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Eulers Formula:

e^(iθ) = (cosθ + isinθ)

In the complex number plane, the real horizontal axis is labeled +real for the positive axis pointing to the right, and -real for the negative axis pointing to the right . The imaginary vertical axis is labeled +imaginary for the positive axis pointing up, and -imaginary for the negative axis pointing down. The values for theta(θ) range from 0 to 2π. So for each value of theta(θ) the corresponding point in the complex number plane has the coordinates: (cosθ,isinθ).

Where cosθ corresponds to the real axis, and isinθ corresponds to the imaginary axis.

When you plug in values for theta(θ) from 0 to 2π and plot the coordinates in the complex number plane you get a graph of a unit circle with the radius equal to one on the real axis.

For: θ = 0°/ e^(iθ) = (1,0) = 1

For: θ = 90°/ e^(iθ) = (0,i) = i

For: θ = 180°/ e^(iθ) = (-1,0) = -1

For: θ = 270°/ e^(iθ) = (0,-i) = -i

You can see from these four points that the function traces a circle in the complex number plane which moves counterclockwise as the value for theta(θ) increases.

When we say the function traces a circle in the complex number plane, we are describing a rotating vector of magnitude 1, that points away from the center, and rotates counterclockwise as the value for theta(θ) increases.

The four values(1, i, -1, -i) are called operators, and when the exponential function[e^(iθ)] is multiplied by them, they tell you how much the vector has rotated or moved from its original position on the positive real axis at: θ=0°:

(1)- corresponds to 0° rotation

(i)- corresponds to 90° rotation 

(-1)- corresponds to 180° rotation 

(-i)- corresponds to 270° rotation 

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