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# Complex exponential magnitude

A closer look at the complex exponential term in Euler's Formula. We see that it represents a complex number, a distance of 1 from the origin of the complex plane. Created by Willy McAllister.

## Want to join the conversation?

• at You said the Y-value squared is sine of theta squared, but it shouldn't be the case here because sine of theta is on the imaginary axis. at least, the magnitude of e to the j theta squared should be cos of theta squared minus sin of theta squared which is not equal to one.
• What is the relationship between wt and theta? I am looking how e^jwt turns into e^j.theta at the beginning of video.
(1 vote)
• We start with the most general exponential, e^jtheta, where theta is any angle. Then we say, "let's suppose we let the angle theta be a function of time." That is, we say theta = f(t). We choose a specific f(t) = wt. That means as time increases, the angle theta increases. The extra term w acts as a scale factor for time. It has units of radians/second.
• I want to talk about some implications of the rect-polar conversion. I saw You and Sal drawing a unit circle (center= origin) that intersect both the real and imaginary axis. The unit circle intersects the imaginary axis twice, and so, if I measure the distance between the origin and one of the intersection, I will get a distance of 1. And since I can measure a distance of 1 along the imaginary axis, then the number 1 must be also part of the imaginary numbers. I am confused.
(1 vote)
• You are merging the concepts of distance and direction. Let's use a simpler rectangular system: a regular map. If you walk 10 steps East the distance is 10 steps. If you walk 10 steps North, the distance is the same: 10 steps. If you walk 10 steps in any direction the distance is always the same. The distance you walk does not have "East-ness" or "North-ness", it's just distance. The direction is a separate idea. It works the same way for the unit circle. The radius of a circle has a certain length. The length of the radius line does not change if you point it in different directions.