Euler's Identity Is Pretty
by isabella2296, Jun 19, 2010, 1:59 PM
Today we're going to talk about Euler's identity.
In particular, its prettyfulness.
So you're probably familiar with old Euler (such a friendly old chap, even if his name does make you think it's pronounced you-ler when it's really oiler) and his identity,
. But if you're not, here's a quick explanation. 
is just this random constant no one actually cares about (seriously, why does no one memorize its digits?) and it's approximately 
. It's used a lot when you compound interest continuously, just because. Then 
, well, if you don't know that, you just shouldn't be here. You just shouldn't. Finally, 
stands for "izzy". It's really awesome (like me) and is equal to 
. 1 is a number. It's like having one apple. 0 is also a number. It's like I just ate your apple.
Once you get over the fact that I ate your apple, you may be thinking that it's pretty freaking cool how these three random constants just happen to work out like that. Guess what? Me too, except I have an apple and you don't!
Now, obviously I'm not just going to show you how prettyful Euler's identity is and just leave it at that. Do you really think I'm such a horrible person? I'm not, honest. So Euler's identity stems from Euler's formula (he never got so creative with the namings of these things, you know?), which says
. It was pretty cool.
But I'm not just going to show you Euler's formula (remember that phrase is not awkwardly because Euler is not you-ler or me-ler) and leave it that, either. Psh. We're going to talk about (so it would seem) the Taylor series. The Taylor series says that
and it sort of goes on forever.
For
, we have 
and 
.
Now, remember that Euler's formula's got an in it; specifically, 
. So we'll multiply all our 
's by 's for the fun of it. This gives us 
. However, this can be simplified because 
, etc.
This gives us
. Let's do some clever rearranging stuff: 
and - oh man! Does that look familiar to you?
This is simply. Pretty swish, right? Now we just let 
and...
![\[ e^{\pi i} + 1 = 0 \]](//latex.artofproblemsolving.com/7/5/b/75bfc5711ad9ce072c00c1392202b1e061739ea9.png)
And the moral of the story is... (go ahead. comment. you know you want to.)
In particular, its prettyfulness.
So you're probably familiar with old Euler (such a friendly old chap, even if his name does make you think it's pronounced you-ler when it's really oiler) and his identity,
. But if you're not, here's a quick explanation. 
is just this random constant no one actually cares about (seriously, why does no one memorize its digits?) and it's approximately 
. It's used a lot when you compound interest continuously, just because. Then 
, well, if you don't know that, you just shouldn't be here. You just shouldn't. Finally, 
stands for "izzy". It's really awesome (like me) and is equal to 
. 1 is a number. It's like having one apple. 0 is also a number. It's like I just ate your apple.Once you get over the fact that I ate your apple, you may be thinking that it's pretty freaking cool how these three random constants just happen to work out like that. Guess what? Me too, except I have an apple and you don't!
Now, obviously I'm not just going to show you how prettyful Euler's identity is and just leave it at that. Do you really think I'm such a horrible person? I'm not, honest. So Euler's identity stems from Euler's formula (he never got so creative with the namings of these things, you know?), which says
. It was pretty cool.But I'm not just going to show you Euler's formula (remember that phrase is not awkwardly because Euler is not you-ler or me-ler) and leave it that, either. Psh. We're going to talk about (so it would seem) the Taylor series. The Taylor series says that
and it sort of goes on forever.For
, we have 
and 
.Now, remember that Euler's formula's got an
in it; specifically, 
. So we'll multiply all our 
's by 's for the fun of it. This gives us 
. However, this can be simplified because 
, etc.This gives us
. Let's do some clever rearranging stuff: 
and - oh man! Does that look familiar to you?This is simply
. Pretty swish, right? Now we just let 
and...![\[ e^{\pi i} + 1 = 0 \]](http://latex.artofproblemsolving.com/7/5/b/75bfc5711ad9ce072c00c1392202b1e061739ea9.png)
And the moral of the story is... (go ahead. comment. you know you want to.)
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