Geometric Representation of Euler's Formula
This applet enables the user to explore a geometric representation of Euler's formula, .
The left window of this applet considers the left hand side of the formula, . Recall that a defining property of the function is that it is its own derivative. Thus, where is some constant. When we extend the exponential function to include imaginary numbers, this property must hold. Now, consider a particle moving in the complex plane whose position is defined parametrically by the function . Then, its velocity, , is given by . Note that the velocity is the position multiplied by and multiplying by is equivalent to rotating through a right angle. Since the initial position of the particle is and its initial velocity is , the particle starts at and begins moving upward. Using your mouse drag the blue point in the left window in the direction of in order to draw the motion of the particle. As you move the point, consider the following questions.
- How would you describe the path the particle travels?
- Why does the particle travel along this path?
- What relationships exist between the real and imaginary parts and and ?
- Why do these relationships exist?