Most Elegant Math Equations Ever

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In your opinion, what equations, solutions, or sets are the most elegant math equations ever solved or conceived of?

Personally, I like Euler's identity. That is, e^(i*pi) +1 = 0. The way it is proven is also something to marvel at too, since it comes directly out of complex analysis.

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$dF=0$

$d*F=0$

Maxwell's equations in vacuum.

Elegant because these are really the start of modern physics. From them came Lorentz invariance, gauge symmetries and duality.

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Hi,

I like this one :

$\sum_{i=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$

That's a wonderful thing that $\pi$ appears in the result of this sum. And i find the way you can prove that e.g. by Fourier series is quite elegant.

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In your opinion, what equations, solutions, or sets are the most elegant math equations ever solved or conceived of?

Personally, I like Euler's identity. That is, e^(i*pi) +1 = 0. The way it is proven is also something to marvel at too, since it comes directly out of complex analysis.

I like Euler's Identity too, $e^{\pi i} + 1 = 0$. It's one of those equations which would strike people's mind when we say 'Mathematics' just as when we say 'Physics', people's mind will get Einstein's equation $e=mc^2$.

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I know this sounds like a smartass answer, but I still have to say:

1 + 1 = 2

Foundation of all mathematical abstraction.

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I know this sounds like a smartass answer, but I still have to say:

1 + 1 = 2

I wouldn't consider that expression mathematical elegance, rather it is a trivial one.

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see, elegance is a subjective property. i agree with sisyphus, it is elegant it is both simple, functional, correct and the rest of mathematics stems from it. all others would not exist if it were not for that equation.

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see, elegance is a subjective property. i agree with sisyphus, it is elegant it is both simple, functional, correct and the rest of mathematics stems from it. all others would not exist if it were not for that equation.

I don't think so. Just because it is simple, functional, and correct does not make it elegant. Saying that the expression 1+1 = 2 has elegance is like comparing a badly drawn stick figure to the Mona Lisa painting.

Mathematical elegance is, in and of itself, a work of art. And it's something I take very seriously.

It is for that, and other reasons that Euler's identity can be considered elegant, while -1+1=0 is not, even though $e^{\pi i} + 1$ $\equiv -1+1$ = 0.

Conclusion: there is nothing particularly elegant or special about 1+1=2. It is trivial, just like all the other expressions like it.

I like Euler's Identity too, $e^{\pi i} + 1 = 0$. It's one of those equations which would strike people's mind when we say 'Mathematics' just as when we say 'Physics', people's mind will get Einstein's equation $e=mc^2$.

I agree

Edited by Deja Vu
multiple post merged
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I couldn't disagree more. There is no equation less trivial than 1+1=2. It is the seed of all mathematical abstraction, the philosophical representation of mathematics itself. It represents, in simple, elegant notation, the very idea of the abstract unit, and all that its existence implies.

Compared with that, Euler's Identity is just a neat trick.

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I would say 9 + 1 = 10 is very important as well. Without the concept of using zero as a power of ten placeholder, the Indian numerals would be no better than roman

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Elegance is the attribute of being unusually effective and simple.

1+1=2 is

1/ very effective, it is the root of all mathematics

2/ very simple, it is the first equation you learn.

it fits quite elegantly if i do say so myself.

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Sin^2A + Cos^2A = 1 is sort of nice in a way.

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see, elegance is a subjective property. i agree with sisyphus, it is elegant it is both simple, functional, correct and the rest of mathematics stems from it. all others would not exist if it were not for that equation.

I agree that elegance is subjective. In my opinion, I feel that simplicity is the most elegant thing in mathematics. Complexity can lead to more room for errors. But...that is just my opinion. Your opinion may differ, his opinion may differ, their opinion may differ. Again, its subjective.

My sentimental fave has always been a$^{2}$ + b$^{2}$ = c$^{2}$

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I wouldn't consider that expression mathematical elegance, rather it is a trivial one.

Well, try and prove this.

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$1+1=2$ is very elegant but for me my favorite is the pythagorean theorem, $a^{2}+b^{2}=c^{2}$ I like it because it is elegant, simple and can be adapted to proven or solve many things like $cos^{2}+sin^{2}=1$ or the distance theorem.

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Well, try and prove this.

I just read this and it made me actually laugh out loud. Love that one, ajb.

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I want a proof starting from the Peano axioms (or equivalent). I don't think it is very difficult, but I claim that $1+1=2$ is not really completely empty or trivial.

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• 4 weeks later...

$\frac{d}{dx}[e^x] = e^x$

Even though $e$ was defined to fit this very equation, I still love the fact a simple exponential function has this property.

"Who has not been amazed to learn that the function y = e^x, like a phoenix rising from its own ashes, is its own derivative?"

~ Francois le Lionnais

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I want a proof starting from the Peano axioms (or equivalent). I don't think it is very difficult, but I claim that $1+1=2$ is not really completely empty or trivial.
I know that you were just saying that to make a point but:

I mean, not that I have any idea what that says.

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• 4 years later...

I'm voting for Eratosthenes, the Greek scholar (circa 276 -195 BC) who measured the circumference of the world by observing the angle of the shadow cast by a vertical stick at noon, and applying Euclid's theorem of equal angles formed by a straight line intersecting two parallel lines. This angle was measured on a particular day in Alexandria which was about 5,000 stades directly north of Syene (now called Aswan). A stade is the length of a Greek stadium.

Now, on that sunny day, while there was a shadow in Alexandria at noon, a strange phenomenon was taking place down in Syene. There were absolutely no shadows. Poor Eratosthenes, no one ever told him about winter and summer solstices.

But solstice or no solstice, Eratosthenes surmised, from empirical observations, that this phenomenon was because the surface of the earth was curved. Therefore he reasoned that if the rays from the sun striking two vertical sticks, one at Alexandria and the other at Syrene, are parallel; then according to Euclid's theorem, the angle of the shadow at Alexandria must be identical to the angle where the imaginary lines of the two sticks on the surface of the earth meet. In a flash, Eratosthenes has deflated the rubbish of a flat earth being held up on the aching shoulders of Atlas and has pin-pointed the centre of the earth having no hi-tech instruments.

And by simple 6-grade arithmetic, he calculates the angle of the shadow at Alexandria with the known distance (5,000 stades) from Syene where there is no shadow, and completes it to make a full 360 degrees circle. The postscript is, the actual circumference of the earth is 24,900 miles and Eratosthenes, with his crude calculation is within a few percentage off.

Hey, let's give this guy a posthumous Nobel prize for the most elegant math equation ever!

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I want a proof starting from the Peano axioms (or equivalent). I don't think it is very difficult, but I claim that [math]1+1=2 is not really completely empty or trivial.

IIRC, from Peano axioms, it's just a matter of definition.

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I don't know about equations but this is the best number 8008135...

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I don't know about equations but this is the best number 8008135...

We always used 5318008 - and flipped

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In case you're wondering how Eratosthenes measured the circumference of the world, this diagram might help.

circumference ot world.pdf

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My other nominee is Richard Feyman. Richard was a theoretical physicist who work on the Manhattan Project that led to the end of World War ii. The problem with Richard is that he detested intellectual pretentiousness with a passion because he believed that the language of science does not have to be so complex and pedantic - so above the heads of those outside the cabal.

"Pompous fools," he called some of his colleagues after returning from a scientific symposium. "They drive me up a wall. Ordinary fools are alright, you can talk to them and help them. But pompous fools - guys who are covering it all over and impressing people as to how wonderful they are will this hocus-pocus - that I cannot stand!"

He once sat at a table with other investigators in Washington as a member of the presidential commission studying the disastrous flight of the space shuttle "Challenger"; and surprised everyone (and probably saved several million dollars in the investigation) by calling for a glass of ice water. When it came, he took a piece of the infamous gasket called the O-ring and dunked it in the ice water. After a short time, he took it out and pinched it with a small clamp. It was as hard as a rock. It did not bounce back to its original shape.

Richard, the theoretical physicist, without burdening the commission with the tedium of scientific explanation. had shown that the O-ring as unsuitable for cold weather in less than a minute.

That, to me , is the epitome of academic elegance.

(Richard Feyman died in Los Angeles in 1988, age 69).

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