In Lesson 1 we used complex numbers to represent a static characteristic- the size of a photograph.
For real world applications however, the characteristics of interest are often dynamic in nature, such as the mechanical vibration of a bearing or the alternating current through a device.
We’ll now show how complex numbers are used to represent these types of dynamic states. We will use rowers in a crew boat to illustrate the technique.
Meet our first row crew, Stan and Calvin, rowing in a smooth harmonic motion. Stan exhibits a perfect stroke and Calvin tries to follow. Our challenge is to precisely describe Calvin’s stroke relative to Stan’s using a single complex number.
Follow the exercise in the "PicoViewer" (left) to learn interactively about this challenge, then press on below to dig in.
Before trying to describe Calvin with a complex number, we introduce the Purist rowers, Rita and Ivan. Rita has a "Purely Real" dynamic response and Ivan has a "Purely Imaginary" dynamic response. Together, they will act as a component pair to build Calvin’s complex number description. Sound confusing? No worries, just jump into the PicoViewer below to get an interactive picture, starting with Rita.
Now let’s assemble the Purist components to construct a complex personality, one having both real and imaginary parts. We show Calvin’s complex personality below, represented by an open circle with a cross on the complex plane. Jump into the Picoviewer below to see how the Real and Imaginary components compose his personality.
We illustrated using complex numbers to describe dynamic output relative to input. The Real part of the the complex output represents the motion component which is in perfect timing with the input. The Imaginary part represents the motion component which is a quarter cycle ahead of the input.
In Lesson 3, we’ll use these concepts to characterize the vibration system from our intro. This application opens the secret into how imaginary numbers are actually used in real world situations.
1. Mapping Imaginary to Physical – Lesson 1
2. Complex Numbers and Dynamic Systems – Lesson 2
3. An Imaginary Number Application – Lesson 3
B. The Physical Link of the Imaginary Unit – Bonus Lesson