# What is so beautiful about Euler's Identity?

1. Dec 11, 2008

### joelio36

I'm a pretty novice Physicist/Mathematician, but I've got a few offers for good universities, to show you my general level of knowledge.

Could someone please explain in terms I will understand why this equation is considered so perfect and beautiful?

2. Dec 11, 2008

### HallsofIvy

Staff Emeritus
What is so beautiful about the Mona Lisa?

Euler's equation, $e^{i\pi}+ 1= 0$, which can also be written $e^{i\pi}= -1, combines five fundamental constants, 0, 1 (or -1), e, i, and [itex]\pi$ into a single, simple, equation. Simplicity and depth make for beauty.

Last edited: Dec 13, 2008
3. Dec 11, 2008

### mathman

Error! Should be + 1 = 0, not -1.

4. Dec 11, 2008

'Tis true... both of the above.

5. Dec 12, 2008

### Fredrik

Staff Emeritus
I think that the "beauty" is in the fact that the constants are from very different branches of mathematics. 0, 1 and i are from algebra, e is from calculus/analysis, and $\pi$ is from geometry.

6. Dec 12, 2008

### Integral

Staff Emeritus
And in combining those fundamental constants it uses each of the 4 fundamental math operations: Addition, multiplication, exponentiation and equality.

All to arrive at a result that seems impossible.

How can that be anything but beautiful?

7. Dec 12, 2008

### Denton

Why does no one mention the i, is there nothing special about imaginary numbers or something?

8. Dec 15, 2008

### maze

Am I the only one who isn't in awe of this equation?

When I first saw it, it seemed random and just didn't make any sense, like those infinite sum formulas of Ramanujan (...one over pi equals WHAT?). But then after I studied complex analysis, and the more I learn in math, the more pedestrian and booring it becomes. It seems to just be a random consequence of much bigger ideas, and it doesn't lead to any insights by itself.

I've thought about this a few times and tried to "see the beauty" but as far as I can tell all the awe is based purely on shock value and nothing deeper.

9. Dec 15, 2008

### Tac-Tics

It's easy to remember and makes a lot of otherwise tough math easy.

10. Dec 15, 2008

### mathwonk

why so hung upon the word beautiful? try unbelievable, or wacky, or unexpected, or sexy, or what ever, but at least it ain't boring.

11. Dec 16, 2008

### WarPhalange

I've never seen an equation that put me in "awe", but this is a pretty cool one. And it only gets better when you find uses for it.

12. Dec 16, 2008

### Doodle Bob

I've never found anything in mathematics to be beautiful. The concept of beauty in mathematics traces back to Hardy's A Mathematician's Apology and is based on a more-or-less Late 19th/Early 20th Century sense of aesthetics.

Nevertheless, this equation has always intrigued me, since it gives us a sneak peek into the structural integrity of Mathematics as an academic discipline.

13. Dec 17, 2008

### Primitive

I always dont get it. To me, if it can solve problems, and extend new ideas, then i like it. I don t bother with 'beauty'.

14. Dec 17, 2008

### daytripper

If you ask me, something is beautiful when it's stimulating and seemingly simple (women excluded of course! eyo!!)
Euler's identity is, to me, a 7. $$A\; =\; \pi r^{2}$$ is like a 3. $$e=mc^{2}$$ is about a 9. The Lorentz factor is a perfect 10, if you ask me. =]
Simplified complexity.... mmmm....