Woah

Oops, I meant except when a=1. Are there still values for a and b that don't work?

If you want a visualization, it has to do with the way exponentiation works in the complex plane.

youtube.com/watch?v=mvmuCPvRoWQ

I bothers me that this guy mentioned how the behavior of i has the preserve the structure of (x^y)(x^y)=x^2y, but didn't mention how it has to preserve (x^y)^y=x^(y^2). This is much more important when explaining complex exponentials because we know that i^2=-1, and are familiar with the behavior of x^-1.

This. Do a Taylor Expansion for cos(x), sin(x) and e^(ix) at x=0 and you'll end up with the following equations
[math]\mathrm {sin} (x) = \displaystyle \sum _ {n = 0} ^{\infty} (-1)^n \left ( \frac {x^{2n + 1}}{(2n+1)!} \right )[/math]
[math]\mathrm {cos} (x) = \displaystyle \sum _ {n = 0} ^{\infty} (-1)^n \left ( \frac {x^{2n}}{(2n)!} \right )[/math]
[math]e^{i x} = \displaystyle \sum _ {n = 0} ^{\infty} (-1)^n \left ( \frac {x^{2n}}{(2n)!} + i \frac {x^{2n+1}}{(2n+1)!} \right ) \Rightarrow e^{i x} = \mathrm {cos} (x) + i \cdot \mathrm {sin} (x)[/math]

It's not at all how Euler derived it, which was what he was talking about.

It's derived from a ton of properties of sine and cosine, 30 years after someone else had already discovered it using only one property of arctangent.

We aren't really talking about exponents any more. There are more equations behind the scenes that you don't see in this identity

How do you take a number and multiply it by itself an imaginary and irrational amount of times?

>How do you take a number and multiply it by itself an imaginary and irrational amount of times?
It's simple: that's not the real definition of exponent. The real definition is
(x^y)(x^y)=x^2y
and
(x^y)^y=x^yy

That's not a definition

By using the limit

woah dude how u display equations like that?
(i dont frequent this board, so sorry if this is common knowledge)

...

Latex. Check out the sticky.

[math]\LaTeX[/math]

>This isn't even true. It's derived from angles in a complex plane

Complex numbers are a extension of real numbers to the plane and there is nothing "imaginary" about them. It is a totally coherent extension of mathematical rules which can be applied to a lot of physical things.

The whole error of this was to call them "imaginary" in the first place. The square root of -1 is a rotation of pi/2 around the coordinate origin, therefore two rotations put you at the real axis again. It is a whole new form of seeing numbers. A better one.

Here is a explanation for dummies, and a very good and intuitive one I think

youtube.com/watch?v=-dhHrg-KbJ0

Yeah, I really liked the approach of this video.

They're not imaginary numbers, they're lateral numbers. Also, there are no negative and positive numbers, they are inverse and direct numbers.

You got it right. Honestly, I didn't think about it that way until I read something one day on the internet.

It is a shame, but very few people know how to teach things the right way.

How hard could it be to show students a simple diagram of the complex plane? That alone would dispel tons of common misconceptions about [math]i[/math].

That's not enough. You need to change the way of seeing numbers and operations...only then you can made real sense of things like Euler's identity.

And in my opinion, unless you have the mind of the mathematician, you need to understand at least some of the applications that has in the real world.

The geometric aspects of the complex numbers are a consequence of Euler's formula, which requires calc 2 to fully understand.

Imaginary numbers aren't difficult to accept conceptually if explain them through analogy with the irrational number or the negative numbers. The biggest problem is the asinine name which everyone takes too literally.

>The geometric aspects of the complex numbers are a consequence of Euler's formula

The other way around works too.

[math]e^{\pi i} = i^2[/math]

[math]e^{\pi i/2} = i[/math]

*inhales*

DUDE

You've got it backwards. Expressions are a description of the geometry not the other way around. Thinking that geometry comes from the expression is how creationists think.

>You've got it backwards. Expressions are a description of the geometry not the other way around.

I am agree. But read what that guy said:

>The geometric aspects of the complex numbers are a consequence of Euler's formula

What the fuck did you just fucking say about complex numbers, you little bitch? I’ll have you know I stopped caring about math when I was introduced to the concept of imaginary numbers, and I’ve been involved in numerous secret raids on Al-Gebra, and I have over 300 crocks of shit. I am trained in equations that can only be solved by inventing numbers that can't exist and I’m the top math deity in the entire US academic forces. You are nothing to me but fucking wrong. I will wipe you the fuck out with math the flaws of which have never been seen before on this Earth, mark my fucking words. You think you can get away with saying that shit to me over the Internet? Think again, fucker. As we speak I am contacting my secret network of algebra solutions across the USA and your IP is being traced right now so you better say "the correct answer is whatever the correct answer is", maggot. The math that says the pathetic little thing transcribed to words. You’re fucking dead, kid. I can be anywhere, anytime, and I can mark you wrong in over seven hundred ways, and that’s just if you write it down in english instead of ancient math runes. Not only am I extensively trained in unarmed combat, but I have access to the entire arsenal of the United States Logical Math Corps and I will use numbers that never lie to their full extent to wipe your miserable ass off the face of the continent, you little shit. If only you could have known what unholy flaws your little “clever” human construct was about to bring down upon you, maybe you would have held your fucking tongue. But you couldn’t, you didn’t, and now you’re paying the price, you goddamn idiot. I will shit complex numbers all over you and you will drown in it. You’re fucking dead, kiddo.