θ is a real number => jθ is an imaginary number
=> e^ (jθ) also has to be an imaginary number.
Could you explain why there is a real component in e^ (jθ)? I don't have any intuitive about this?Not true. This is a complex number in general. It can be real, imaginary or complex, depending on the value of θ.
So this is where you are making your mistake. The simplest way to see your mistake is to take the case of θ=0. Here you have exp(0)=1. Obviously, this is real and not imaginary.
Apparently, e^ (jθ) has both real and imaginary components not just imaginary component like what I think.
Yes.. now you got it.
Numerical example:
2e^ (j3) = 2*cos(3) + 2*j sin(3)
EDIT: The equation is actually called "Eulers Formula".
Thanks steveB.
Could you explain why there is a real component in e^ (jθ)? I don't have any intuitive about this?
Could you explain why there is a real component in e^ (jθ)? I don't have any intuitive about this?
That is a nice explanation of complex exponentials. Does it answer your question? If so, what part answered it?
Hi,Hi again,
Here is a very nice 3d geometrical interpretation showing what e^(i*x) and it's rotation i*e^(i*x) means...
https://hightechavenue.blogspot.com/2013/07/analog-avenue-helix-rotations.html
Note that 'x' is the vertical axis and that would be normal for 't' in relativity theory.
But i also already mentioned that 'i' times something makes everything imaginary real, and everything real imaginary, and when we go from real to imaginary or imaginary to real that's a rotation in the complex plane, and you can see the helixes starting out 90 degrees out of phase with each other (90 degrees in the complex plane). So when we multiply by 'i' we are really rotating the helix by 90 degrees. By that simple rule then when we multiply by 'i' a second time, we effectively rotate the helix another 90 degrees so we should see a new helix that starts out 180 degrees ahead of the first. One more multiplication by 'i' and we move ahead another 90 degrees yet, and one last multiplication and we are a full 360 degrees ahead which is often interpreted as no rotation. This follows the pattern i explained previously where we have four possibilities but that's all.
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