I understood it with Euler's formula and 3Blue1Brown's video.
But I still can't accept it with my heart.
Is there any more clear explanation than below?
$e^{it} = \cos(t) + i \sin(t)$
so $x$ and $y$ component changes by time, showing us rotation.
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4$\begingroup$ What is stopping this truth from entering your heart? How do you define complex exponentiation to begin with? If you want physical motivation, maybe you could study solutions to the harmonic oscillator $\frac{d^2 x(t)}{dt^2} + x(t) = 0$. To quote John von Neumann: "Young man, in mathematics you don't understand things. You just get used to them." If it's a matter of accepting it in the heart, maybe you can be comforted by the fact that one day this will be as natural as one plus one makes two is now for you. $\endgroup$– AlkaKadriCommented Jan 10 at 8:08
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$\begingroup$ Wow thats a very touching sentence! I think youre right. I'll get used to it soon $\endgroup$– COTHECommented Jan 11 at 4:44
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2 Answers
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Prerequisite
Consider a vector on the complex plane $\mathbf{v}_{1}=x+jy$. When we multiply this vector with the imaginary number $j$, we obtain another vector $\mathbf{v}_{2}=-y+jx$. A particularly important property is that $\mathbf{v}_{2}$ is simply the vector $\mathbf{v}_{1}$, rotated $90$ degrees counterclockwise:
Complex Exponent
Now consider a position vector $\mathbf{r}=e^{jt}$. When we differentiate this vector with respect to $t$ (or time), we obtain the velocity vector $\frac{d\mathbf{r}}{dt}=j\cdot e^{jt}$.
Now remember that multiplying a vector on the complex plane by $j$ rotates the vector and makes it perpendicular to the initial vector. What happens when you always move in the direction perpendicular to your position? You move on a circle!
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The expression $e^{it}$, where $i$ is the imaginary unit and $t$ is a real number, is a way of representing a point on the unit circle in the complex plane. This representation comes from Euler's formula, which states that $\mathrm{cis}(t)=e^{it} = \cos(t) + i \sin(t)$. In simpler terms, $e^{it}$ gives us a complex number with a real part $\cos(t)$ and an imaginary part $\sin(t)$.
Now, think of the complex plane as having a unit circle (a circle with a radius of 1) centered at the origin. As $t$ varies, the point $e^{it}$ moves along this unit circle. The key point is that this movement is like tracing a path on the circle.
When $t$ increases, the argument of $e^{it}$ increases, causing the point to move counterclockwise along the unit circle. Therefore, $e^{it}$ is often used to describe rotations in mathematics and physics.
In essence, $e^{it}$ gives us a dynamic way to represent points on the unit circle, and as $t$ changes, it corresponds to the continuous rotation of a point around the center of the circle in the complex plane.
