In the introductory complex analysis course I am taking, nearly every theorem relates to rotation and argument. Why do complex numbers love doing this so much? I can understand why these theorems w...
This works over any field - the power of the Complex numbers stems from the fact that this process always works in the complex number - every polynomial has a root.
The square root of i is (1 + i)/sqrt (2). [Try it out my multiplying it by itself.] It has no special notation beyond other complex numbers; in my discipline, at least, it comes up about half as often as the square root of 2 does --- that is, it isn't rare, but it arises only because of our prejudice for things which can be expressed using small integers.
82 The square root is not a well defined function on complex numbers. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number.
The exponentiation of complex numbers involves some very beautiful geometry! We could do this through some algebraic manipulation easily enough, but it is more interesting for us to try and see what's going on!
Then it is well defined on the complex plane as well and Euler's formula gives us its real and imaginary parts. However Euler's formula still valid for any complex number and makes bridges between ordinary trigonometric functions, hyperbolic trigonometric functions, and complex logarithmic function.
I know this is an old question, but I would just like to expand a bit. What about for when complex number a+bi while a=0 and b<0. Would the argument always be 3pi/2? (New to Math SE, haven't got all the markdown yet. Sorry)
In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can. However, is there a geometric interpretation for the multiplication of two complex numbers?
