Understanding Euler's Formula | BetterExplained

Companion video to the article at http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula. Key analogies:

* Multiplication as a transformation
* Exponential growth as a continuous change
* Imaginary numbers as rotation
* Exponential, imaginary growth as continuous rotation
* Radians as distance traveled around the circle

Combining the analogies, you can see e^ix as a rotation around the circle, equivalent to cos(x) + i*sin(x), which is the position in a "grid system".

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[resal81]

Simply awesome.

[betterexplained]

@resal81 Thanks!

[PhilosophyAnimation]

Very nice job of explaining -- this combined with the article that it supplements are great.

[betterexplained]

@PhilosophyAnimation Thanks, I appreciate it!

[pilioff]

thanks, very helpful

[betterexplained]

@pilioff You're welcome :)

[dwightivany]

I've been to a lot of math lectures with many experienced professors. This young man has done an amazing presentation. The best 10 minutes on an academic topic I have ever witnessed.

[betterexplained]

@MonicaKn17 Thanks!

[MonicaKn17]

Awesome. I love it when math is explained so visually!

[churde]

happy math

ROR

[betterexplained]

@EclecticSceptic Hi, sorry it didn't work for you! Analogies are personal -- for me, I see the equation as relating linear motion (sine + cosine) to rotation (move out, and rotate up).