Had pic related in my lecture today and I did not get it at all.

>>8166798
>So basically this is a circle with radius=1 on the complex plane or what?
Basically yes. e^{it} forms a spiral of constant radius 1 through the t axis. Which is why the equation pictured is a really shallow thing to be spread as some mathematical miracle - the whole picture is more beautiful to me.

>>8166808
Okay, that understanding helps.
Thank you.

>>8166808
you clearly don't understand it then.
Look at what if fucking says mate.
if you take a function solution of {y' = y, y(0)=1}, take its value at 1, raise it to a complex power involving pi, you get -1, which is a simple number.
It just how simplicity can stem from apparently very complicated functions and numbers: e is transcendental, pi is transcendental.
This is rare enough to be mentionned.

we don't even fucking know if pi^e is transcendental.

e^ix is just a way to express a rotational vector, where x is the phase
if you put x=pi than you have that the phase is =-180°, thus being -1

Take the definition of [math] e^x [/math]

[math]e^x=1+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...[/math]

Substitute [math] x [/math] with [math] i\theta [/math]

[math]e^{i\theta}=1+\frac{(i\theta)^1}{1!}+\frac{(i\theta)^2}{2!}+\frac{(i\theta)^3}{3!}+...[/math]

This factorises to give

[math]1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\frac{\theta^6}{6!}+...+i[\frac{\theta}{1!}-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-...][/math]

Therefore

[math]e^{i\theta}=cis(\theta)[/math]

So

[math]e^{i\pi}=cis(\pi)=-1[/math]

Just think of [math]e^{i\theta}[/math] as another form to express complex numbers in. Don't get hung up on that "math is beautiful" shit in your picture. I could say "[math]e^{\frac{i\pi}{2}}[/math] is beautiful because it equals [math]i[/math] look at how beautiful math is!".

It is a pretty neat form though. I like how you can show why two negative numbers multiply to give a positive one using the form.

[math]-1*-1[/math]
[math]=e^{i\pi}*e^{i\pi}[/math]
[math]=e^{2i\pi}[/math]
[math]=e^{0i}[/math]
[math]=1[/math]

>>8166946
Fucked up tex
Therefore

[math]e^{i \theta}=cis(\theta)[/math]

So

[math]e^{i \pi}=cis(\pi)=-1[/math]

>>8166946
>I could say "eiπ2eiπ2 is beautiful because it equals ii

Mathematical immaturity

File: 1450756605002.jpg (35KB, 548x415px) Image search: [Google]

35KB, 548x415px

>>8166957
Ah yes, and making petty insults on an anonymous chink image board sure does show the world how mature you are!

>>8166946
>It is a pretty neat form though. I like how you can show why two negative numbers multiply to give a positive one using the form.
Right, but you don't need all of that to interpret multiplication by -1 as rotation by pi/reflection of the real line.

>>8166798

The formula in the OP (and its equivalent statement involving zero for maximum math memes) is also a special case of Euler's formula (a very generic, ambiguous phrase, but wiki leads directly to this context when you look it up).

In a restricted (integer) sense, De Moivre's formula (look this one up too), which more obviously describes a unit circle from a young student's point of view, is related to the above Euler's formula, but it does not comprise the general case.

De Moivre's formula is useful for beginning to explore the effect of raising complex numbers to (integer) powers. Basically, the angles wrap around the origin n times, so as n gets bigger there' more "swirly". :^)