... he said one day we would prove him wrong. That's probably because he too wasn't really satisfied with the theory of relativity... it just worked for him, and has worked for us for a long time, but it is time to refine his work, and many are working on expanding his ideas or as he put it "proving him wrong".

(Once again IceCreamTruck realizes that he doesn't really exists, and vanishes in a puff of logic!)

PS. If the above math is wrong please correct me... I can make drastic mistakes at times.
tazone
Ideal gas law
p*V=n*R*T

p=pressure of gas [Pa]
V=volume of gas [m^3]
n=amount of particles in gas [mol]
R=universal gas constant [J/molK]
T=temperature [K]
IceCreamTruck
 tazone wrote: Ideal gas law p*V=n*R*T p=pressure of gas [Pa] V=volume of gas [m^3] n=amount of particles in gas [mol] R=universal gas constant [J/molK] T=temperature [K]

nice... it means you can apply this to figure out the maximum and minimum temperatures for storing any gas, if you know the max/min pressure of your storage tanks and the measurement of gas you have. Well, you'd have a good approximation

I just asked because I asked someone one time when we were carrying a couple different kinds of gas around in a black van for work. It got cold and hot in the van depending on what time of year and how much direct sunlight. He was right in that we never had any problems, so we were within acceptable tolerance, apparently. I just like to know this kind of information and not guess. Warnings were posted all over the tanks, but there wasn't a shred of useful information on any of them.

The maximum pressure and volume are printed on most tanks... I just wanted to know how the pressure changes when temperature (T in the equation) goes up! You've given me something to play with, so I thank you.

Furthermore, I was thinking, if I take the temperature of the gas (room temperature) and the pressure, then I can fill in everything to solve for the number of particles "n". Pretty cool. Yes, I wear my geek on my sleeve.
saberlivre
The Einstein equations for the gravitational field. Fantastic!!
IceCreamTruck
 saberlivre wrote: The Einstein equations for the gravitational field. Fantastic!!

Like this? http://en.wikipedia.org/wiki/Einstein_field_equations
saberlivre
Yes, its beauty lies in the fact that they relate physical properties (energy-matter) to purely geometrical ones (space-time).
IceCreamTruck
 saberlivre wrote: Yes, its beauty lies in the fact that they relate physical properties (energy-matter) to purely geometrical ones (space-time).

What I like is our universe is very similar to the mathematics of a hologram in that one must understand the science of the very small in order to understand the science of the very large, and to my knowledge we are still working on unifying the picture that holds both the very small and the very large into account so that the beauty of the hologram that is our universe can be more accurately known.

I'm wanting to give your equations there a go, but I'll save that as an idea for another time. Good stuff though.
Dennise
This infinite series - that calculates the natural logarithm's base 'e' - is a real beauty too:

e = 1+ 1/1! + 1/2! + 1/3! + 1/4! + 1/5! ......

It was discovered by Newton in 1665.

And Euler's beautiful series for pi:

pi^2/6 = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 ......

That those two irrational constants can be computed using such elegant structured series' is almost breathtaking to me!

(size adjusted to normal - Bikerman)
I'd ask posters to please leave the main body text size as it is on default - otherwise we end up with 'shouty' postings.